Semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries equations

Abstract

The ubiquitous Korteweg–de Vries equation is widely used in oceanic, atmospheric dynamics, and fluid mechanics and can explain almost all kinds of physical phenomena of waves, while explicit solutions are studied in nonlinear mechanics and optics. We investigate explicit solutions for the semi-discrete local and nonlocal Frobenius-coupled complex modified Korteweg–de Vries (cmKdV) equations, which describe sound waves in an inharmonic lattice. A semi-discrete four-component Frobenius-coupled cmKdV equation is presented by utilizing a zero-curvature equation. We use two different reduction methods to get a semi-discrete local Frobenius-coupled cmKdV equation and a semi-discrete nonlocal Frobenius-coupled cmKdV equation. In the paper, we study the above two new types of semi-discrete Frobenius-coupled cmKdV equations. Then the bi-Hamiltonian structures of the above two types of equations are constructed, which show that the two types of equations possess Liouville integrability. The covariant properties of the both new equations are shown by constructing their corresponding Darboux transformation (DT), respectively. Soliton solutions, semi-rational soliton solutions, and rogue wave solutions of the above two types of equations are constructed by N-fold DT and generalized (n,N−n)-fold DT. Three-dimensional plots and density profiles of these explicit solutions are presented. The characteristics of these figures demonstrate the relationship between the above two types of equations.