Diverse soliton solutions and dynamical analysis of the discrete coupled mKdV equation with 4×4 Lax pair

Abstract

Under consideration in this study is the discrete coupled modified Korteweg–de Vries (mKdV) equation with 4 × 4 Lax pair. Firstly, through using continuous limit technique, this discrete equation can be mapped to the coupled KdV and mKdV equations, which may depict the development of shallow water waves, the optical soliton propagation in cubic nonlinear media and the Alfven wave in a cold collision-free plasma. Secondly, the discrete generalized (r, N–r)-fold Darboux transformation is constructed and extended to solve this discrete coupled equation with the fourth-order linear spectral problem, from which diverse exact solutions including usual multi-soliton and semi-rational soliton solutions on the vanishing background, higher-order rational soliton and mixed hyperbolic-rational soliton solutions on the non-vanishing background are derived, and the limit states of some soliton and rational soliton solutions are analyzed by the asymptotic analysis technique. Finally, the numerical simulations are used to explore the dynamical behaviors of some exact soliton solutions. These results may be helpful for understanding some physical phenomena in fields of shallow water wave, optics, and plasma physics.