Cauchy matrix approach for generalized semi-discrete lattice potential Korteweg–de Vries equations

Abstract

Starting from a new scattering relation, we generate a generalized semi-discrete lattice potential Korteweg–de Vries(gsd-lpKdV) equation from a determining equation set(DES) and its Lax presentation. Further, the gsd-lpKdV equation with self-consistent sources(gsd-lpKdVSCS) is worked out by introducing two discrete variables to the plane wave factors. Specially, explicit formulae for kink solitons, Jordan-block solutions and diagonal-Jordan-block solutions of gsd-lpKdVSCS are obtained based on the Cauchy matrix approach. Moreover, We illustrate the dynamics of one-kink soliton, two-kink soliton, high order Jordan-block solutions and the diagonal-Jordan-block solution.