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

We study the Cauchy problem for the integrable nonlocal modified Korteweg-de Vries (mKdV) equation with a step-like initial data: q(x, 0) = q(0)(x), where q(0)(x) = o(1) as x -> -infinity and q(0)(x) = A + o(1), as x -> infinity, A > 0. We construct the solution of the nonlocal mKdV equation via the solution of a 2 x 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. (c) 2021 Elsevier Ltd. All rights reserved.

Automated summary

The study focuses on the Cauchy problem for the integrable nonlocal modified KdV equation, constructing the solution through solving a 2 x 2 matrix Riemann-Hilbert problem in the complex plane. Additionally, an explicit form of the one-soliton solution is presented in terms of the Riemann-Hilbert problem.