Interaction with an obstacle in the 2d focusing nonlinear Schr\"odinger equation

TL;DR

This paper numerically investigates how a convex obstacle affects the behavior of solutions to the 2D focusing nonlinear Schrödinger equation, including blow-up phenomena, solution splitting, and the influence of obstacle size.

Contribution

It introduces the concepts of weak and strong interactions with obstacles and analyzes their impact on solution dynamics, including blow-up thresholds and new blow-up initial data.

Findings

Obstacle influences solution behavior, causing splitting into transmitted and reflected parts.

The sharp blow-up threshold matches the Euclidean case, despite the obstacle presence.

New initial data leading to finite-time blow-up after strong obstacle interaction.

Abstract

We present a numerical study of solutions to the $2d$ cubic and quintic focusing nonlinear Schr\"odinger equation in the exterior of a smooth, compact and strictly convex obstacle (a disk) with Dirichlet boundary condition. We first investigate the effect of the obstacle on the behavior of solutions traveling toward the obstacle at different angles and with different velocities directions. We introduce a new concept of weak and strong interactions of the solutions with the obstacle. Next, we study the existence of blow-up solutions depending on the type of the interaction and show how the presence of the obstacle changes the overall behavior of solutions (e.g., from blow-up to global existence), especially in the strong interaction case, as well as how it affects the shape of solutions compared to their initial data, (e.g., splitting into transmitted and reflected parts). We also investigate the influence of the size of the obstacle on the eventual existence of blow-up solutions in the strong interaction case in terms of the transmitted and the reflected parts of the mass. Moreover, we show that the sharp threshold for global existence vs. finite time blow-up solutions in the mass critical case in the presence of the obstacle is the same as the one given by Weinstein for {\rm{NLS}} in the whole Euclidean space $\R^d$. Finally, we construct new Wall-type initial data that blows up in finite time after a strong interaction with an obstacle and having a very distinct dynamics compared with all other blow-up scenarios and dynamics for the {\rm{NLS}} in the whole Euclidean space $\R^d$.