The Cauchy problem for Hartree-Fock time-dependent equations
The Cauchy problem for the coupled Klein-Gordon-Schrödinger system
We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural...
The Cauchy problem for the Gross–Pitaevskii equation
The Cauchy problem for the Schrödinger equation in dimension three with concentrated nonlinearity
The cubic Szegő equation
We consider the following Hamiltonian equation on the Hardy space on the circle,where is the Szegő projector. This equation can be seen as a toy model for totally non dispersive evolution equations. We display a Lax pair structure for this equation. We prove that it admits an infinite sequence of conservation laws in involution, and that it can be approximated by a sequence of finite dimensional completely integrable Hamiltonian systems. We establish several instability phenomena illustrating...
The effect of the graph topology on the existence of multipeak solutions for nonlinear Schrödinger equations.
The existence of global solutions to the elliptic-hyperbolic Davey-Stewartson system with small initial data
The Ginzburg-Landau equations of superconductivity and the one-phase Stefan problem
The Hirota method and soliton solutions to the multidimensional nonlinear Schrödinger equation.
The nonrelativistic limit of the nonlinear Dirac equation
The scattering problem for a noncommutative nonlinear Schrödinger equation.
The spaces of quantum states for some harmonic oscillator potentials on the punctured plane C∖{0}
We consider equivariant solutions of Schrödinger equations on C∖{0} with harmonic oscillator potentials. We determine the spaces of equivariant quantum states in three cases: for an isotropic and anisotropic harmonic oscillator potential centered at 0, and for a potential not centered at 0.
The trajectory-coherent approximation and the system of moments for the Hartree type equation.
The WKB method and geometric instability for nonlinear Schrödinger equations on surfaces
In this paper we are interested in constructing WKB approximations for the nonlinear cubic Schrödinger equation on a Riemannian surface which has a stable geodesic. These approximate solutions will lead to some instability properties of the equation.
Theoretical and numerical aspects of stochastic nonlinear Schrödinger equations
We describe several results obtained recently on stochastic nonlinear Schrödinger equations. We show that under suitable smoothness assumptions on the noise, the nonlinear Schrödinger perturbed by an additive or multiplicative noise is well posed under similar assumptions on the nonlinear term as in the deterministic theory. Then, we restrict our attention to the case of a focusing nonlinearity with critical or supercritical exponent. If the noise is additive, smooth in space and non degenerate,...
Time averaging for the strongly confined nonlinear Schrödinger equation
Transverse evolution operator for the Gross-Pitaevskii equation in semiclassical approximation.
Transverse nonlinear instability for two-dimensional dispersive models
Traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity: some results and open problems
This text is a survey of recent results on traveling waves for nonlinear Schrödinger equations with nonzero conditions at infinity. We present the existence, nonexistence and stability results and we describe the main ideas used in proofs.
Travelling wave solutions for the KdV-Burgers-Kuramoto and nonlinear Schrödinger equations which describe pseudospherical surfaces.