Transient asymptotics of the modified Camassa-Holm equation

TL;DR

This paper analyzes the long-term behavior of solutions to the modified Camassa-Holm equation across three transition zones, revealing Painlevé-type and Jacobi theta function asymptotics through advanced Riemann-Hilbert problem techniques.

Contribution

It provides the first detailed asymptotic descriptions in multiple transition zones for the modified Camassa-Holm equation under low regularity conditions.

Findings

Painlevé-type asymptotics in the first two transition regions

Jacobi theta function asymptotics in the collisionless shock region

Application of $ar{ ext{D}}$ nonlinear steepest descent method

Abstract

We investigate long time asymptotics of the modified Camassa-Holm equation in three transition zones under a nonzero background. The first transition zone lies between the soliton region and the first oscillatory region, the second one lies between the second oscillatory region and the fast decay region, and possibly, the third one, namely, the collisionless shock region, that bridges the first transition region and the first oscillatory region. Under a low regularity condition on the initial data, we obtain Painlev\'e-type asymptotic formulas in the first two transition regions, while the transient asymptotics in the third region involves the Jacobi theta function. We establish our results by performing a $\bar{\partial}$ nonlinear steepest descent analysis to the associated Riemann-Hilbert problem.