Long Time Asymptotic Behavior of the Focusing Nonlinear Schrodinger Equation

TL;DR

This paper analyzes the long-time behavior of solutions to the focusing nonlinear Schrödinger equation, revealing that solutions asymptotically resemble modulated multi-solitons within specific space-time cones, using advanced mathematical techniques.

Contribution

It extends the nonlinear steepest descent method to compute precise long-time asymptotics for the focusing NLS with minimal initial data assumptions.

Findings

Asymptotic solutions are multi-solitons modulated by interactions.

Residual error in asymptotics is of order O(t^(-3/4)).

Results apply to initial data with minimal regularity and no spectral singularities.

Abstract

We study the Cauchy problem for the focusing nonlinear Schrodinger (NLS) equation. Using the DBAR generalization of the nonlinear steepest descent method we compute the long time asymptotic expansion of the solution in any fixed space-time cone x_1 + v_1 t <= x <= x_2 + v_2 t with v_1 <= v_2 up to an (optimal) residual error of order O(t^(-3/4)). In each (x,t) cone the leading order term in this expansion is a multi-soliton whose parameters are modulated by soliton-soliton and soliton-radiation interactions as one moves through the cone. Our results only require that the initial data possess one L^2(R) moment and (weak) derivative and that it not generate any spectral singularities (embedded eigenvalues).