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Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential. (English) Zbl 0613.35076

The authors consider the problem of existence of stationary solutions to a one-dimensional nonlinear Schrödinger equation of the form

$\left(1\right)\phantom{\rule{1.em}{0ex}}ih{u}_{t}=-\left({h}^{2}/2m\right){u}_{xx}+Vu-a{|u|}^{2}·u$

with a bounded smooth potential V, $a>0$, h small. They show that for every critical point of V there is an ${h}_{0}$ such that for all $h\in \left(0,{h}_{0}\right)$ one has a nontrivial stationary solution to (1) which lies in a neighborhood of a translate around the critical point ${x}_{0}$ of the ground state solution ${u}_{0}$ of the free equation (that is equation (1) without potential); this means that the solution is concentrated near ${x}_{0}$ and in some sense nonspreading. The method the authors use in their proof is inspired by procedures to find instanton solutions for the Yang-Mills equation, namely a Lyapunov-Schmidt reduction to finite dimensions combined with a crucial argument using the nondegeneracy of the ground state ${u}_{0}$ (i.e. the kernel of the Frechet derivative at ${u}_{0}$ of the nonlinear operator belonging to (1) should be spanned by the components of grad ${u}_{0}\right)$.

Reviewer: H.Lange

##### MSC:
 35Q99 PDE of mathematical physics and other areas 35K55 Nonlinear parabolic equations 35B40 Asymptotic behavior of solutions of PDE