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We consider semiclassically scaled, weakly nonlinear Schrödinger equations with external confining potentials and additional angular-momentum rotation term. This type of model arises in the Gross–Pitaevskii theory of trapped, rotating quantum gases. We construct asymptotic solutions in the form of semiclassical wave packets, which are concentrated in both space and in frequency around an classical Hamiltonian phase-space flow. The rotation term is thereby seen to alter this flow, but not the corresponding classical action. [ABSTRACT FROM AUTHOR]

We prove the existence of a class of orbitally stable bound state solutions to nonlinear Schrödinger equations with super-quadratic confinement in two and three spatial dimensions. These solutions are given by time-dependent rotations of a non-radially symmetric spatial profile which in itself is obtained via a doubly constrained energy minimization. One of the two constraints imposed is the total mass, while the other is given by the expectation value of the angular momentum around the z-axis. Our approach also allows for a new description of the set of minimizers subject to only a single mass constraint. [ABSTRACT FROM AUTHOR]

We study the cubic-quartic nonlinear Schrödinger equation (NLS) in two and three spatial dimension. This equation arises in the mean-field description of Bose-Einstein condensates with Lee-Huang-Yang correction. We first prove global existence of solutions in natural energy spaces which allow for the description of self-bound quantum droplets with vorticity. Existence of such droplets, described as central vortex states in 2D and 3D, is then proved using an approach via constrained energy minimizers. In 2D we also obtain a sharp lower bound on their mass. Even though these vortex states are known to be unstable in general, we prove that they are indeed orbitally stable (as a set) under the flow associated to the NLS with repulsive inverse-square potential in 2D. [ABSTRACT FROM AUTHOR]

We consider the mean-field dynamics of Bose–Einstein condensates in rotating harmonic traps and establish several stability and instability properties for the corresponding solution. We particularly emphasize the difference between the situation in which the trap is symmetric with respect to the rotation axis and the one where this is not the case. [ABSTRACT FROM AUTHOR]

We revisit the time-adiabatic theorem of quantum mechanics and show that it can be extended to weakly nonlinear situations, that is to nonlinear Schrödinger equations in which either the nonlinear coupling constant or, equivalently, the solution is asymptotically small. To this end, a notion of criticality is introduced at which the linear bound states stay adiabatically stable, but nonlinear effects start to show up at leading order in the form of a slowly varying nonlinear phase modulation. In addition, we prove that in the same regime a class of nonlinear bound states also stays adiabatically stable, at least in terms of spectral projections. [ABSTRACT FROM AUTHOR]

The present work is devoted to the study of dynamical features of Bohmian measures, recently introduced by the authors. We rigorously prove that for sufficiently smooth wave functions the corresponding Bohmian measure furnishes a distributional solution of a nonlinear Vlasov-type equation. Moreover, we study the associated defect measures appearing in the classical limit. In one space dimension, this yields a new connection between mono-kinetic Wigner and Bohmian measures. In addition, we shall study the dynamics of Bohmian measures associated to so-called semi-classical wave packets. For these type of wave functions, we prove local in-measure convergence of a rescaled sequence of Bohmian trajectories towards the classical Hamiltonian flow on phase space. Finally, we construct an example of wave functions whose limiting Bohmian measure is not mono-kinetic but nevertheless equals the associated Wigner measure. [ABSTRACT FROM AUTHOR]

We consider the non-relativistic Hartree model in the gravitational case, i.e. with attractive Coulomb-Newton interaction. For a given mass M > 0, we construct stationary states with non-zero temperature T by minimizing the corresponding free energy functional. It is proved that minimizers exist if and only if the temperature of the system is below a certain threshold T* > 0 (possibly infinite), which itself depends on the specific choice of the entropy functional. We also investigate whether the corresponding minimizers are mixed or pure quantum states and characterize a critical temperature $${T_c \in (0, T*)}$$ above which mixed states appear. [ABSTRACT FROM AUTHOR]

As a simple model for piezoelectricity we consider a gas of infinitely many non-interacting electrons subject to a slowly time-dependent periodic potential. We show that in the adiabatic limit the macroscopic current is determined by the geometry of the Bloch bundle. As a consequence we obtain the King-Smith and Vanderbilt formula up to errors smaller than any power of the adiabatic parameter. [ABSTRACT FROM AUTHOR]

The aim of this paper is the accurate numerical study of the Kadomtsev–Petviashvili (KP) equation. In particular, we are concerned with the small dispersion limit of this model, where no comprehensive analytical description exists so far. To this end, we first study a similar highly oscillatory regime for asymptotically small solutions, which can be described via the Davey–Stewartson system. In a second step, we investigate numerically the small dispersion limit of the KP model in the case of large amplitudes. Similarities and differences to the much better studied Korteweg–de Vries situation are discussed as well as the dependence of the limit on the additional transverse coordinate. [ABSTRACT FROM AUTHOR]

We consider a generalized DiracFock type evolution equation deduced from nophoton Quantum Electrodynamics, which describes the selfconsistent timeevolution of relativistic electrons, the observable ones as well as those filling up the Dirac sea. This equation has been originally introduced by Dirac in 1934 in a simplified form. Since we work in a Hartree-Fock type approximation, the elements describing the physical state of the electrons are infinite rank projectors. Using the Bogoliubov-Dirac-Fock formalism, introduced by ChaixIracane (J. Phys. B., 22, 37913814, 1989), and recently established by Hainzl-Lewin-Séré, we prove the existence of globalintime solutions of the considered evolution equation. [ABSTRACT FROM AUTHOR]

We study the simultaneous semi-classical and adiabatic asymptotics for a class of (weakly) nonlinear Schrödinger equations with a fast periodic potential and a slowly varying confinement potential. A rigorous two-scale WKB–analysis, locally in time, is performed. The main nonlinear phenomenon is a modification of the Berry phase. [ABSTRACT FROM AUTHOR]