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11 results on '"Rota Nodari"'

1. On a nonlinear Schrödinger equation for nucleons in one space dimension

Author

Simona Rota Nodari, Christian Klein, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), French National Research Agency (ANR)ANR-17-CE40-0035ANR-17-CE40-0016isite BFC project NAANoDEuropean Commission778010EITAG project - FEDER de BourgogneRegion Bourgogne-Franche-ComteEUR EIPHIANR-17-EURE-0002 EIPHI, ANR-17-CE40-0035,ANuI,Approches analytiques, numériques et des systèmes intégrables pour les équations aux dérivées partielles dispersives nonlinéaires(2017), ANR-17-CE40-0016,DYRAQ,Dynamique des systèmes quantiques relativistes(2017), ANR-17-EURE-0002,EIPHI,Ingénierie et Innovation par les sciences physiques, les savoir-faire technologiques et l'interdisciplinarité(2017), French National Research Agency (ANR)ANR-17-CE40-0035French National Research Agency (ANR)ANR-17-CE40-0016isite BFC project NAANoD European Commission778010EITAG project - FEDER de Bourgogne Region Bourgogne-Franche-Comte EUR EIPHI ANR-17-EURE-0002 EIPHI, European Project: 778010,IPaDEGAN, and Université de Bourgogne (UB)-Centre National de la Recherche Scientifique (CNRS)

Subjects
numerical study, Space dimension, Nonlinear Schrö, 010103 numerical & computational mathematics, Nonlinear Schrödinger equations, 01 natural sciences, Stability (probability), symbols.namesake, Mathematics - Analysis of PDEs, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Numerical Analysis, 0101 mathematics, [MATH]Mathematics [math], dinger equations, Nonlinear Schrödinger equation, Mathematics, MSC 35Q55, 35C08, 65M70, Numerical Analysis, Applied Mathematics, 010102 general mathematics, Time evolution, ground states, Computational Mathematics, Classical mechanics, Modeling and Simulation, Atomic nucleus, symbols, Particle, Nucleon, Analysis, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

We study a 1D nonlinear Schrödinger equation appearing in the description of a particle inside an atomic nucleus. For various nonlinearities, the ground states are discussed and given in explicit form. Their stability is studied numerically via the time evolution of perturbed ground states. In the time evolution of general localized initial data, they are shown to appear in the long time behaviour of certain cases.

Published
2021
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2. Contributions to the mathematical study of models from classical and quantum physics

Author

Rota Nodari, Simona, Université Bourgogne Franche-Comté [COMUE] (UBFC), Université Bourgogne Franche-Comté, Sylvie Benzoni-Gavage, and ANR-17-CE40-0016,DYRAQ,Dynamique des systèmes quantiques relativistes(2017)

Subjects
EDP en optique non linéaire, Variational methods, Gaz de Coulomb et de Riesz, EDP non linéaires, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], PDE in nonlinear optics, Méthodes mathématiques en physique quantique, Mathematical methods in quantum physics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Coulomb and Riesz gases, Nonlinear PDEs, Méthodes variationnelles
Published
2021

3. Equidistribution of Jellium Energy for Coulomb and Riesz Interactions

Author

Mircea Petrache, Simona Rota Nodari, Max Planck Institute for Mathematics (MPIM), Max-Planck-Gesellschaft, Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), S. Rota Nodari was partially supported by PEPS – Jeunes Chercheur-e-s – 2016 (CNRS), and M. Petrache was supported by an EPDI fellowship., Max Planck Institute for Mathematics ( Bonn ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), and Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS )

Subjects
Scale (ratio), General Mathematics, Jellium, Dimension (graph theory), FOS: Physical sciences, [ MATH.MATH-FA ] Mathematics [math]/Functional Analysis [math.FA], Coulomb gases, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], 01 natural sciences, Microscopic scale, Equidistribution, MSC: 82B05, 82B21, 82D05, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], 0103 physical sciences, Coulomb, FOS: Mathematics, 0101 mathematics, 010306 general physics, Mathematical Physics, Mathematics, Mathematical physics, Conjecture, Numerical analysis, 010102 general mathematics, Mathematical analysis, Probability (math.PR), [ MATH.MATH-MP ] Mathematics [math]/Mathematical Physics [math-ph], Mathematical Physics (math-ph), Functional Analysis (math.FA), Renormalized energy, Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Computational Mathematics, Riesz gases, Crystallization, [ MATH.MATH-PR ] Mathematics [math]/Probability [math.PR], Analysis, Energy (signal processing), Mathematics - Probability
Abstract

For general dimension $d$ we prove the equidistribution of energy at the micro-scale in $\mathbb R^d$, for the optimal point configurations appearing in Coulomb gases at zero temperature. At the microscopic scale, i.e. after blow-up at the scale corresponding to the interparticle distance, in the case of Coulomb gases we show that the energy concentration is precisely determined by the macroscopic density of points, independently of the scale. This uses the "jellium energy'' which was previously shown to control the next-order term in the large particle number asymptotics of the minimum energy. As a corollary, we obtain sharp error bounds on the discrepancy between the number of points and its expected average of optimal point configurations for Coulomb gases, extending previous results valid only for $2$-dimensional log-gases. For Riesz gases with interaction potentials $g(x)=|x|^{-s}, s\in]\min\{0,d-2\},d[$ and one-dimensional log-gases, we prove the same equidistribution result under an extra hypothesis on the decay of the localized energy, which we conjecture to hold for minimizing configurations. In this case we use the Caffarelli-Silvestre description of the non-local fractional Laplacians in $\mathbb R^d$ to localize the problem.

Published
2018
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4. The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications

Author

Simona Rota Nodari, Mathieu Lewin, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Université Paris sciences et lettres (PSL), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), ANR-17-CE40-0016,DYRAQ,Dynamique des systèmes quantiques relativistes(2017), European Project: 725528,MDFT, Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS), and Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Pure mathematics, Conjecture, Applied Mathematics, 010102 general mathematics, Orbital stability, 01 natural sciences, Schrödinger equation, Mathematics - Spectral Theory, 010101 applied mathematics, symbols.namesake, Mathematics - Analysis of PDEs, FOS: Mathematics, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Uniqueness, 0101 mathematics, Degeneracy (mathematics), Spectral Theory (math.SP), Nonlinear Schrödinger equation, Analysis, Analysis of PDEs (math.AP), Mathematics, [MATH.MATH-SP]Mathematics [math]/Spectral Theory [math.SP]
Abstract

In this paper we first prove a general result about the uniqueness and non-degeneracy of positive radial solutions to equations of the form $$\Delta u+g(u)=0$$ . Our result applies in particular to the double power non-linearity where $$g(u)=u^q-u^p-\mu u$$ for $$p>q>1$$ and $$\mu >0$$ , which we discuss with more details. In this case, the non-degeneracy of the unique solution $$u_\mu $$ allows us to derive its behavior in the two limits $$\mu \rightarrow 0$$ and $$\mu \rightarrow \mu _*$$ where $$\mu _*$$ is the threshold of existence. This gives the uniqueness of energy minimizers at fixed mass in certain regimes. We also make a conjecture about the variations of the $$L^2$$ mass of $$u_\mu $$ in terms of $$\mu $$ , which we illustrate with numerical simulations. If valid, this conjecture would imply the uniqueness of energy minimizers in all cases and also give some important information about the orbital stability of $$u_\mu $$ .

Published
2020
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5. Heteroclinic Structure of Parametric Resonance in Fibers with Periodic Dispersion

Author

Matteo Conforti, Alexandre Kudlinski, Stefano Trillo, Guillaume Dujardin, Arnaud Mussot, Stephan De Bièvre, Simona Rota Nodari, Andrea Armaroli, Laboratoire de Physique des Lasers, Atomes et Molécules - UMR 8523 (PhLAM), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Fonctions Optiques pour les Technologies de l'informatiON (FOTON), Université de Rennes (UR)-Université européenne de Bretagne - European University of Brittany (UEB)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École Nationale Supérieure des Sciences Appliquées et de Technologie (ENSSAT)-Télécom Bretagne-Centre National de la Recherche Scientifique (CNRS), Engineering Department [Ferrara], and Università degli Studi di Ferrara = University of Ferrara (UniFE)

Subjects
Physics, Optical fiber, Nonlinear optics, Resonance, Mechanics, fibers, law.invention, NO, Modulational instability, Nonlinear system, Classical mechanics, Instabilities and chaos, law, Modulation, Nonlinear optics, fibers, Nonlinear optics, four-wave mixing, Dispersion (optics), Parametric oscillator, four-wave mixing, [MATH]Mathematics [math], Parametric statistics
Abstract

International audience; We investigate the nonlinear stage of modulational instability in dispersion oscillating fibers in normal dispersion regime. We unveil a heteroclinic structure leading to the excitability of superbreathers and parametric amplification outside the linear gain bandwidth.

Published
2016
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6. Heteroclinic structure of parametric resonance in the nonlinear Schrödinger equation

Author

Andrea Armaroli, Arnaud Mussot, S. Rota Nodari, Alexandre Kudlinski, Matteo Conforti, S. De Bièvre, Stefano Trillo, Guillaume Dujardin, Laboratoire de Physique des Lasers, Atomes et Molécules - UMR 8523 ( PhLAM ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), Laboratoire Paul Painlevé - UMR 8524 ( LPP ), Quantitative methods for stochastic models in physics ( MEPHYSTO ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ) -Université de Lille-Centre National de la Recherche Scientifique ( CNRS ) -Université Libre de Bruxelles [Bruxelles] ( ULB ) -Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Fonctions Optiques pour les Technologies de l'informatiON ( FOTON ), Université de Rennes 1 ( UR1 ), Université de Rennes ( UNIV-RENNES ) -Université de Rennes ( UNIV-RENNES ) -Université européenne de Bretagne ( UEB ) -Institut National des Sciences Appliquées - Rennes ( INSA Rennes ) -École Nationale Supérieure des Sciences Appliquées et de Technologie ( ENSSAT ) -Centre National de la Recherche Scientifique ( CNRS ) -Télécom Bretagne, Engineering Department [Ferrara], University of Ferrara [Ferrara], IRCICA. Grant number: USR 3380 CNRS, PRIN 2012BFNWZ2, ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions ( 2011 ), ANR-11-EQPX-0017/11-EQPX-0017,FLUX,Fibres optiques pour les hauts Flux ( 2011 ), ANR-14-ACHN-0014,NoAWE,Dynamique non-linéaire d’ondes extrêmes ( 2014 ), ANR-13-JS04-0004,TOPWAVE,Fibres optiques topographiques : nouvelles perspectives en optique non-linéaire guidée ( 2013 ), ANR-12-JS09-0005,FOPAFE,Amplificateur paramétrique fibré d'impulsions femtosecondes à haute énergie ( 2012 ), Laboratoire de Physique des Lasers, Atomes et Molécules - UMR 8523 (PhLAM), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), Laboratoire Paul Painlevé (LPP), Quantitative methods for stochastic models in physics (MEPHYSTO), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université libre de Bruxelles (ULB), Fonctions Optiques pour les Technologies de l'informatiON (FOTON), Université de Rennes (UR)-Université européenne de Bretagne - European University of Brittany (UEB)-Institut National des Sciences Appliquées - Rennes (INSA Rennes), Institut National des Sciences Appliquées (INSA)-Institut National des Sciences Appliquées (INSA)-École Nationale Supérieure des Sciences Appliquées et de Technologie (ENSSAT)-Télécom Bretagne-Centre National de la Recherche Scientifique (CNRS), Università degli Studi di Ferrara = University of Ferrara (UniFE), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), ANR-11-EQPX-0017,FLUX,Fibres optiques pour les hauts Flux(2011), ANR-14-ACHN-0014,NoAWE,Dynamique non-linéaire d'ondes extrêmes(2014), ANR-13-JS04-0004,TOPWAVE,Fibres optiques topographiques : nouvelles perspectives en optique non-linéaire guidée(2013), ANR-12-JS09-0005,FOPAFE,Amplificateur paramétrique fibré d'impulsions femtosecondes à haute énergie(2012), Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université libre de Bruxelles (ULB)-Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Université de Rennes 1 (UR1), Université de Rennes (UNIV-RENNES)-Université de Rennes (UNIV-RENNES)-Université européenne de Bretagne - European University of Brittany (UEB)-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 Nationale Supérieure des Sciences Appliquées et de Technologie (ENSSAT)-Télécom Bretagne-Centre National de la Recherche Scientifique (CNRS), and Università degli Studi di Ferrara (UniFE)

Subjects
optical fibers, Breather, breathers, General Physics and Astronomy, 01 natural sciences, Instability, NO, 010305 fluids & plasmas, symbols.namesake, [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], 0103 physical sciences, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010306 general physics, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Parametric statistics, Physics, [PHYS.PHYS.PHYS-OPTICS]Physics [physics]/Physics [physics]/Optics [physics.optics], PACS numbers: 42.65.Ky, 42.65.Yj,42.65.Re, 42.81.Dp, [ PHYS.PHYS.PHYS-OPTICS ] Physics [physics]/Physics [physics]/Optics [physics.optics], Nonlinear Sciences - Exactly Solvable and Integrable Systems, modulational instability, Mathematical analysis, parametric resonance, nonlinearity, parametric resonance, modulational instability, nonlinearity, optical fibers, breathers, Nonlinear Sciences - Pattern Formation and Solitons, Numerical integration, Modulational instability, Nonlinear system, symbols, Parametric oscillator, Physics - Optics
Abstract

International audience; We show that the nonlinear stage of modulational instability induced by parametric driving in the defocusing nonlinear Schrödinger equation can be accurately described by combining mode truncation and averaging methods, valid in the strong driving regime. The resulting integrable oscillator reveals a complex hidden heteroclinic structure of the instability. A remarkable consequence, validated by the numerical integration of the original model, is the existence of breather solutions separating different Fermi-Pasta-Ulam recurrent regimes. Our theory also shows that optimal parametric amplification unexpectedly occurs outside the bandwidth of the resonance (or Arnold tongues) arising from the linearized Floquet analysis.

Published
2016
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7. Orbital stability via the energy-momentum method: the case of higher dimensional symmetry groups

Author

Simona Rota Nodari, Stephan De Bièvre, Laboratoire Paul Painlevé - UMR 8524 ( LPP ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ), Quantitative methods for stochastic models in physics ( MEPHYSTO ), Université de Lille-Centre National de la Recherche Scientifique ( CNRS ) -Université de Lille-Centre National de la Recherche Scientifique ( CNRS ) -Université Libre de Bruxelles [Bruxelles] ( ULB ) -Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ), Institut de Mathématiques de Bourgogne [Dijon] ( IMB ), Université de Bourgogne ( UB ) -Centre National de la Recherche Scientifique ( CNRS ), FEDER (PIA-LABEX-CEMPI 42527), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions ( 2011 ), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Méthodes quantitatives pour les modèles aléatoires de la physique (MEPHYSTO-POST), Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Institut de Mathématiques de Bourgogne [Dijon] (IMB), Université de Bourgogne (UB)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Centre National de la Recherche Scientifique (CNRS), ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille, Centre National de la Recherche Scientifique (CNRS)-Université de Franche-Comté (UFC), and Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université Bourgogne Franche-Comté [COMUE] (UBFC)-Université de Bourgogne (UB)

Subjects
Lyapunov function, hamiltonian relative equilibria, Dynamical systems theory, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], [ MATH.MATH-DS ] Mathematics [math]/Dynamical Systems [math.DS], Banach space, Mathematics::Analysis of PDEs, Energy–momentum relation, [ MATH.MATH-SG ] Mathematics [math]/Symplectic Geometry [math.SG], Dynamical Systems (math.DS), Symmetry group, 01 natural sciences, symbols.namesake, Mathematics - Analysis of PDEs, Mathematics (miscellaneous), [ MATH.MATH-AP ] Mathematics [math]/Analysis of PDEs [math.AP], FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Mathematics - Dynamical Systems, 0101 mathematics, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical physics, Physics, Mechanical Engineering, 010102 general mathematics, persistence, standing waves, states, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], 010101 applied mathematics, Nonlinear system, Mathematics - Symplectic Geometry, instabilities, solitary waves, symbols, Symplectic Geometry (math.SG), systems, Hamiltonian (quantum mechanics), Analysis, Schrödinger's cat, Analysis of PDEs (math.AP)
Abstract

International audience; We consider the orbital stability of relative equilibria of Hamiltonian dynamical systems on Banach spaces, in the presence of a multi-dimensional invariance group for the dynamics. We prove a persistence result for such relative equilibria, present a generalization of the Vakhitov-Kolokolov slope condition to this higher dimensional setting, and show how it allows to prove the local coercivity of the Lyapunov function, which in turn implies orbital stability. The method is applied to study the orbital stability of relative equilibria of nonlinear Schrödinger and Manakov equations. We provide a comparison of our approach to the one by Grillakis-Shatah-Strauss.

Published
2016
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8. Uniqueness and non-degeneracy for a nuclear nonlinear Schrödinger equation

Author

Mathieu Lewin, Simona Rota Nodari, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS), European Project: 258023,EC:FP7:ERC,ERC-2010-StG_20091028,MNIQS(2010), and Laboratoire Paul Painlevé - UMR 8524 (LPP)

Subjects
Mathematics::Analysis of PDEs, 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], Uniqueness, 0101 mathematics, 010306 general physics, Nonlinear Schrödinger equation, Nonlinear Sciences::Pattern Formation and Solitons, Mathematical Physics, Mathematical physics, Mathematics, Applied Mathematics, 010102 general mathematics, Mathematical analysis, Sigma, Nonlinear system, Dirac equation, symbols, Nucleon, Degeneracy (mathematics), Analysis, Schrödinger's cat
Abstract

We prove the uniqueness and non-degeneracy of positive solutions to a cubic nonlinear Schrodinger (NLS) type equation that describes nucleons. The main difficulty stems from the fact that the mass depends on the solution itself. As an application, we construct solutions to the \({\sigma}\)–\({\omega}\) model, which consists of one Dirac equation coupled to two Klein–Gordon equations (one focusing and one defocusing).

Published
2015

9. Orbital stability: analysis meets geometry

Author

Stephan De Bièvre, Simona Rota Nodari, François Genoud, Quantitative methods for stochastic models in physics (MEPHYSTO), Laboratoire Paul Painlevé (LPP), Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Inria Lille - Nord Europe, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Université libre de Bruxelles (ULB), Université de Lille-Centre National de la Recherche Scientifique (CNRS), Faculty of Mathematics [Vienna], University of Vienna [Vienna], This work was supported in part by the Labex CEMPI (ANR-11-LABX-0007-01). F.G. thanks CEMPI and the Lab. Paul Painlevé for their hospitality during his one-month visit to the Université Lille 1 in September 2013. He also acknowledges the support of the ERC Advanced Grant 'Nonlinear studies of water flows with vorticity'. The authors are grateful to V. Combet, A. De Laire, S. Keraani, G. Rivi`ere, B. Tumpach and G. Tuynman for stimulating discussions on the subject matter of these notes., ANR-11-LABX-0007,CEMPI,Centre Européen pour les Mathématiques, la Physique et leurs Interactions(2011), European Project: 267116,EC:FP7:ERC,ERC-2010-AdG_20100224,NWFV(2011), Laboratoire Paul Painlevé - UMR 8524 (LPP), Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Centre National de la Recherche Scientifique (CNRS)-Université de Lille-Inria Lille - Nord Europe, and Centre National de la Recherche Scientifique (CNRS)-Université de Lille

Subjects
Hamiltonian mechanics, Dynamical systems theory, 010102 general mathematics, [MATH.MATH-DS]Mathematics [math]/Dynamical Systems [math.DS], Banach space, Geometry, Wave equation, 01 natural sciences, [MATH.MATH-SG]Mathematics [math]/Symplectic Geometry [math.SG], symbols.namesake, Classical mechanics, 0103 physical sciences, Manakov system, symbols, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], 010307 mathematical physics, 0101 mathematics, Hamiltonian (quantum mechanics), Nonlinear Schrödinger equation, Mathematics, Symplectic geometry
Abstract

International audience; We present an introduction to the orbital stability of relative equilibria of Hamiltonian dynamical systems on (finite and infinite dimensional) Banach spaces. A convenient formulation of the theory of Hamiltonian dynamics with symmetry and the corresponding momentum maps is proposed that allows us to highlight the interplay between (symplectic) geometry and (functional) analysis in the proofs of orbital stability of relative equilibria via the so-called energy-momentum method. The theory is illustrated with examples from finite dimensional systems, as well as from Hamiltonian PDE's, such as solitons, standing and plane waves for the nonlinear Schrödinger equation, for the wave equation, and for the Manakov system.

Published
2015
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10. Renormalized Energy Equidistribution and Local Charge Balance in 2D Coulomb Systems

Author

Simona Rota Nodari, Sylvia Serfaty, Analyse, Géométrie et Modélisation (AGM - UMR 8088), Centre National de la Recherche Scientifique (CNRS)-CY Cergy Paris Université (CY), Laboratoire Jacques-Louis Lions (LJLL), and Université Pierre et Marie Curie - Paris 6 (UPMC)-Université Paris Diderot - Paris 7 (UPD7)-Centre National de la Recherche Scientifique (CNRS)

Subjects
General Mathematics, Modulo, Probability (math.PR), Mathematical analysis, FOS: Physical sciences, Mathematical Physics (math-ph), Plasma, [MATH.MATH-FA]Mathematics [math]/Functional Analysis [math.FA], Functional Analysis (math.FA), Mathematics - Functional Analysis, [MATH.MATH-PR]Mathematics [math]/Probability [math.PR], Equidistributed sequence, symbols.namesake, [MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph], FOS: Mathematics, Coulomb, symbols, Boundary value problem, Minification, [MATH]Mathematics [math], Hamiltonian (quantum mechanics), Mathematical Physics, Mathematics - Probability, ComputingMilieux_MISCELLANEOUS, Mathematics
Abstract

We consider two related problems: the first is the minimization of the "Coulomb renormalized energy" of Sandier-Serfaty, which corresponds to the total Coulomb interaction of point charges in a uniform neutralizing back- ground (or rather variants of it). The second corresponds to the minimization of the Hamiltonian of a two-dimensional "Coulomb gas" or "one-component plasma", a system of n point charges with Coulomb pair interaction, in a con- fining potential (minimizers of this energy also correspond to "weighted Fekete sets"). In both cases we investigate the microscopic structure of minimizers, i.e. at the scale corresponding to the interparticle distance. We show that in any large enough microscopic set, the value of the energy and the number of points are "rigid" and completely determined by the macroscopic density of points. In other words, points and energy are "equidistributed" in space (modulo appropriate scalings). The number of points in a ball is in particular known up to an error proportional to the radius of the ball. We also prove a result on the maximal and minimal distances between points. Our approach involves fully exploiting the minimality by reducing to minimization problems with fixed boundary conditions posed on smaller subsets.

Published
2014
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11. Perturbation Method for Particle-like Solutions of the Einstein-Dirac-Maxwell Equations

Author

Simona Rota Nodari, CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Dipartimento di Matematica 'Federico Enriques', and Università degli Studi di Milano [Milano] (UNIMI)

Subjects
Coupling constant, Condensed Matter::Quantum Gases, Spinor, 010308 nuclear & particles physics, Particle-like solutions, perturbation method, Dirac (software), General Medicine, Fermion, 01 natural sciences, Einstein-Dirac-Maxwell equations, 010305 fluids & plasmas, symbols.namesake, Mathematics - Analysis of PDEs, Maxwell's equations, Dirac equation, 0103 physical sciences, symbols, FOS: Mathematics, [MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP], Circular symmetry, Einstein, Mathematical physics, Mathematics, Analysis of PDEs (math.AP)
Abstract

International audience; The aim of this Note is to prove by a perturbation method the existence of solutions of the coupled Einstein-Dirac-Maxwell equations for a static, spherically symmetric system of two fermions in a singlet spinor state and with the electromagnetic coupling constant $\left(\frac{e}{m}\right)^2

Published
2010
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