Multiplicity and stability of standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential

In this paper, we study the multiplicity, stability, and quantitative properties of normalized standing waves for the nonlinear Schrödinger‐Poisson equation with a harmonic potential.     

Asymptotic stability of ground states in 3D nonlinear Schrödinger equation including subcritical cases

We consider a class of nonlinear Schrödinger equation in three space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in L²) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that “shadows” the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations.     

Sharp upper and lower bounds on the blow-up rate for nonlinear Schrödinger equation with potential

We consider the blow-up solutions of the Cauchy problem for critical nonlinear Schrödinger equation with a repulsive harmonic potential. In terms of Merle and Raphaël’s recent arguments as well as Carles’ transform, the sharp upper and lower bounds of the blow-up rate are obtained.     

Infinitely many solutions for a resonant sublinear Schrödinger equation

In this paper, we consider a nonlinear sublinear Schrödinger equation at resonance in . By using bounded domain approximation technique, we prove that the problem has infinitely many solutions. Copyright © 2013 John Wiley & Sons, Ltd.     

An inverse scattering problem for the Schrödinger equation in a semiclassical process

We study a direct and an inverse scattering problem for a pair of Hamiltonians (H(h),H₀(h)) on , where H₀(h)=−h²Δ and H(h)=H₀(h)+V, V is a short-range potential and h is the semiclassical parameter. First, we show that if two potentials are equal in the classical allowed region for a fixed non-trapping energy, the associated scattering matrices coincide up to O(h^∞) in . Then, for potentials with a regular behaviour at infinity, we study the inverse scattering problem. We show that in dimension n3, the knowledge of the scattering operators S(h), , up to O(h^∞) in , and which are localized near a fixed energy λ>0, determine the potential V at infinity.     

Extended Lagrangian approach for the defocusing nonlinear Schrödinger equation

We study the defocusing nonlinear Schrödinger (NLS) equation written in hydrodynamic form through the Madelung transform. From the mathematical point of view, the hydrodynamic form can be seen as the Euler–Lagrange equations for a Lagrangian submitted to a differential constraint corresponding to the mass conservation law. The dispersive nature of the NLS equation poses some major numerical challenges. The idea is to introduce a two‐parameter family of extended Lagrangians, depending on a greater number of variables, whose Euler–Lagrange equations are hyperbolic and accurately approximate NLS equation in a certain limit. The corresponding hyperbolic equations are studied and solved numerically using Godunov‐type methods. Comparison of exact and asymptotic solutions to the one‐dimensional cubic NLS equation (“gray” solitons and dispersive shocks) and the corresponding numerical solutions to the extended system was performed. A very good accuracy of such a hyperbolic approximation was observed.     

Multi solitary waves for nonlinear Schrödinger equations

We consider the nonlinear Schrödinger equation in for any d1, with a nonlinearity such that solitary waves exist and are stable. Let R_k(t,x) be K arbitrarily given solitary waves of the equation with different speeds v₁,v₂,…,v_K. In this paper, we prove that there exists a solution u(t) of the equation such that

     

A multisymplectic integrator for the periodic nonlinear Schrödinger equation

A multisymplectic integrator for the periodic nonlinear Schrödinger equation is presented in this paper. Its accuracy is proved. By introducing a norm, we investigate its nonlinear stability. We also discuss the relationship between this multisymplectic integrator and two variational integrators which are derived by using the discrete multisymplectic field theory and the finite element method.     

Increasing stability for the inverse problem for the Schrödinger equation

In this article, we study the increasing stability property for the determination of the potential in the Schrödinger equation from partial data. We shall assume that the inaccessible part of the boundary is flat, and homogeneous boundary condition is prescribed on this part. In contrast to earlier works, we are able to deal with the case when potentials have some Sobolev regularity and also need not be compactly supported inside the domain.     

Asymptotics for the third‐order nonlinear Schrödinger equation in the critical case

We consider the Cauchy problem for the third‐order nonlinear Schrödinger equation where and is the Fourier transform. Our purpose in this paper is to prove the large time asymptoitic behavior of solutions for the defocusing case λ > 0 with a logarithmic correction under the non zero mass condition Copyright © 2016 John Wiley & Sons, Ltd.     

On the fractional Schrödinger problem with non‐symmetric potential

We study the semilinear equation where 0 < s < 1, , V(x) is a sufficiently smooth non‐symmetric potential with , and ? > 0 is a small number. Letting U be the radial ground state of (?Δ)^sU + U ? U^p=0 in , we build solutions of the form for points ?_j,j = 1,?,m, using a Lyapunov–Schmidt variational reduction. Copyright © 2014 John Wiley & Sons, Ltd.     

Global wellposedness for 1D non-linear Schrödinger equation for data with an infinite L norm

We prove global wellposedness for the one-dimensional cubic non-linear Schrödinger equation in a space of distributions which is invariant under Galilean transformations and includes L². This space arises naturally in the study of the restriction properties of the Fourier transform to curved surfaces. The L^p bounds, p≠2, for the extension operator, dual to the restricition one, plays a fundamental role in our approach.