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 \[ (1)\quad ihu_ t=-(h^ 2/2m)u_{xx}+Vu-a| u|^ 2\cdot u \] 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 \(h\in (0,h_ 0)\) 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


35Q99 Partial differential equations of mathematical physics and other areas of application
35K55 Nonlinear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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