Nonlinear Schrodinger equation and frequency saturation

TL;DR

This paper introduces a frequency cut-off method to stabilize solutions of the nonlinear Schrödinger equation, ensuring global well-posedness across Sobolev spaces and providing error estimates based on solution regularity.

Contribution

It presents a novel approximation approach using frequency cut-offs to prevent instability in nonlinear Schrödinger equations, achieving global well-posedness.

Findings

Global well-posedness in Sobolev spaces achieved

Error estimates depend on solution regularity

Method effectively avoids instability phenomena

Abstract

We propose an approach that permits to avoid instability phenomena for the nonlinear Schrodinger equations. We show that by approximating the solution in a suitable way, relying on a frequency cut-off, global well-posedness is obtained in any Sobolev space with nonnegative regularity. The error between the exact solution and its approximation can be measured according to the regularity of the exact solution, with different accuracy according to the cases considered.