Construction of solitary wave solution of the nonlinear focusing schr\"odinger equation outside a strictly convex obstacle for the $L^2$-supercritical case

TL;DR

This paper constructs solitary wave solutions for the focusing L^2-supercritical Schrödinger equation outside a convex obstacle, extending known results to arbitrary velocities using advanced analytical techniques.

Contribution

It introduces a novel method combining compactness and topological arguments to construct solitary waves with arbitrary velocities outside convex obstacles.

Findings

Existence of solitary wave solutions with arbitrary nonzero velocity.

Solutions are close to the scattering threshold established in previous work.

Construction applies to the exterior of strictly convex obstacles in R^3.

Abstract

We consider the focusing $L^2$-supercritical Schr\"odinger equation in the exterior of a smooth, compact, strictly convex obstacle. We construct a solution behaving asymptotically as a solitary waves on $R^3$, as large time. When the velocity of the solitary wave is high, the existence of such a solution can be proved by a classical fixed point argument. To construct solutions with arbitrary nonzero velocity, we use a compactness argument similar to the one that was introduced by F.Merle in 1990 to construct solution of NLS blowing up at several blow-up point together with a topological argument using Brouwer's theorem to control the unstable direction of the linearized operator at soliton. These solutions are arbitrarily close to the scattering threshold given by a previous work of R.Killip, M.Visan and X.Zhang which is the same as the one on whole Euclidean space.