Long-time asymptotics for a complex cubic Camassa-Holm equation

TL;DR

This paper analyzes the long-time behavior of solutions to a complex cubic Camassa-Holm equation using the ar-steepest descent method, providing detailed asymptotic expansions in different space-time regions.

Contribution

It introduces a novel application of the ar-steepest descent method to derive long-time asymptotics for the complex cubic Camassa-Holm equation.

Findings

Asymptotic solutions are characterized by solitons with specific residual errors.

Different space-time regions exhibit distinct asymptotic behaviors.

The analysis identifies stationary phase points affecting the solution's decay rate.

Abstract

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa-Holm (ccCH) equation $$m_{t}=b u_{x}+\frac{1}{2}\left[m\left(|u|^{2}-\left|u_{x}\right|^{2}\right)\right]_{x}-\frac{1}{2} m\left(u \bar{u}_{x}-u_{x} \bar{u}\right), \quad m=u-u_{x x},$$ where $b>0$ is an arbitrary positive real constant. Long-time asymptotics of the equation is obtained through the $\bar{\partial}$-steepest descent method. Firstly, based on the spectral analysis of the Lax pair and scattering matrix, the solution of the equation is able to be constructed via solving the corresponding Riemann-Hilbert (RH) problem. Then, we present different long time asymptotic expansions of the solution $u(y,t)$ in different space-time solitonic regions of $\xi=y/t$. The half-plane ${(y,t):-\infty <y< \infty, t > 0}$ is divided into four asymptotic regions: $\xi \in(-\infty,-1)$, $\xi \in (-1,0)$, $\xi \in (0,\frac{1}{8})$ and $\xi \in (\frac{1}{8},+\infty)$. When $\xi$ falls in $(-\infty,-1)\cup (\frac{1}{8},+\infty)$, no stationary phase point of the phase function $\theta(z)$ exists on the jump profile in the space-time region. In this case, corresponding asymptotic approximations can be characterized with an $N(\Lambda)$-solitons with diverse residual error order $O(t^{-1+2\varepsilon})$. There are four stationary phase points and eight stationary phase points on the jump curve as $\xi \in (-1,0)$ and $\xi \in (0,\frac{1}{8})$, respectively. The corresponding asymptotic form is accompanied by a residual error order $O(t^{-\frac{3}{4}})$.