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1. Error estimates of local energy regularization for the logarithmic Schrodinger equation

Author: Chunmei Su, Rémi Carles, Qinglin Tang, Weizhu Bao, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Yau Mathematical Sciences Center, Tsinghua University, School of Mathematics, State Key Laboratory of Hydraulics and Mountain River Engineering, Chengdu University of Technology (CDUT), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), and Technische Universität München [München] (TUM)
Subjects: Quadratic growth, Polynomial, Partial differential equation, Logarithm, Applied Mathematics, 010102 general mathematics, Numerical Analysis (math.NA), 16. Peace & justice, 01 natural sciences, Regularization (mathematics), 010101 applied mathematics, Nonlinear system, Rate of convergence, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, FOS: Mathematics, Applied mathematics, Mathematics - Numerical Analysis, 0101 mathematics, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics, Energy functional
Abstract: The logarithmic nonlinearity has been used in many partial differential equations (PDEs) for modeling problems in various applications.Due to the singularity of the logarithmic function, it introducestremendous difficulties in establishing mathematical theories, as well asin designing and analyzing numerical methods for PDEs with such nonlinearity. Here we take the logarithmic Schr\"odinger equation (LogSE)as a prototype model. Instead of regularizing $f(\rho)=\ln \rho$ in theLogSE directly and globally as being done in the literature, we propose a local energy regularization (LER) for the LogSE byfirst regularizing $F(\rho)=\rho\ln \rho -\rho$ locally near $\rho=0^+$ with a polynomial approximation in the energy functional of the LogSE and then obtaining an energy regularized logarithmic Schr\"odinger equation (ERLogSE) via energy variation. Linear convergence is established between the solutions of ERLogSE and LogSE in terms of a small regularization parameter $0, Comment: 31 pages, 10 figures, final version. More explanations and some proofs are more detailed
Published: 2022
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2. Logarithmic Schrödinger equation with quadratic potential

Author: Rémi Carles, Guillaume Ferriere, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), AIS, Rennes Metropole, Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Kullback–Leibler divergence, Logarithmic Schrödinger equation, General Physics and Astronomy, Harmonic (mathematics), 01 natural sciences, Solitons, Quadratic equation, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 010306 general physics, Mathematical Physics, Mathematics, Mathematical physics, Harmonic potential, Applied Mathematics, 010102 general mathematics, Statistical and Nonlinear Physics, Dispersion, Nonlinear system, Logarithmic sobolev inequality, Sign (mathematics)
Abstract: We analyze dynamical properties of the logarithmic Schr{\"o}dinger equation under a quadratic potential. The sign of the nonlinearity is such that it is known that in the absence of external potential, every solution is dispersive, with a universal asymptotic profile. The introduction of a harmonic potential generates solitary waves, corresponding to generalized Gaussons. We prove that they are orbitally stable, using an inequality related to relative entropy, which may be thought of as dual to the classical logarithmic Sobolev inequality. In the case of a partial confinement, we show a universal dispersive behavior for suitable marginals. For repulsive harmonic potentials, the dispersive rate is dictated by the potential, and no universal behavior must be expected., Comment: More explanations
Published: 2021
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3. Separation of scales: Dynamical approximations for complex quantum systems

Author: Irene Burghardt, Caroline Lasser, Benjamin Lasorne, Clotilde Fermanian Kammerer, Rémi Carles, Institute of Physical and Theoretical Chemistry [Frankfurt am Main], Goethe-Universität Frankfurt am Main, Institut de Recherche Mathématique de Rennes (IRMAR), Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Analyse et Mathématiques Appliquées (LAMA), Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS)-Université Gustave Eiffel, Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM ICMMM), Université Montpellier 2 - Sciences et Techniques (UM2)-Université Montpellier 1 (UM1)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM), Zentrum Mathematik, M5 Technische Universität München, CNRS 80|prime, Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Université de Rennes (UNIV-RENNES)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Institut Charles Gerhardt Montpellier - Institut de Chimie Moléculaire et des Matériaux de Montpellier (ICGM), Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Zentrum Mathematik, Technische Universität München, Technische Universität Munchen - Université Technique de Munich [Munich, Allemagne] (TUM), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Centre National de la Recherche Scientifique (CNRS)-Université de Montpellier (UM)-Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Institut de Chimie du CNRS (INC), Technische Universität München [München] (TUM), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Université Montpellier 1 (UM1)-Université Montpellier 2 - Sciences et Techniques (UM2)-Ecole Nationale Supérieure de Chimie de Montpellier (ENSCM)-Institut de Chimie du CNRS (INC)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Statistics and Probability, FOS: Physical sciences, General Physics and Astronomy, Context (language use), 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Physics - Chemical Physics, 0103 physical sciences, FOS: Mathematics, Taylor series, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Statistical physics, 010306 general physics, 10. No inequality, Wave function, Mathematical Physics, Mathematics, Ansatz, Chemical Physics (physics.chem-ph), Collocation, 010304 chemical physics, Statistical and Nonlinear Physics, Observable, Mathematical Physics (math-ph), Function (mathematics), Action (physics), ddc, [CHIM.THEO]Chemical Sciences/Theoretical and/or physical chemistry, Modeling and Simulation, symbols, [PHYS.PHYS.PHYS-CHEM-PH]Physics [physics]/Physics [physics]/Chemical Physics [physics.chem-ph], Analysis of PDEs (math.AP)
Abstract: We consider composite quantum-dynamical systems that can be partitioned into weakly interacting subsystems, similar to system-bath type situations. Using a factorized wave function ansatz, we mathematically characterize dynamical scale separation.Specifically, we investigate a coupling r{\'e}gime that is partially flat, i.e., slowly varying with respect to one set of variables, for example, those of the bath. Further, we study the situation where one of the sets of variables is semiclassically scaled and derive a quantum-classical formulation. In both situations, we propose two schemes of dimension reduction: one based on Taylor expansion (collocation) and the other one based on partial averaging (mean-field). We analyze the error for the wave function and for the action of observables, obtaining comparable estimates for both approaches. The present study is the first step towards a general analysis of scale separation in the context of tensorized wavefunction representations., Comment: Title changed, Section 3.3 reorganized, new Section 6 (numerical example)
Published: 2021
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4. Norm inflation for nonlinear Schrodinger equations in Fourier-Lebesgue and modulation spaces of negative regularity

Author: Rémi Carles, Divyang G. Bhimani, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Rennes Métropole., ANR-11-LABX-0020,LEBESGUE,Centre de Mathématiques Henri Lebesgue : fondements, interactions, applications et Formation(2011), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
Subjects: Modulation space, Partial differential equation, Applied Mathematics, General Mathematics, 010102 general mathematics, 020206 networking & telecommunications, 02 engineering and technology, Lebesgue integration, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Schrödinger equation, symbols.namesake, Nonlinear system, Fourier transform, Mathematics - Analysis of PDEs, Fourier analysis, Norm (mathematics), 0202 electrical engineering, electronic engineering, information engineering, symbols, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Analysis, Mathematics
Abstract: We consider nonlinear Schr{\"o}dinger equations in Fourier-Lebesgue and modulation spaces involving negative regularity. The equations are posed on the whole space, and involve a smooth power nonlinearity. We prove two types of norm inflation results. We first establish norm inflation results below the expected critical regularities. We then prove norm inflation with infinite loss of regularity under less general assumptions. To do so, we recast the theory of multiphase weakly nonlinear geometric optics for nonlinear Schr{\"o}dinger equations in a general abstract functional setting., Comment: 27 pages. Two references added, some typos fixed
Published: 2020
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5. WKB analysis of non-elliptic nonlinear Schrodinger equations

Author: Rémi Carles, Clément Gallo, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, and Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)
Subjects: Work (thermodynamics), Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematics::Analysis of PDEs, Order (ring theory), 010103 numerical & computational mathematics, Radius, 01 natural sciences, WKB approximation, Schrödinger equation, Sobolev space, symbols.namesake, Nonlinear system, Mathematics - Analysis of PDEs, Nonlinear Sciences::Exactly Solvable and Integrable Systems, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], symbols, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Order system, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Mathematics
Abstract: We justify the WKB analysis for generalized nonlinear Schr{\"o}dinger equations (NLS), including the hyperbolic NLS and the Davey-Stewartson II system. Since the leading order system in this analysis is not hyperbolic, we work with analytic regularity, with a radius of analyticity decaying with time, in order to obtain better energy estimates. This provides qualitative information regarding equations for which global well-posedness in Sobolev spaces is widely open., Comment: arXiv admin note: text overlap with arXiv:1612.04149
Published: 2020
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6. Superexponential growth or decay in the heat equation with a logarithmic nonlinearity

Author: Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Logarithm, Applied Mathematics, Gaussian, 010102 general mathematics, Mathematical analysis, Double exponential function, Ode, 01 natural sciences, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, Line (geometry), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, Uniqueness, 0101 mathematics, Analysis, Mathematics
Abstract: We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Finally, we prove that such phenomena remain, in the case of compactlysupported initial data., Comment: 14 pages
Published: 2017
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7. Norm-inflation with Infinite Loss of Regularity for Periodic NLS Equations in Negative Sobolev Spaces

Author: Thomas Kappeler and Rémi Carles
Subjects: General Mathematics, 010102 general mathematics, Mathematical analysis, 01 natural sciences, Schrödinger equation, 010101 applied mathematics, Sobolev space, symbols.namesake, Norm (mathematics), symbols, 0101 mathematics, Nonlinear Schrödinger equation, Mathematics, Mathematical physics
Abstract: In this paper we consider Schrodinger equations with nonlinearities of odd order 2σ + 1 on Td. We prove that for σd ≥2, they are strongly illposed in the Sobolev space Hs for any s < 0, exhibiting norm-inflation with infinite loss of regularity. In the case of the one-dimensional cubic nonlinear Schrödinger equation and its renormalized version we prove such a result for Hs with s < −2/3.
Published: 2017
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8. Regularized numerical methods for the logarithmic Schrodinger equation

Author: Weizhu Bao, Chunmei Su, Rémi Carles, Qinglin Tang, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics, University of Innsbruck, School of Mathematics, Sichuan University, Sichuan University [Chengdu] (SCU), Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Universität Innsbruck [Innsbruck]
Subjects: Logarithm, Discretization, Applied Mathematics, Numerical analysis, Finite difference method, Order (ring theory), Numerical Analysis (math.NA), 010103 numerical & computational mathematics, 16. Peace & justice, 01 natural sciences, 010101 applied mathematics, Computational Mathematics, Nonlinear system, Mathematics - Analysis of PDEs, Integrator, FOS: Mathematics, Applied mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, Energy (signal processing), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Analysis of PDEs (math.AP), Mathematics
Abstract: We present and analyze two numerical methods for the logarithmic Schr{\"o}dinger equation (LogSE) consisting of a regularized splitting method and a regularized conservative Crank-Nicolson finite difference method (CNFD). In order to avoid numerical blow-up and/or to suppress round-off error due to the logarithmic nonlinearity in the LogSE, a regularized logarithmic Schr{\"o}dinger equation (RLogSE) with a small regularized parameter 0 < $\epsilon$ $\ll$ 1 is adopted to approximate the LogSE with linear convergence rate O($\epsilon$). Then we use the Lie-Trotter splitting integrator to solve the RLogSE and establish its error bound O($\tau$ 1/2 ln($\epsilon$ --1)) with $\tau$ > 0 the time step, which implies an error bound at O($\epsilon$ + $\tau$ 1/2 ln($\epsilon$ --1)) for the LogSE by the Lie-Trotter splitting method. In addition, the CNFD is also applied to discretize the RLogSE, which conserves the mass and energy in the discretized level. Numerical results are reported to confirm our error bounds and to demonstrate rich and complicated dynamics of the LogSE., Comment: 23 pages, 8 colored figures
Published: 2019
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9. Error estimates of a regularized finite difference method for the logarithmic Schr\'odinger equation

Author: Rémi Carles, Chunmei Su, Weizhu Bao, Qinglin Tang, Department of Mathematics [Singapore], National University of Singapore (NUS), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Department of Mathematics, University of Innsbruck, MOE2015-T2-2-146, Ministry of Education - Singapore, YJ201807, Fundamental Research Funds for the Central Universities, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Universität Innsbruck [Innsbruck]
Subjects: Numerical Analysis, Logarithm, Logarithmic Schrödinger equation, AMS subject classifications. 35Q40, 35Q55, 65M15, 81Q05, Applied Mathematics, Mathematical analysis, Finite difference method, 010103 numerical & computational mathematics, regularized logarithmic Schrödinger equation, 01 natural sciences, Computational Mathematics, Nonlinear system, convergence rate, 35Q40, 35Q55, 65M15, 81Q05, Rate of convergence, error estimates, Mathematics - Numerical Analysis, 0101 mathematics, semi-implicit finite difference method, logarithmic nonlinearity, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: We present a regularized finite difference method for the logarithmic Schr\"odinger equation (LogSE) and establish its error bound. Due to the blow-up of the logarithmic nonlinearity, i.e. $\ln \rho\to -\infty$ when $\rho\rightarrow 0^+$ with $\rho=|u|^2$ being the density and $u$ being the complex-valued wave function or order parameter, there are significant difficulties in designing numerical methods and establishing their error bounds for the LogSE. In order to suppress the round-off error and to avoid blow-up, a regularized logarithmic Schr\"odinger equation (RLogSE) is proposed with a small regularization parameter $0, Comment: 23 pages, 4 figures
Published: 2018
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10. On semi-classical limit of nonlinear quantum scattering

Author: Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-12-JS01-0005,SchEq,Equations de Schrödinger et applications(2012), and ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012)
Subjects: Scattering, General Mathematics, 010102 general mathematics, Scattering length, Planck constant, 01 natural sciences, Classical limit, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Quantum mechanics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Coherent states, Scattering theory, Statistical physics, 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mathematics
Abstract: We consider the nonlinear Schr{\"o}dinger equation with a short-range external potential, in a semi-classical scaling. We show that for fixed Planck constant, a com-plete scattering theory is available, showing that both the potential and the nonlinearity are asymptotically negligible for large time. Then, for data under the form of coherent state, we show that a scattering theory is also available for the approximate envelope of the propagated coherent state, which is given by a nonlinear equation. In the semi-classical limit, these two scattering operators can be compared in terms of classical scattering the-ory, thanks to a uniform in time error estimate. Finally, we infer a large time decoupling phenomenon in the case of finitely many initial coherent states., Comment: 41 pages
Published: 2016
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11. Complementary study of the standing wave solutions of the Gross-Pitaevskii equation in dipolar quantum gases

Author: Hichem Hajaiej, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), King Saud University [Riyadh] (KSU), ANR-12-JS01-0005,SchEq,Equations de Schrödinger et applications(2012), and ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013)
Subjects: Condensed Matter::Quantum Gases, Condensed Matter::Other, General Mathematics, 010102 general mathematics, Condensation, 01 natural sciences, Symmetry (physics), 010101 applied mathematics, Standing wave, Gross–Pitaevskii equation, Dipole, Mathematics - Analysis of PDEs, Classical mechanics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Orbit (control theory), Ground state, Mathematical Physics, Mathematics
Abstract: We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions., Comment: 10 pages
Published: 2015
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12. Explicit solutions in evolutionary genetics and applications

Author: Rémi Carles and Matthieu Alfaro
Subjects: Human evolutionary genetics, Calculus, Applied mathematics, Heat equation, General Medicine, Finite time, Mathematics
Abstract: We show that the solution to a nonlocal reaction–diffusion equation, present in evolutionary genetics, can be related explicitly to the solution of the heat equation with the same initial data. As a consequence, we show different possible scenario for the solution: it can be either well-defined for all time, or become extinct in finite time, or even be defined for no positive time. In the former case, we give the leading-order asymptotic behavior of the solution for large time, which is universal.
Published: 2015
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13. On the Cauchy problem for the Hartree type equation in the Wiener algebra

Author: Lounes Mouzaoui and Rémi Carles
Subjects: Pure mathematics, Integrable system, Applied Mathematics, General Mathematics, 010102 general mathematics, Wiener algebra, Space (mathematics), 01 natural sciences, 010101 applied mathematics, symbols.namesake, Fourier transform, Square-integrable function, Hartree equation, Kernel (statistics), symbols, Standard probability space, 0101 mathematics, Mathematics
Abstract: We consider the mass-subcritical Hartree equation with a homogeneous kernel, in the space of square integrable functions whose Fourier transform is integrable. We prove a global well-posedness result in this space. On the other hand, we show that the Cauchy problem is not even locally well-posed if we simply work in the space of functions whose Fourier transform is integrable. Similar results are proven when the kernel is not homogeneous, and is such that its Fourier transform belongs to some Lebesgue space.
Published: 2014
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14. Splitting methods for the nonlocal Fowler equation

Author: Rémi Carles, Afaf Bouharguane, Equations aux Dérivées Partielles (EDP), Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and ANR-08-BLAN-0301,MathOcean,Analyse mathématique en océanographie et applications(2008)
Subjects: Discrete-time Fourier transform, Numerical time integration, 010103 numerical & computational mathematics, 01 natural sciences, symbols.namesake, FOS: Mathematics, Mathematics - Numerical Analysis, Nonlocal operator, Operator splitting, Pseudo-spectral method, 65M15, 35K59, 86A05, 0101 mathematics, Mathematics, Fourier transform on finite groups, Algebra and Number Theory, Applied Mathematics, Mathematical analysis, Fourier inversion theorem, Finite difference method, Numerical Analysis (math.NA), 010101 applied mathematics, Split-step method, Split-step Fourier method, Computational Mathematics, Fourier transform, Error analysis, Fourier analysis, symbols, Stability, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: We consider a nonlocal scalar conservation law proposed by Andrew C. Fowler to describe the dynamics of dunes, and we develop a numerical procedure based on splitting methods to approximate its solutions. We begin by proving the convergence of the well-known Lie formula, which is an approximation of the exact solution of order one in time. We next use the split-step Fourier method to approximate the continuous problem using the fast Fourier transform and the finite difference method. Our numerical experiments confirm the theoretical results., Comment: 20 pages, 3 figures. Presentation modified, some errors fixed
Published: 2013
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15. On Fourier time-splitting methods for nonlinear Schrodinger equations in the semi-classical limit II. Analytic regularity

Author: Clément Gallo, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), and ANR-12-JS01-0005,SchEq,Equations de Schrödinger et applications(2012)
Subjects: Discretization, Applied Mathematics, Mathematical analysis, 010103 numerical & computational mathematics, 16. Peace & justice, Planck constant, 01 natural sciences, Classical limit, WKB approximation, Schrödinger equation, 010101 applied mathematics, Computational Mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), Mathematics - Numerical Analysis, 0101 mathematics, Nonlinear Schrödinger equation, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: We consider the time discretization based on Lie-Trotter splitting, for the nonlinear Schrodinger equation, in the semi-classical limit, with initial data under the form of WKB states. We show that both the exact and the numerical solutions keep a WKB structure, on a time interval independent of the Planck constant. We prove error estimates, which show that the quadratic observables can be computed with a time step independent of the Planck constant. The functional framework is based on time-dependent analytic spaces, in order to overcome a previously encountered loss of regularity phenomenon., Comment: 21 pages
Published: 2017
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16. WKB analysis of generalized derivative nonlinear Schrodinger equations without hyperbolicity

Author: Clément Gallo, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), and ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013)
Subjects: Applied Mathematics, 010102 general mathematics, Mathematical analysis, Derivative, Space (mathematics), 01 natural sciences, WKB approximation, Euler equations, Schrödinger equation, 010101 applied mathematics, symbols.namesake, Nonlinear system, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Modeling and Simulation, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mathematics
Abstract: We consider the semi-classical limit of nonlinear Schrodinger equations in the presence of both a polynomial nonlinearity and thederivative in space of a polynomial nonlinearity. By working in a class of analytic initial data, we do not have to assume any hyperbolicstructure on the (limiting) phase/amplitude system. The solution, its approximation, and the error estimates are considered in time dependent analyticregularity., Comment: 13 pages
Published: 2016
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17. Replicator-mutator equations with quadratic fitness

Author: Matthieu Alfaro, Rémi Carles, Institut Montpelliérain Alexander Grothendieck (IMAG), and Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Work (thermodynamics), Applied Mathematics, General Mathematics, Gaussian, 010102 general mathematics, Harmonic potential, 01 natural sciences, Term (time), 010101 applied mathematics, symbols.namesake, Quadratic equation, Mathematics - Analysis of PDEs, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Applied mathematics, Heat equation, 0101 mathematics, Finite time, Mathematics
Abstract: This work completes our previous analysis on models arising in evolutionary genetics. We consider the so-called replicator-mutator equation, when the fitness is quadratic. This equation is a heat equation with a harmonic potential, plus a specific nonlocal term. We give an explicit formula for the solution, thanks to which we prove that when the fitness is non-positive (harmonic potential), solutions converge to a universal stationary Gaussian for large time, whereas when the fitness is non-negative (inverted harmonic potential), solutions always become extinct in finite time., Comment: 12 pages
Published: 2016
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18. Universal dynamics for the defocusing logarithmic Schrodinger equation

Author: Isabelle Gallagher, Rémi Carles, Institut Montpelliérain Alexander Grothendieck ( IMAG ), Université de Montpellier ( UM ) -Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Jussieu ( IMJ ), Université Pierre et Marie Curie - Paris 6 ( UPMC ) -Université Paris Diderot - Paris 7 ( UPD7 ) -Centre National de la Recherche Scientifique ( CNRS ), Département de Mathématiques et Applications - ENS Paris ( DMA ), École normale supérieure - Paris ( ENS Paris ) -Centre National de la Recherche Scientifique ( CNRS ), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion. ( 2013 ), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques ( 2012 ), Institut Montpelliérain Alexander Grothendieck (IMAG), Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Jussieu (IMJ), Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS), Département de Mathématiques et Applications - ENS Paris (DMA), Centre National de la Recherche Scientifique (CNRS)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), ANR-13-BS01-0009,BoND,Frontières, numérique, dispersion.(2013), ANR-12-MONU-0007,BECASIM,Simulation numérique avancée pour les condensats de Bose-Einstein : Modèles numériques déterministes et stochastiques, Calcul haute performance, Simulation d'expériences physiques(2012), École normale supérieure - Paris (ENS Paris), and Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Fokker-Planck operator, Logarithm, General Mathematics, Gaussian, global attractor, 01 natural sciences, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Analysis of PDEs, 35Q40, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Euler equation, 0101 mathematics, nonlinear Schrödinger equation, Nonlinear Schrödinger equation, logarithmic nonlinearity, Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, Euler equations, 010101 applied mathematics, Sobolev space, Fokker–Planck operator, 35Q55, Nonlinear system, Sobolev norms, symbols, Compressibility
Abstract: We consider the nonlinear Schrodinger equation with a logarithmic nonlinearity in a dispersive regime. We show that the presence of the nonlinearity affects the large time behavior of the solution: the dispersion is faster than usual by a logarithmic factor in time and the positive Sobolev norms of the solution grow logarithmically in time. Moreover, after rescaling in space by the dispersion rate, the modulus of the solution converges to a universal Gaussian profile. These properties are suggested by explicit computations in the case of Gaussian initial data, and remain when an extra power-like nonlinearity is present in the equation. One of the key steps of the proof consists in using the Madelung transform to reduce the equation to a variant of the isothermal compressible Euler equation, whose large time behavior turns out to be governed by a parabolic equation involving a Fokker-Planck operator., Comment: Final version
Published: 2016
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19. Semiclassical wave packet dynamics in Schrödinger equations with periodic potentials

Author: Rémi Carles, Christof Sparber, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), University of Illinois System-University of Illinois System, and ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008)
Subjects: Applied Mathematics, Wave packet, 010102 general mathematics, Semiclassical physics, 01 natural sciences, Periodic potential, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, Classical mechanics, Effective mass (solid-state physics), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Discrete Mathematics and Combinatorics, 010307 mathematical physics, 0101 mathematics, Adiabatic process, MSC : 81Q20, 35A35, 35Q40, 81Q05, Mathematical Physics, Mathematics
Abstract: We consider semiclassically scaled Schrodinger equations with an external potential and a highly oscillatory periodic potential. We construct asymptotic solutions in the form of semiclassical wave packets. These solutions are concentrated (both, in space and in frequency) around the effective semiclassical phase-space flow obtained by Peierl's substitution, and involve a slowly varying envelope whose dynamics is governed by a homogenized Schrodinger equation with time-dependent effective mass. The corresponding adiabatic decoupling of the slow and fast degrees of freedom is shown to be valid up to Ehrenfest time scales., Comment: 15 pages. More explanations. An error fixed in the main result, to include the Berry phase
Published: 2012
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20. Energy cascades for NLS on the torus

Author: Rémi Carles and Erwan Faou
Subjects: FOS: Physical sciences, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, Primary 35Q55, Secondary 35C20, 37K55, FOS: Mathematics, Discrete Mathematics and Combinatorics, Countable set, 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mathematics, Geometrical optics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Torus, Mathematical Physics (math-ph), 010101 applied mathematics, Arbitrarily large, Nonlinear system, Energy cascade, symbols, Reduction (mathematics), Analysis, Analysis of PDEs (math.AP)
Abstract: We consider the nonlinear Schrodinger equation with cubic (focusing or defocusing) nonlinearity on the multidimensional torus. For special small initial data containing only five modes, we exhibit a countable set of time layers in which arbitrarily large modes are created. The proof relies on a reduction to multiphase weakly nonlinear geometric optics, and on the study of a particular two-dimensional discrete dynamical system., Comment: 15 pages, 2 figures
Published: 2012
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21. Finite Time Extinction by Nonlinear Damping for the Schrödinger Equation

Author: Rémi Carles, Clément Gallo, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008)
Subjects: Sublinear function, Applied Mathematics, Weak solution, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Space (mathematics), 01 natural sciences, Manifold, Term (time), Schrödinger equation, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Analysis, Mathematics
Abstract: We consider the Schrodinger equation on a compact manifold, in the presence of a nonlinear damping term, which is homogeneous and sublinear. For initial data in the energy space, we construct a weak solution, defined for all positive time, which is shown to be unique. In the one-dimensional case, we show that it becomes zero in finite time. In the two and three-dimensional cases, we prove the same result under the assumption of extra regularity on the initial datum., Comment: 13 pages. Title changed. Final version to appear in CPDE
Published: 2011
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22. WKB analysis for the Gross–Pitaevskii equation with non-trivial boundary conditions at infinity

Author: Thomas Alazard, Rémi Carles, Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)
Subjects: Applied Mathematics, media_common.quotation_subject, 010102 general mathematics, Mathematical analysis, Function (mathematics), 16. Peace & justice, Infinity, 01 natural sciences, WKB approximation, Sobolev space, Gross–Pitaevskii equation, Mathematics - Analysis of PDEs, Singularity, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Boundary value problem, Limit (mathematics), 0101 mathematics, 010306 general physics, Mathematical Physics, Analysis, Mathematics, media_common
Abstract: We consider the semi-classical limit for the Gross-Pitaevskii equation. In order to consider non-trivial boundary conditions at infinity, we work in Zhidkov spaces rather than in Sobolev spaces. For the usual cubic nonlinearity, we obtain a point-wise description of the wave function as the Planck constant goes to zero, so long as no singularity appears in the limit system. For a cubic-quintic nonlinearity, we show that working with analytic data may be necessary and sufficient to obtain a similar result., Comment: 20 pages
Published: 2009
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23. A Poisson Formula for the Schrödinger Operator

Author: Tohru Ozawa, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Mathematics, Hokkaido University, and Hokkaido University [Sapporo, Japan]
Subjects: Schrödinger group, Partial differential equation, Applied Mathematics, General Mathematics, 010102 general mathematics, Mathematical analysis, Poisson summation formula, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 16. Peace & justice, Poisson distribution, 01 natural sciences, 010101 applied mathematics, Nonlinear system, symbols.namesake, Operator (computer programming), symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Direct proof, 0101 mathematics, Nonlinear Schrödinger equation, Analysis, Mathematics
Abstract: 7 pages; International audience; We prove a nonlinear Poisson type formula for the Schrodinger group. Such a formula had been derived in a previous paper by the authors, as a consequence of the study of the asymptotic behavior of nonlinear wave operators for small data. In this note, we propose a direct proof, and extend the range allowed for the power of the nonlinearity to the set of all short range nonlinearities. Moreover, energy-critical nonlinearities are allowed.
Published: 2008
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24. On the Cauchy problem in Sobolev spaces for nonlinear Schrödinger equations with potential

Author: Rémi Carles
Subjects: Cauchy problem, 35B33, 35B65, General Mathematics, media_common.quotation_subject, Mathematical analysis, Mathematics::Analysis of PDEs, 35A07, Infinity, 35A05, Sobolev inequality, 35Q55, Sobolev space, symbols.namesake, Nonlinear system, Mathematics - Analysis of PDEs, symbols, Initial value problem, Nonlinear Schrödinger equation, media_common, Mathematics, Sign (mathematics)
Abstract: We consider the Cauchy problem for nonlinear Schrodinger equations in the presence of a smooth, possibly unbounded, potential. No assumption is made on the sign of the potential. If the potential grows at most linearly at infinity, we construct solutions in Sobolev spaces (without weight), locally in time. Under some natural assumptions, we prove that the $H^1$-solutions are global in time. On the other hand, if the potential has a super-linear growth, then the Sobolev regularity of positive order is lost instantly, not matter how large it is, unless the initial datum decays sufficiently fast at infinity.
Published: 2008
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25. Monotonicity properties of the blow-up time for nonlinear Schrödinger equations: Numerical evidence

Author: Cristophe Besse, Norbert J. Mauser, Rémi Carles, and Hans Peter Stimming
Subjects: Coupling constant, Oscillation, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Monotonic function, 01 natural sciences, 010101 applied mathematics, Nonlinear system, symbols.namesake, Quadratic equation, Damping factor, symbols, Discrete Mathematics and Combinatorics, 0101 mathematics, Wave function collapse, Nonlinear Schrödinger equation, Mathematics
Abstract: We consider the focusing nonlinear Schrodinger equation, in the $L^2$-critical and supercritical cases. We investigate numerically the dependence of the blow-up time on a parameter in three cases: dependence upon the coupling constant, when the initial data are fixed; dependence upon the strength of a quadratic oscillation in the initial data when the equation and the initial profile are fixed; finally, dependence upon a damping factor when the initial data are fixed. It turns out that in most situations monotonicity in the evolution of the blow-up time does not occur. In the case of quadratic oscillations in the initial data, with critical nonlinearity, monotonicity holds; this is proven analytically.
Published: 2008
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26. NUMERICAL ASPECTS OF NONLINEAR SCHRÖDINGER EQUATIONS IN THE PRESENCE OF CAUSTICS

Author: Laurent Gosse, Rémi Carles, Institut CNRS-PAULI (ICP), INST WOLFGANG PAULI-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Istituto per le Applicazioni del Calcolo 'Mauro Picone' (IAC), and Consiglio Nazionale delle Ricerche [Roma] (CNR)
Subjects: 35B33, Field (physics), 010103 numerical & computational mathematics, 01 natural sciences, Schrödinger equation, symbols.namesake, 81Q20, caustics, Point (geometry), Mathematics - Numerical Analysis, 0101 mathematics, Nonlinear Schrodinger equation, time-splitting scheme, 35P25, Mathematics, Cusp (singularity), Fourier scheme, WKB expansion, Applied Mathematics, 010102 general mathematics, 65T50, 35Q55, Nonlinear system, Classical mechanics, Modeling and Simulation, symbols, Caustic (optics), Heuristics, Focus (optics), [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract: The aim of this text is to develop on the asymptotics of some 1-D nonlinear Schrodinger equations from both the theoretical and the numerical perspectives, when a caustic is formed. We review rigorous results in the field and give some heuristics in cases where justification is still needed. The scattering operator theory is recalled. Numerical experiments are carried out on the focus point singularity for which several results have been proven rigorously. Furthermore, the scattering operator is numerically studied. Finally, experiments on the cusp caustic are displayed, and similarities with the focus point are discussed., Comment: 20 pages. To appear in Math. Mod. Meth. Appl. Sci
Published: 2007
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27. Semi-classical limit of Schrödinger–Poisson equations in space dimension n⩾3

Author: Thomas Alazard and Rémi Carles
Subjects: Pointwise convergence, Integrable system, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Planck constant, 01 natural sciences, Classical limit, 010101 applied mathematics, Momentum, symbols.namesake, Quadratic equation, Doping profile, Position (vector), symbols, Schrödinger–Poisson, Semi-classical analysis, 0101 mathematics, Analysis, Mathematics
Abstract: We prove the existence of solutions to the Schrodinger–Poisson system on a time interval independent of the Planck constant, when the doping profile does not necessarily decrease at infinity, in the presence of a subquadratic external potential. The lack of integrability of the doping profile is resolved by working in Zhidkov spaces, in space dimension at least three. We infer that the main quadratic quantities (position density and modified momentum density) converge strongly as the Planck constant goes to zero. When the doping profile is integrable, we prove pointwise convergence.
Published: 2007
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28. Scattering pour l'équation de Schrödinger en présence d'un potentiel répulsif

Author: Laurent Michel, Jean-François Bony, Rémi Carles, and Dietrich Häfner
Subjects: symbols.namesake, Partial differential equation, symbols, General Medicine, Scattering theory, Hamiltonian (quantum mechanics), Mathematics, Mathematical physics, Schrödinger equation
Abstract: Resume On etudie la theorie de la diffusion pour l'equation de Schrodinger avec hamiltonien de reference −Δ−〈x〉α, dans R n , avec 0
Published: 2004
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29. Focusing of a pulse with arbitrary phase shift for a nonlinear wave equation

Author: David Lannes and Rémi Carles
Subjects: Nonlinear wave equation, General Mathematics, Geometry, Humanities, Mathematics
Abstract: Nous considerons un systeme de deux equations des ondes lineaires conservatives, couplees non-lineairement, en dimension trois d'espace. Pour des donnees initiales radiales de type impulsions courtes, les solutions focalisent a l'origine lorsque la longueur d'onde tend vers zero. Le caractere conservatif de l'equation fait que la traversee de la caustique n'est pas triviale : nous l'analysons pour des donnees initiales particulieres. Il ressort que le dephasage entre l'onde entrante (avant focalisation) et l'onde sortante (apres focalisation) se comporte en In e, ou e represente la longueur d'onde.
Published: 2003
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30. Changing blow-up time in nonlinear Schrödinger equations

Author: Rémi Carles
Subjects: Split-step method, symbols.namesake, Nonlinear system, Classical mechanics, Stark effect, symbols, Harmonic potential, General Medicine, Mathematical physics, Mathematics, Schrödinger equation
Published: 2003
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31. On nonlinear Schrodinger type equations with nonlinear damping

Author: Rémi Carles, Paolo Antonelli, Christof Sparber, Scuola Normale Superiore di Pisa (SNS), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Mathematics, Statistics and Computer Science [Chicago] (UIC), University of Illinois [Chicago] (UIC), and University of Illinois System-University of Illinois System
Subjects: Scattering, General Mathematics, 010102 general mathematics, Mathematical analysis, Zero (complex analysis), Type (model theory), 01 natural sciences, 010101 applied mathematics, Nonlinear system, symbols.namesake, Mathematics - Analysis of PDEs, Quadratic equation, Control theory, FOS: Mathematics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Finite time, Schrödinger's cat, Analysis of PDEs (math.AP), Mathematics
Abstract: We consider equations of nonlinear Schrodinger type augmented by nonlinear damping terms. We show that nonlinear damping prevents finite time blow-up in several situations, which we describe. We also prove that the presence of a quadratic confinement in all spatial directions drives the solution of our model to zero for large time. In the case without external potential we prove that the solution may not go to zero for large time due to (non-trivial) scattering., 17 pages. The case of partial confinement was removed, due to a flaw in the proof
Published: 2015
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32. Remarques sur l'équation de Schrödinger non linéaire avec potentiel harmonique

Author: Rémi Carles
Subjects: General Medicine, Humanities, Mathematics
Abstract: Resume La condensation de Bose–Einstein fait apparaitre des equations de Schrodinger non lineaires avec potentiel harmonique. Nous etudions le probleme de Cauchy pour ces equations, en particulier le phenomene d'explosion en temps fini. Pour cela, nous etablissons une loi d'evolution analogue a la loi de conservation pseudo-conforme dans le cas sans potentiel. Nous montrons alors que sous un critere simple, autorisant en particulier une plage de valeurs positives pour l'energie, la solution explose en temps fini. Pour citer cet article : R. Carles, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 763–766.
Published: 2002
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33. Absorption d'impulsions non linéaires radiales focalisantes dans

Author: Jeffrey Rauch and Rémi Carles
Subjects: General Medicine, Atomic physics, Absorption (electromagnetic radiation), Mathematics, Pulse (physics)
Abstract: Resume Nous etudions la validite de l'optique geometrique pour des impulsions ultra-courtes solutions d'equations d'ondes en dimension trois d'espace, qui focalisent en un point. Si l'amplitude des donnees initiales est suffisamment grande, ces ondes subissent des effets non lineaires tres forts. Dans le cas des equations dissipatives, les impulsions courtes sont absorbees. Dans le cas des equations accretives, la famille des impulsions devient non bornee.
Published: 2001
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34. Diffusion d'impulsions non linéaires radiales focalisantes dans

Author: Jeffrey Rauch and Rémi Carles
Subjects: Cauchy problem, Partial differential equation, Exact solutions in general relativity, Radial diffusion, Scattering operator, Existence theorem, General Medicine, Impulse (physics), Mathematics, Mathematical physics
Abstract: Resume Nous etudions la validite de l'optique geometrique pour des impulsions ultra-courtes solutions d'equations d'ondes en dimension trois d'espace, qui focalisent en un point. Pour une amplitude critique des donnees initiales, l'optique geometrique lineaire est valide avant le point focal, et apres. La transition entre ces deux regimes est decrite par un operateur de diffusion, qui elargit le support des ondes.
Published: 2001
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35. Comportements Précisés Près D'Une Caustique En Optique Géométrique Non Linéaire

Author: Rémi Carles
Subjects: Applied Mathematics, Humanities, Analysis, Mathematics
Published: 1998
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36. On the orbital stability of Gaussian solitary waves in the log-KdV equation

Author: Rémi Carles, Dmitry E. Pelinovsky, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Mathematics and Statistics [Hamilton], and McMaster University [Hamilton, Ontario]
Subjects: Logarithm, Gaussian, FOS: Physical sciences, General Physics and Astronomy, Pattern Formation and Solitons (nlin.PS), symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, Korteweg–de Vries equation, Nonlinear Sciences::Pattern Formation and Solitons, Eigenvalues and eigenvectors, Mathematical Physics, Mathematics, Applied Mathematics, Operator (physics), Mathematical analysis, Time evolution, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Nonlinear Sciences - Pattern Formation and Solitons, Pulse (physics), Nonlinear Sciences::Exactly Solvable and Integrable Systems, symbols, Analysis of PDEs (math.AP)
Abstract: We consider the logarithmic Korteweg-de Vries (log-KdV) equation, which models solitary waves in anharmonic chains with Hertzian interaction forces. By using an approximating sequence of global solutions of the regularized generalized KdV equation in $H^1(\mathbb{R})$ with conserved $L^2$ norm and energy, we construct a weak global solution of the log-KdV equation in a subset of $H^1(\mathbb{R})$. This construction yields conditional orbital stability of Gaussian solitary waves of the log-KdV equation, provided uniqueness and continuous dependence of the constructed solution holds. Furthermore, we study the linearized log-KdV equation at the Gaussian solitary wave and prove that the associated linearized operator has a purely discrete spectrum consisting of simple purely imaginary eigenvalues in addition to the double zero eigenvalue. The eigenfunctions, however, do not decay like Gaussian functions but have algebraic decay. Nevertheless, using numerical approximations, we show that the Gaussian initial data do not spread out and preserve their spatial Gaussian decay in the time evolution of the linearized log-KdV equation., 21 pages; 4 figures
Published: 2014
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37. Explicit solutions for replicator-mutator equations: extinction vs. acceleration

Author: Matthieu Alfaro, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)
Subjects: 0303 health sciences, Class (set theory), Extinction, [SDV.GEN.GPO]Life Sciences [q-bio]/Genetics/Populations and Evolution [q-bio.PE], Applied Mathematics, Gaussian, Acceleration (differential geometry), Function (mathematics), 01 natural sciences, 010101 applied mathematics, 03 medical and health sciences, symbols.namesake, Mathematics - Analysis of PDEs, symbols, Applied mathematics, Quantitative Biology::Populations and Evolution, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Heat equation, 0101 mathematics, Finite time, Quantitative Biology - Populations and Evolution, 030304 developmental biology, Mathematics
Abstract: We consider a class of nonlocal reaction-diffusion problems, referred to as replicator-mutator equations in evolutionary genetics. By using explicit changes of unknown function, we show that they are equivalent to the heat equation and, therefore, compute their solution explicitly. Based on this, we then prove that, in the case of beneficial mutations in asexual populations, solutions dramatically depend on the tails of the initial data: they can be global, become extinct in finite time or, even, be defined for no positive time. In the former case, we prove that solutions are accelerating, and in many cases converge for large time to some universal Gaussian profile. This sheds light on the biological relevance of such models., Comment: 15 pages
Published: 2014
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38. Madelung, Gross-Pitaevskii and Korteweg

Author: Jean-Claude Saut, Rémi Carles, Raphaël Danchin, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM), Laboratoire de Mathématiques d'Orsay (LM-Orsay), Centre National de la Recherche Scientifique (CNRS)-Université Paris-Sud - Paris 11 (UP11), ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Université Paris-Sud - Paris 11 (UP11)-Centre National de la Recherche Scientifique (CNRS), ANR-08-JCJC-0124-01,R.A.S,Régimes Asymptotiques pour l'équation de Schrödinger(2008), and Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS)
Subjects: Applied Mathematics, 010102 general mathematics, FOS: Physical sciences, General Physics and Astronomy, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), Type (model theory), 01 natural sciences, Compressible flow, Connection (mathematics), 010101 applied mathematics, symbols.namesake, Classical mechanics, Mathematics - Analysis of PDEs, Quantum hydrodynamics, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Analysis of PDEs (math.AP), Mathematics
Abstract: This paper surveys various aspects of the hydrodynamic formulation of the nonlinear Schrodinger equation obtained via the Madelung transform in connexion to models of quantum hydrodynamics and to compressible fluids of the Korteweg type., 32 pages
Published: 2012
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39. Nonlinear Schrodinger equation and frequency saturation

Author: Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008)
Subjects: Primary 35Q55, Secondary 35A01, 35B30, 35B45, 35B65, 35B30, 01 natural sciences, Instability, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, well-posedness, 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, nonlinear Schrödinger equation, approximation, Nonlinear Schrödinger equation, Mathematics, Numerical Analysis, 35A01, 35B65, Applied Mathematics, 010102 general mathematics, Mathematical analysis, 35B45, 35Q55, Sobolev space, Nonlinear system, Exact solutions in general relativity, symbols, 010307 mathematical physics, Saturation (chemistry), Analysis, Well posedness
Abstract: We propose an approach that permits to avoid instability phenomena for the nonlinear Schrodinger equations. We show that by approximating the solution in a suitable way, relying on a frequency cut-off, global well-posedness is obtained in any Sobolev space with nonnegative regularity. The error between the exact solution and its approximation can be measured according to the regularity of the exact solution, with different accuracy according to the cases considered., Comment: 15 pages: appendix added
Published: 2012
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40. On Schrodinger equations with modified dispersion

Author: Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008)
Subjects: Applied Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, 01 natural sciences, 010101 applied mathematics, Multiplier (Fourier analysis), Split-step method, Sobolev space, symbols.namesake, Fourier transform, Mathematics - Analysis of PDEs, Ordinary differential equation, Bounded function, symbols, Initial value problem, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Schrödinger equation, Analysis, Mathematics
Abstract: We consider the nonlinear Schrodinger equation with a modified spatial dispersion, given either by an homogeneous Fourier multiplier, or by a bounded Fourier multiplier. Arguments based on ordinary differential equations yield ill-posedness results which are sometimes sharp. When the Fourier multiplier is bounded, we infer that no Strichartz-type estimate improving on Sobolev embedding is available. Finally, we show that when the symbol is bounded, the Cauchy problem may be ill-posed in the case of critical regularity, with arbitrarily small initial data. The same is true when the symbol is homogeneous of degree one, where scaling arguments may not even give the right critical value., Comment: 10 pages. References and comments added
Published: 2011
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41. Numerical aspects of the nonlinear Schrödinger equation in the semiclassical limit in a supercritical regime

Author: Bijan Mohammadi, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008)
Subjects: Numerical Analysis, Applied Mathematics, 010102 general mathematics, Semiclassical physics, 010103 numerical & computational mathematics, Planck constant, 01 natural sciences, Euler equations, Momentum, Computational Mathematics, symbols.namesake, Classical mechanics, Rate of convergence, Modeling and Simulation, symbols, Limit (mathematics), 0101 mathematics, Spectral method, Nonlinear Schrödinger equation, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Mathematics
Abstract: 29 pages, 18 figures. More explanations, more references, and an extra experience past the breakup time.; International audience; We study numerically the semiclassical limit for the nonlinear Schrödinger equation thanks to a modification of the Madelung transform due to Grenier. This approach allows for the presence of vacuum. Even if the mesh size and the time step do not depend on the Planck constant, we recover the position and current densities in the semiclassical limit, with a numerical rate of convergence in accordance with the theoretical results, before shocks appear in the limiting Euler equation. By using simple projections, the mass and the momentum of the solution are well preserved by the numerical scheme, while the variation of the energy is not negligible numerically. Experiments suggest that beyond the critical time for the Euler equation, Grenier's approach yields smooth but highly oscillatory terms.
Published: 2011
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42. Minimal blow-up solutions to the mass-critical inhomogeneous NLS equation

Author: Valeria Banica, Rémi Carles, Thomas Duyckaerts, Département de Mathématiques [Evry], Université d'Évry-Val-d'Essonne (UEVE), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Analyse, Géométrie et Modélisation (AGM - UMR 8088), CY Cergy Paris Université (CY)-Centre National de la Recherche Scientifique (CNRS), ANR-08-JCJC-0124,R.A.S.,Régimes Asymptotiques pour l'équation de Schrödinger(2008), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), and Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, 01 natural sciences, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, Critical point (thermodynamics), [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Point (geometry), 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mathematics, Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, Mathematical Physics (math-ph), Manifold, 010101 applied mathematics, Nonlinear system, symbols, Ground state, Analysis, Analysis of PDEs (math.AP)
Abstract: We consider the mass-critical focusing nonlinear Schrodinger equation in the presence of an external potential, when the nonlinearity is inhomogeneous. We show that if the inhomogeneous factor in front of the nonlinearity is sufficiently flat at a critical point, then there exists a solution which blows up in finite time with the maximal (unstable) rate at this point. In the case where the critical point is a maximum, this solution has minimal mass among the blow-up solutions. As a corollary, we also obtain unstable blow-up solutions of the mass-critical Schrodinger equation on some surfaces. The proof is based on properties of the linearized operator around the ground state, and on a full use of the invariances of the equation with an homogeneous nonlinearity and no potential, via time-dependent modulations., 36 pages. More explanations, references updated, statement of Theorem 1.1 corrected. FInal version
Published: 2009
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43. On the time evolution of Wigner measures for Schrodinger equations

Author: Rémi Carles, Norbert J. Mauser, Clotilde Fermanian-Kammerer, Hans Peter Stimming, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Laboratoire d'Analyse et de Mathématiques Appliquées (LAMA), Université Paris-Est Marne-la-Vallée (UPEM)-Fédération de Recherche Bézout-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Wolfgang Pauli Institute (WPI), University of Vienna [Vienna], Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), and Centre National de la Recherche Scientifique (CNRS)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Fédération de Recherche Bézout-Université Paris-Est Marne-la-Vallée (UPEM)
Subjects: FOS: Physical sciences, 01 natural sciences, Schrödinger equation, Wigner measures, symbols.namesake, Quadratic equation, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Limit (mathematics), 0101 mathematics, ill-posedness, Mathematical Physics, Mathematics, Mathematical physics, Pointwise, Applied Mathematics, 010102 general mathematics, WKB methods, Time evolution, General Medicine, Mathematical Physics (math-ph), caustic, 010101 applied mathematics, symbols, Caustic (optics), eigenvalue crossing, Analysis, Ill posedness, Analysis of PDEs (math.AP)
Abstract: In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails., Comment: Survey, 26 pages
Published: 2009
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44. Multiphase weakly nonlinear geometric optics for Schrodinger equations

Author: Christof Sparber, Eric Dumas, Rémi Carles, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Institut Fourier (IF ), Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS), Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge [UK] (CAM), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Institut Fourier (IF), Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), and Department of Applied Mathematics and Theoretical Physics / Centre for Mathematical Sciences (DAMTP/CMS)
Subjects: [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Plane wave, FOS: Physical sciences, 01 natural sciences, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Schrödinger equation, Mathematical Physics, Mixing (physics), Mathematics, Euclidean space, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Torus, Mathematical Physics (math-ph), 010101 applied mathematics, Sobolev space, Computational Mathematics, Nonlinear system, Classical mechanics, symbols, Analysis, Analysis of PDEs (math.AP)
Abstract: We describe and rigorously justify the nonlinear interaction of highly oscillatory waves in nonlinear Schrodinger equations, posed on Euclidean space or on the torus. Our scaling corresponds to a weakly nonlinear regime where the nonlinearity affects the leading order amplitude of the solution, but does not alter the rapid oscillations. We consider initial states which are superpositions of slowly modulated plane waves, and use the framework of Wiener algebras. A detailed analysis of the corresponding nonlinear wave mixing phenomena is given, including a geometric interpretation on the resonance structure for cubic nonlinearities. As an application, we recover and extend some instability results for the nonlinear Schrodinger equation on the torus in negative order Sobolev spaces., Comment: 29 pages
Published: 2009
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45. Pointwise Description of the Wave Function

Author: Rémi Carles
Subjects: Pointwise, Mathematical analysis, Function (mathematics), Mathematics
Published: 2008
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46. On the Gross-Pitaevskii equation for trapped dipolar quantum gases

Author: Peter A. Markowich, Rémi Carles, Christof Sparber, Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Applied Mathematics and Theoretical Physics (DAMTP), University of Cambridge [UK] (CAM), 'KAUST Investigator Award' (P. Markowich). 'Wolfson Research Merit Award' of the Royal Society (P. Markowich). 'APART grant' of the Austrian Academy of Science (C. Sparber)., and ANR-05-BLAN-0163,SCASEN,Singularités et comportement asymptotique des solutions d'Euler et de Navier-Stokes(2005)
Subjects: dipole interaction, Quantum gas, dimension reduction, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], FOS: Physical sciences, General Physics and Astronomy, 01 natural sciences, law.invention, Schrödinger equation, symbols.namesake, law, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], Uniqueness, 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Mathematics, Mathematical physics, Condensed Matter::Quantum Gases, Condensed Matter::Other, Applied Mathematics, 010102 general mathematics, Condensation, Bose-Einstein condensates, Statistical and Nonlinear Physics, Mathematical Physics (math-ph), 010101 applied mathematics, Dipole, Nonlinear system, Gross–Pitaevskii equation, symbols, Bose–Einstein condensate, Gross-Pitaevskii equation
Abstract: We study the time-dependent Gross-Pitaevskii equation describing Bose-Einstein condensation of trapped dipolar quantum gases. Existence and uniqueness as well as the possible blow-up of solutions are studied. Moreover, we discuss the problem of dimension-reduction for this nonlinear and nonlocal Schrodinger equation., 21 pages
Published: 2008
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47. Scattering theory for radial nonlinear Schrödinger equations on hyperbolic space

Author: Valeria Banica, Gigliola Staffilani, Rémi Carles, Département de Mathématiques [Evry], Université d'Évry-Val-d'Essonne (UEVE), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)-Centre National de la Recherche Scientifique (CNRS), Department of Mathematics [MIT], Massachusetts Institute of Technology (MIT), and Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM)
Subjects: Hyperbolic space, 010102 general mathematics, Mathematical analysis, Mathematics::Analysis of PDEs, Type (model theory), Mathematical proof, 01 natural sciences, Symmetry (physics), Schrödinger equation, Range (mathematics), symbols.namesake, Nonlinear system, 0103 physical sciences, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, Geometry and Topology, Scattering theory, 0101 mathematics, Analysis, Mathematics
Abstract: Some comments and references added in Section 6.; International audience; We study the long time behavior of radial solutions to nonlinear Schrödinger equations on hyperbolic space. We show that the usual distinction between short range and long range nonlinearity is modified: the geometry of the hyperbolic space makes every power-like nonlinearity short range. The proofs rely on weighted Strichartz estimates, which imply Strichartz estimates for a broader family of admissible pairs, and on Morawetz type inequalities. The latter are established without symmetry assumptions.
Published: 2008
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48. On the wave operators for the critical nonlinear Schrodinger equation

Author: Rémi Carles, Tohru Ozawa, Institut CNRS-PAULI (ICP), INST WOLFGANG PAULI-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques et de Modélisation de Montpellier (I3M), Centre National de la Recherche Scientifique (CNRS)-Université Montpellier 2 - Sciences et Techniques (UM2)-Université de Montpellier (UM), Department of Mathematics, Hokkaido University, and Hokkaido University [Sapporo, Japan]
Subjects: Similarity (geometry), 35B33, General Mathematics, [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph], Mathematics::Analysis of PDEs, Inverse, Derivative, Type (model theory), 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, 35B40, 35Q55, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Mathematical physics, Mathematics, 010102 general mathematics, Mathematical analysis, Mathematics::Spectral Theory, Split-step method, Nonlinear system, Fourier transform, symbols, 010307 mathematical physics
Abstract: We prove that for the mass critical nonlinear Schrodinger equations, the wave operators and their inverse are related explicitly in terms of the Fourier transform. We discuss some consequences of this property. In the one-dimensional case, we show a precise similarity between the mass critical nonlinear Schrodinger equation and a nonlinear Schrodinger equation of derivative type., Comment: 8 pages
Published: 2007
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49. On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^2$-critical case

Author: Rémi Carles, Sahbi Keraani, Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Appliquées ( MAB ), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques Appliquées de Bordeaux (MAB), Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1, Guillemer, Marie-Annick, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Centre National de la Recherche Scientifique (CNRS)
Subjects: General Mathematics, Space (mathematics), 01 natural sciences, Schrödinger equation, 35Q55, 35B40, 35B05, symbols.namesake, Quadratic equation, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], Conformal symmetry, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, [MATH.MATH-AP] Mathematics [math]/Analysis of PDEs [math.AP], nonlinear Schrödinger equation, Nonlinear Schrödinger equation, Mathematics, bow up, Applied Mathematics, Operator (physics), 010102 general mathematics, Mathematical analysis, scattering, 16. Peace & justice, 010101 applied mathematics, Split-step method, Nonlinear system, symbols, semi-classical analysis
Abstract: International audience; We consider a nonlinear semi-classical Schrödinger equation for which quadratic oscillations lead to focusing at one point, described by a nonlinear scattering operator. The relevance of the nonlinearity was discussed by R. Carles, C. Fermanian-Kammerer and I. Gallagher for $L^2$-supercritical power-like nonlinearities and more general initial data. The present results concern the $L^2$-critical case, in space dimensions 1 and 2; we describe the set of non-linearizable data, which is larger, due to the conformal invariance. As an application, we precise a result by F. Merle and L. Vega concerning finite time blow up for the critical Schrödinger equation. The proof relies on linear and nonlinear profile decompositions.
Published: 2007

50. Linear vs. nonlinear effects for nonlinear Schrödinger equations with potential

Author: Rémi Carles, Institut de Recherche Mathématique de Rennes (IRMAR), AGROCAMPUS OUEST, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université de Rennes 2 (UR2), Université de Rennes (UNIV-RENNES)-École normale supérieure - Rennes (ENS Rennes)-Centre National de la Recherche Scientifique (CNRS)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Université de Rennes (UNIV-RENNES)-Institut National des Sciences Appliquées (INSA), Laboratoire de Mathématiques Appliquées de Bordeaux (MAB), Centre National de la Recherche Scientifique (CNRS)-Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1, Université de Rennes (UR)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École normale supérieure - Rennes (ENS Rennes)-Université de Rennes 2 (UR2)-Centre National de la Recherche Scientifique (CNRS)-INSTITUT AGRO Agrocampus Ouest, Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro)-Institut national d'enseignement supérieur pour l'agriculture, l'alimentation et l'environnement (Institut Agro), Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1 (UB)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique de Rennes ( IRMAR ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -AGROCAMPUS OUEST-École normale supérieure - Rennes ( ENS Rennes ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National des Sciences Appliquées ( INSA ) -Université de Rennes 2 ( UR2 ), Université de Rennes ( UNIV-RENNES ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire de Mathématiques Appliquées ( MAB ), and Université Bordeaux Segalen - Bordeaux 2-Université Sciences et Technologies - Bordeaux 1-Centre National de la Recherche Scientifique ( CNRS )
Subjects: Polynomial, General Mathematics, FOS: Physical sciences, Mathematical proof, 01 natural sciences, Primary: 35Q55, Schrödinger equation, symbols.namesake, Mathematics - Analysis of PDEs, Secondary: 35A05, 35B30, 35B35, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, Applied mathematics, Order (group theory), [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 0101 mathematics, Nonlinear Schrodinger equation, Mathematical Physics, Mathematics, Scattering, Applied Mathematics, 010102 general mathematics, scattering, Mathematical Physics (math-ph), Mehler's formula, 010101 applied mathematics, Nonlinear system, symbols, semi-classical analysis, Finite time, blow-up, Analysis of PDEs (math.AP), 35Q55, 35A05, 35B30, 35B35
Abstract: We review some recent results on nonlinear Schrodinger equations with potential, with emphasis on the case where the potential is a second order polynomial, for which the interaction between the linear dynamics caused by the potential, and the nonlinear effects, can be described quite precisely. This includes semi-classical regimes, as well as finite time blow-up and scattering issues. We present the tools used for these problems, as well as their limitations, and outline the arguments of the proofs., Comment: 20 pages; survey of previous results
Published: 2005
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