## 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

### MSC:

 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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### References:

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