Long time asymptotics for the nonlocal mKdV equation with finite density initial data

Abstract

We consider the Cauchy problem for an integrable real nonlocal (reverse space–time) mKdV equation with nonzero boundary conditions qt(x,t)−6σq(x,t)q(−x,−t)qx(x,t)+qxxx(x,t)=0,q(x,0)=q0(x),limx→±∞q0(x)=q±, where |q±|=1 and q+=δq−, σδ=−1. Based on the spectral analysis of the Lax pair, we express the solution of the Cauchy problem of the nonlocal mKdV equation in terms of a Riemann–Hilbert problem. In a fixed space–time solitonic region −6<x/t<6, we apply ∂̄-steepest descent method to analyze the long-time asymptotic behavior of the solution q(x,t). We find that the long time asymptotic behavior of q(x,t) can be characterized with an N(Λ)-soliton on discrete spectrum and leading order term O(t−1/2) on continuous spectrum up to a residual error order O(t−1).