Long-Time Asymptotics for the Modified Camassa-Holm Equation with Nonzero Boundary Conditions

((u2 - u2 We consider the modified Camassa-Holm (mCH) equation mt + x)m)x = 0 with m := u - uxx on the line -infinity < x < +infinity, where u(x, t) is subject to nonzero boundary conditions at infinity: u(x, t) -> 1 as x -> +/-infinity. The paper aims at studying the long-time asymptotics of solutions of the initial value problems for this problem, using the Riemann-Hilbert for-malism recently developed in [3]. The emphasis is made on the asymptotics in two sectors of the (x, t) half-plane (t > 0), where the main asymptotic terms are given in terms of modulated, decaying (as t-1/2) trigonometric oscillations, as well as in a sector where solitons dominate the long time behavior of the solution of the initial value problem.