Long-time asymptotic behaviour for the fifth order modified Korteweg-de Vries equation

TL;DR

This paper analyzes the long-time behavior of solutions to the fifth order modified Korteweg-de Vries equation using advanced mathematical techniques, providing detailed asymptotic descriptions based on inverse scattering and Riemann-Hilbert problems.

Contribution

It applies the Deift-Zhou nonlinear steepest descent method to derive long-time asymptotics for the 5th order MKdV, extending previous methods to higher-order equations.

Findings

Asymptotic formulas expressed via parabolic cylinder functions

Transformation of the inverse scattering problem into a Riemann-Hilbert problem

Reduction of phase function degree to facilitate analysis

Abstract

Following Deift-Zhou's nonlinear steepest descent method, the long-time asymptotic behavior for the Cauchy problem of the 5th order modified Korteweg-de Vries equation is analyzed. Based on the inverse scattering transform, the 5th order MKdV is transformed to a 2 by 2 oscillatory Riemann-Hilbert problem, then by manipulating the Cauchy operator and reducing the degree of the phase function, the long-time asymptotics of the solution is given in terms of solutions of the parabolic cylinder equation.