N-soliton solutions and their dynamic analysis to the generalized complex mKdV equation

Abstract

A new generalized complex modified Korteweg–de Vries (mKdV) equation is studied by using Riemann-Hilbert approach. Firstly, we derive a Lax pair associated with a 3 × 3 matrix spectral problem for the generalized complex mKdV equation. Then, we can formulate the Riemann-Hilbert problem via the spectral analysis of the $x$-part of the Lax pair. According to the symmetry properties of the potential matrix, we find two cases of zero structures for the Riemann-Hilbert problem. By solving the particular Riemann-Hilbert problem and using the inverse scattering transformation, we obtain the unified formulas of the $N$-soliton solutions for the generalized complex mKdV equation. In addition, the dynamical behaviors of the single-soliton solution and the two-soliton solution are analyzed by choosing appropriate parameters.