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

Abstract

We study the long time asymptotic behavior for the Cauchy problem of the modified Camassa-Holm (mCH) equation with step-like initial valvewhere A 1 , A 2 are two positive constant. Our main technical tool is the representation of the Cauchy problem with an associated matrix Riemann-Hilbert (RH) problem and the consequent asymptotic analysis of this RH problem.Based on the spectral analysis of the Lax pair associated with the mCH equation and scattering matrix, the solution of the Cauchy problem is characterized via the solution of a RH problem in the new scale (y, t). Further using the Deift-Zhou steepest descent method, we derive different long time asymptotic expansion of the solution u(y, t) in different space-time solitonic regions of ξ = y/t and the different choice of the initial value. We divide the half-plane {(y, t) : −∞ < y < ∞, t > 0} into three asymptotic regions by the different choice of g-function. The corresponding asymptotic approximations can be characterized with the plane wave in genus-0 Region I and II, and hyperelliptic

Transient asymptotics of the modified Camassa–Holm equation

“…More precisely, these regions are • a soliton region: {(𝑥, 𝑡) ∶ 𝜉 > 2}, • a fast decay region: {(𝑥, 𝑡) ∶ 𝜉 < −1∕4}, • two oscillatory regions: {(𝑥, 𝑡) ∶ 0 < 𝜉 < 2} ∪ {(𝑥, 𝑡) ∶ −1∕4 < 𝜉 < 0}, where 𝜉 ∶= 𝑥∕𝑡; see Figure 1 for an illustration. Under the condition that the initial value belongs to the Schwartz space, that is, 𝑚(𝑥, 0) is sufficiently smooth and is rapidly decaying as |𝑥| → ∞, large time asymptotics of the mCH equation in the regular asymptotic zones are presented in [7]; we also refer to [35,48] for the studies with step-like boundary conditions.…”