On the asymptotic stability of $N$-soliton solutions of the modified nonlinear Schr\"{o}dinger equation

TL;DR

This paper analyzes the long-time behavior of solutions to the modified nonlinear Schrödinger equation with finite density initial data, confirming the soliton resolution conjecture and deriving precise asymptotics using the $ar{ ext{∂}}$ steepest descent method.

Contribution

It provides the first rigorous verification of the soliton resolution conjecture for the mNLS equation with finite density data and derives detailed asymptotics in the soliton region.

Findings

Asymptotic behavior characterized by finite soliton solutions.

Verification of the soliton resolution conjecture for mNLS.

Residual error term of order $ ext{O}(t^{-3/4})$ in the asymptotics.

Abstract

The Cauchy problem of the modified nonlinear Schr\"{o}dinger (mNLS) equation with the finite density type initial data is investigated via $\overline{\partial}$ steepest descent method. In the soliton region of space-time $x/t\in(5,7)$, the long-time asymptotic behavior of the mNLS equation is derived for large times. Furthermore, for general initial data in a non-vanishing background, the soliton resolution conjecture for the mNLS equation is verified, which means that the asymptotic expansion of the solution can be characterized by finite number of soliton solutions as the time $t$ tends to infinity, and a residual error $\mathcal {O}(t^{-3/4})$ is provided.