Asymptotic behavior for a new higher-order nonlinear Schrödinger equation

TL;DR

This work derives and analyzes a new higher-order nonlinear Schrödinger equation (NHNSE) from a Lax pair and weighted Sobolev initial data. By applying the $\bar{\partial}$-steepest descent method to a formulated Riemann-Hilbert problem, the authors obtain the long-time asymptotics for $q(x,t)$ in the regime $\xi=\frac{x}{t}>\frac{2}{3}$, with leading contributions arising from two stationary points and modelled by parabolic-cylinder functions. The main result is the asymptotic formula $q(x,t)=2i(M_1^{(lo)})_{12}+O(t^{-3/4})$, where the $t^{-1/2}$ decay stems from the local analyses near $z_1$ and $z_2$ and the amplitudes depend on the local scattering data at these points. The paper provides a novel application of the $\bar{\partial}$-steepest descent framework to a new integrable hierarchy, delivering a rigorous nonlinear late-time description for the NHNSE and laying groundwork for further extensions to similar higher-order models.

Abstract

We investigate the Cauchy problem of a new higher-order nonlinear Schrödinger equation (NHNSE) with weighted Sobolev initial data which is derived by ourselves. By applying $\bar{\partial}$-steepest descent method, we derive the long-time asymptotics of the NHNSE. Explicit steps are as follows: first of all, based on the spectral analysis of a Lax pair and scattering matrice, the solution of the NHNSE is exhibted through solving the corresponding Riemann-Hilbert problem. Secondly, by applying some properties of the Riemann-Hilbert problem, we obtain the long-time asymptotics of the solution to the NHNSE. As we know that the properties of the NHNSE presented in the paper have not been found in any scholar journals.

Figures (1)

• Figure : Figure.3.1 The classification of sign Re$(i\theta(z))$. In the white regions, Re$(i\theta(z)) < 0$, so $|e^{2it\theta(z)}| \rightarrow 0$ as $t \rightarrow \infty$. In the pink regions, Re$(i\theta(z)) > 0$, $|e^{-2it\theta(z)}| \rightarrow 0$ as $t \rightarrow \infty$.