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

TL;DR

This paper uses inverse scattering transform to analyze the semi-discrete Gardner equation under non-vanishing boundary conditions, revealing heteropolar solitons, kink solutions, and rogue wave phenomena with detailed collision dynamics.

Contribution

It introduces a novel analysis of heteropolar solitons and kink solutions in the semi-discrete Gardner equation under specific boundary conditions, including rogue wave generation mechanisms.

Findings

Heteropolar solitons with different polarities are obtained.

Collision types include head-on and overtaking collisions.

Rogue waves with amplitudes over twice the background are produced during collisions.

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.