Long-time asymptotics of the Sawada-Kotera equation and Kaup-Kupershmidt equation on the line

TL;DR

This paper analyzes the long-time behavior of the Sawada-Kotera and Kaup-Kupershmidt equations using Riemann-Hilbert problems and inverse scattering, confirming results with numerical simulations.

Contribution

It constructs Riemann-Hilbert problems for SK, KK, and modified SK-KK equations and applies the Deift-Zhou method to derive their asymptotics.

Findings

Asymptotic solutions match numerical simulations

Riemann-Hilbert problem formulation for SK and KK equations

Long-time behavior characterized using steepest-descent method

Abstract

Both Sawada-Kotera (SK) equation and Kaup-Kupershmidt (KK) equation are integrable systems with third-order Lax operator. Moreover, they are related with the same modified nonlinear equation (called modified SK-KK equation) by Miura transformations. This work first constructs the Riemann-Hilbert problem associated with the SK equation, KK equation and modified SK-KK equation by direct and inverse scattering transforms. Then the long-time asymptotics of these equations are studied based on Deift-Zhou steepest-descent method for Riemann-Hilbert problem. Finally, it is shown that the asymptotic solutions match very well with the results of direct numerical simulations.