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

Abstract

In this paper, we investigate the Cauchy problem of the following complex cubic Camassa–Holm equation

$$\begin{aligned} m_{t}=bu_{x}+\frac{1}{2}\left[ m\left( |u|^{2}-\left| u_{x}\right| ^{2}\right) \right] _{x}-\frac{1}{2} m\left( u \bar{u}_{x}-u_{x} \bar{u}\right) , \quad m=u-u_{x x}, \end{aligned}$$

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 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 \(O(t^{-\frac{3}{4}})\).