Threshold solutions in the focusing 3D cubic NLS equation outside a strictly convex obstacle

TL;DR

This paper investigates the behavior of solutions to the focusing 3D cubic nonlinear Schrödinger equation outside a convex obstacle at the critical threshold, proving all such solutions are globally defined and scatter, unlike in the Euclidean case.

Contribution

It extends the understanding of threshold solutions from Euclidean space to exterior domains, showing the non-existence of heteroclinic orbits and establishing scattering for all threshold solutions.

Findings

No heteroclinic orbits exist in exterior domains.

All solutions at the threshold are globally defined.

Solutions scatter, contrasting with Euclidean space results.

Abstract

We study the dynamics of the focusing $3d$ cubic nonlinear Schr\"odinger equation in the exterior of a strictly convex obstacle at the mass-energy threshold, namely, when $ E_{\Omega}[u_0] M_{\Omega}[u_0] = E_{\R^3}[Q] M_{\R^3}[Q] $ and $ \left\| \nabla u_0 \right\|_{L^{2}(\Omega)} \left\|u_0\right\|_{L^{2}(\Omega)}< \left\| \nabla Q \right\|_{L^2(\R^3)} \left\| Q \right\|_{L^2(\R^3)} ,$ where $u_0 \in H^1_0(\Omega)$ is the initial data, $Q$ is the ground state on the Euclidean space, $E$ is the energy and $M$ is the mass. In the whole Euclidean space Duyckaerts and Roudenko (following the work of Duyckaerts and Merle on the energy-critical problem) have proved the existence of a specific global solution that scatters for negative times and converges to the soliton in positive times. We prove that these heteroclinic orbits do not exist for the problem in the exterior domain and that all solutions at the threshold are globally defined and scatter. The main difficulty is the control of the space translation parameter, since the Galilean transformation is not available.