Dynamical analysis of diverse exact solutions, soliton surfaces and continuum limit theory for a semidiscrete Gardner equation

In this paper, our attention is given to the semidiscrete Gardner equation, which is a discrete analogue of the continuous Gardner equation describing the long wave propagation in a two-layer fluid. Firstly, a generalized (n,N-n)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(n, N-n)$$\end{document}-fold Darboux transformation (DT) for this semi-discrete Gardner equation is constructed based on its recognized Lax pair. Secondly, the resulting DT is used to obtain exact solutions including ordinary soliton, rational soliton (RS) and their hybrid solutions within the non-zero seed background, and then analyze their asymptotic states as well as physical characteristics. Numerical simulations are also carried out to exhibit the dynamic characteristics of certain exact solutions. Thirdly, the soliton surface corresponding to this semi-discrete equation is investigated. Finally, employing continuum limit theory, we map the semi-discrete equation to the continuous equation, and obtain corresponding continuum limit for its Lax pair and DT. The findings given in this paper are conducive to a more profound understanding of the physical properties depicted by this equation.