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

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.