The modified Camassa-Holm equation on a nonzero background: large-time asymptotics for the Cauchy problem

Abstract

This paper deals with the Cauchy problem for the modified Camassa-Holm (mCH) equation \begin{alignat*}{4} &m_t+\left((u^2-u_x^2)m\right)_x=0,&\quad&m:= u-u_{xx},&\quad&t>0,&\;&-\infty<x<+\infty,\\ &u(x,0)=u_0(x),&&&&&&-\infty<x<+\infty, \end{alignat*} in the case when the initial data $u_0(x)$ as well as the solution $u(x,t)$ are assumed to approach a nonzero constant as $x\to\pm\infty$. In a recent paper we developed the Riemann--Hilbert formalism for this problem, which allowed us to represent the solution of the Cauchy problem in terms of the solution of an associated Riemann--Hilbert factorization problem. In this paper, we apply the nonlinear steepest descent method, based on this Riemann--Hilbert formalism, to study the large-time asymptotics of the solution of this Cauchy problem. We present the results of the asymptotic analysis in the solitonless case for the two sectors $\frac{3}{4}<\frac{x}{t}<1$ and $1<\frac{x}{t}<3$ (in the $(x,t)$ half-plane, $t>0$), where the leading asymptotic term of the deviation of the solution from the background is nontrivial: this term is given by modulated (with parameters depending on $\frac{x}{t}$), decaying (as $t^{-1/2}$) trigonometric oscillations.