Dynamics of the semi-discrete Gardner equation under two types of non-vanishing boundary conditions: heteropolar solitons and kinks

Abstract

In this work, we will use inverse scattering transform to study the semi-discrete Gardner equation under two types of non-vanishing boundary conditions, and investigate two interesting nonlinear waves in the presence of discrete spectrum, namely heteropolar solitons and kinks. When $u_n\rightarrow -\frac{a}{2b}$ as $n\rightarrow \pm \infty$, this is a symmetric boundary condition, for which the heteropolar solitons, i.e., two kinds of single soliton solutions with different polarities will be obtained. If considering two sets of discrete eigenvalues, there will be two types of soliton collisions, head-on and overtaking collision, depending on the position of discrete spectrum. Interestingly, the energy gathered at the moment of collision with different polarities, producing the so-called rogue wave phenomenon with a large amplitude more than twice the background, and its generation mechanism is briefly analyzed. When $u_n\rightarrow \frac{c_{\pm}\sqrt{ a^2+4b }-a}{2b}$ as $n\rightarrow \pm \infty$, the kink, i.e., the undercompressive dispersive shock wave, will be obtained under the specific step-like boundary condition.