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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

(1)ihu t =-(h 2 /2m)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(0,h 0 ) 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 ).

Reviewer: H.Lange

MSC:
35Q99PDE of mathematical physics and other areas
35K55Nonlinear parabolic equations
35B40Asymptotic behavior of solutions of PDE