Inverse scattering transform and dynamics of soliton solutions for nonlocal focusing modified Korteweg-de Vries equation

TL;DR

This paper derives explicit multi-soliton solutions for the nonlocal mKdV equation using the Riemann-Hilbert method, analyzing their dynamics, symmetry properties, and solution behaviors such as boundedness, singularity, and kink formation.

Contribution

It introduces a novel Riemann-Hilbert approach to obtain and analyze multi-soliton solutions for the nonlocal mKdV equation, revealing new dynamic behaviors and solution phenomena.

Findings

Explicit N-soliton solutions derived

Analysis of solution behaviors including bounded, singular, and kink solutions

Observation of new dynamic behaviors via characteristic line rotation

Abstract

In this work, we mainly study the general $N$-soliton solutions of the nonlocal modified Korteweg-de Vries (mKdV) equation by utilizing the Riemann-Hilbert (RH) method. For the initial value belonging to Schwarz space, we firstly obtain the corresponding eigenfunctions and scattering data in the direct scattering process. Then we successfully establish a suitable RH problem of the nonlocal mKdV equation. The exact expression of the solution for the equation is derived via solving the RH problem. Using the symmetry of scattering data, the phenomena corresponding to different eigenvalues are analyzed, including bounded solutions, singular solutions, position solutions and kink solutions. Finally, the propagation path of the solution is observed, and the characteristic line is further used to analyze the continuity or other phenomena of the solution. The new dynamic behavior of the solution is observed by rotating the characteristic line at a certain angle.