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

The Riemann-Hilbert (RH) problem is developed to study the general N -soliton solutions of the nonlocal modified Korteweg-de Vries (mKdV) equation in this work. For the initial value belonging to Schwarz space, we firstly obtain the corresponding eigenfunctions and scattering data in the direct scattering process. Then the suitable RH problem of the nonlocal mKdV equation is successfully established. The exact expression of the solution for the equation is derived via solving the RH problem. Utilizing the symmetry of scattering data, the abundant phenomena corresponding to different eigenvalues are analyzed, including bounded solutions, singular solutions, degenerate solution and kink solutions. Finally, the propagation behaviors of the solution (including one-, two-, and three soliton) are observed, and the characteristic lines are further used to analyze the continuity and other phenomena of the solution. Mathematics Subject Classification (2010) 35Q15 · 35Q51 · 45Q05