On the long-time asymptotics of the modified Camassa–Holm equation with step-like initial data

Abstract We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data. \begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*} where $c_+$ and $c_-$ are two positive constants. It is shown that the solution of the step-like initial problem can be characterised via the solution of a matrix Riemann–Hilbert (RH) problem in the new scale $(y,t)$ . A double coordinate $(\xi, c)$ with $c=c_+/c_-$ is adopted to divide the half-plane $\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$ into four asymptotic regions. Further applying the Deift–Zhou steepest descent method, we derive the long-time asymptotic expansions of the solution $u(y,t)$ in different space-time regions with appropriate g-functions. The corresponding leading asymptotic approximations are given with the slow/fast decay step-like background wave in genus-0 regions and elliptic waves in genus-2 regions. The second term of the asymptotics is characterised by the Airy function or parabolic cylinder model. Their residual error order is $\mathcal{O}(t^{-2})$ or $\mathcal{O}(t^{-1})$ , respectively.