A unified inverse scattering transform and soliton solutions of the nonlocal modified KdV equation with non-zero boundary conditions

TL;DR

This paper develops a unified inverse scattering transform for nonlocal mKdV equations with non-zero boundary conditions, enabling explicit soliton solutions and detailed analysis of their dynamics.

Contribution

It introduces a uniformization variable and formulates a Riemann-Hilbert problem to solve the inverse scattering for nonlocal mKdV equations with NZBCs, which was previously more complex.

Findings

Explicit soliton solutions for nonlocal mKdV with NZBCs are derived.

The dynamical behaviors of solitons are analyzed for different reflectionless cases.

The method unifies the treatment of focusing and defocusing nonlocal mKdV equations.

Abstract

We present a rigorous theory of a unified and simple inverse scattering transform (IST) for both focusing and defocusing real nonlocal (reverse-space-time) modified Korteweg-de Vries (mKdV) equations with non-zero boundary conditions (NZBCs) at infinity. The IST problems for the nonlocal equations with NZBCs are more complicated then ones for the local equations with NZBCs. The suitable uniformization variable is introduced in order to make the direct and inverse problems be established on a complex plane instead of a two-sheeted Riemann surface. The direct scattering problem establishes the analyticity, symmetries, and asymptotic behaviors of Jost solutions and scattering matrix, and properties of discrete spectra. The inverse problem is formulated and solved by means of a matrix-valued Riemann-Hilbert problem. The reconstruction formula, trace formulae, and theta conditions are obtained. Finally, the dynamical behaviors of solitons for four different cases for the reflectionless potentials for both focusing and defocusing nonlocal mKdV equations with NZBCs are analyzed in detail.