Numerical simulations of the energy- supercritical Nonlinear Schr\"odinger equation

TL;DR

This paper uses numerical simulations to study the energy supercritical nonlinear Schrödinger equation, demonstrating that solutions with certain initial conditions remain bounded and scatter over long times.

Contribution

It provides the first numerical evidence that solutions of the energy supercritical NLS in five dimensions can remain bounded and scatter, supporting theoretical conjectures.

Findings

Solutions with specific initial conditions stay bounded in the $H^2$ norm.

Such solutions exist globally and scatter.

Numerical results align with theoretical predictions for supercritical equations.

Abstract

We present numerical simulations of the defocusing nonlinear Schrodinger (NLS) equation with an energy supercritical nonlinearity. These computations were motivated by recent works of Kenig-Merle and Kilip-Visan who considered some energy supercritical wave equations and proved that if the solution is {a priori} bounded in the critical Sobolev space (i.e. the space whose homogeneous norm is invariant under the scaling leaving the equation invariant), then it exists for all time and scatters. In this paper, we numerically investigate the boundedness of the $H^2$-critical Sobolev norm for solutions of the NLS equation in dimension five with quintic nonlinearity. We find that for a class of initial conditions, this norm remains bounded, the solution exists for long time, and scatters.