Construction of a solitary wave solution of the nonlinear focusing schrödinger equation outside a strictly convex obstacle in the L^2 -supercritical case

We consider the focusing \begin{document}$ L^2 $\end{document}-supercritical Schrödinger equation in exterior of a smooth, compact, strictly convex obstacle id="M3">\begin{document}$ \Theta \subset \mathbb{R}^3 $\end{document}. We construct solution behaving asymptotically as solitary wave on id="M4">\begin{document}$ \mathbb{R}^3, $\end{document} for large times. When velocity is high, existence such can be proved by classical fixed point argument. To solutions with arbitrary nonzero velocity, we use compactness argument similar to one that was introduced F.Merle 1990 NLS blowing up at several points together topological using Brouwer's theorem control unstable direction linearized operator soliton. These are arbitrarily close scattering threshold given previous work R. Killip, M. Visan and X. Zhang, which same whole Euclidean space S. Roundenko J. Holmer radial case authors T. Duyckaerts non-radial case.