On a Riemann–Hilbert problem for the focusing nonlocal mKdV equation with step-like initial data

Abstract

We study the Cauchy problem for the integrable nonlocal modified Korteweg–deVries (mKdV) equation with a step-like initial data: q(x,0)=q0(x), where q0(x)=o(1) as x→−∞ and q0(x)=A+o(1), as x→∞, A>0. We construct the solution of the nonlocal mKdV equation via the solution of a 2 × 2 matrix Riemann–Hilbert problem in the complex plane. Further, The explicit form of the one-soliton solution in a special case is expressed in terms of the Riemann–Hilbert problem.