Construction of multi-solitary waves solution to the focusing nonlinear Schr\"odinger equation outside an obstacle in the $L^2$-subcritical case

Abstract

We consider the focusing $L^2$-subcritical Schr\"odinger equation in the exterior of a smooth, compact, strictly convex obstacle $\Theta \subset \mathbb{R}^d$. We construct a solution that, for large times, behaves asymptotically as a finite sum of solitary waves on $\mathbb{R}^d$, each traveling with sufficiently large and distinct velocities, and satisfying Dirichlet boundary conditions. The construction is achieved via a compactness argument similar to that introduced by F.Merle in 1990 for constructing solutions of the NLS equation that blow up at several points, combined with modulation theory, the coercivity property of the linearized operator, and localized energy estimates.