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.     

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.     

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.     

Optimal regularity for Schrödinger equations

In this paper we study the regularity theory for the Schrödinger equations under proper conditions. Furthermore, it will be verified that these conditions are optimal.     

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.     

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.     

Analysis of finite element method for one-dimensional time-dependent Schrödinger equation on unbounded domain

This paper addresses the theoretical analysis of a fully discrete scheme for the one-dimensional time-dependent Schrödinger equation on unbounded domain. We first reduce the original problem into an initial-boundary value problem in a bounded domain by introducing a transparent boundary condition, then fully discretize this reduced problem by applying Crank–Nicolson scheme in time and linear or quadratic finite element approximation in space. By a rigorous analysis, this scheme has been proved to be unconditionally stable and convergent, its convergence order has also be obtained. Finally, two numerical examples are performed to show the accuracy of the scheme.     

A conservative difference scheme for two‐dimensional nonlinear Schrödinger equation with wave operator

A conservative difference scheme is presented for two‐dimensional nonlinear Schrödinger equation with wave operator. The discrete energy method and an useful technique are used to analyze the difference scheme. It is shown, both theoretically and numerically, that the difference solution is conservative, unconditionally stable and convergent with second order in maximum norm. A numerical experiment indicates that the scheme is very effective. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 862–876, 2016     

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.     

Minimal time for the approximate bilinear control of Schrödinger equations

We consider a quantum particle in a potential V(x) subject to a time‐dependent (and spatially homogeneous) electric field E(t) (the control). Boscain, Caponigro, Chambrion, and Sigalotti proved that, under generic assumptions on V, this system is approximately controllable on the unit sphere, in sufficiently large time T. In the present article, we show that, for a large class of initial states (dense in unit sphere), approximate controllability does not hold in arbitrarily small time. This generalizes our previous result for Gaussian initial conditions. Furthermore, we prove that the minimal time can in fact be arbitrarily large.