Stability Analysis of the Rational Solutions, Periodic Cross-Rational Solutions, Rational Kink Cross-Solutions, and Homoclinic Breather Solutions to the KdV Dynamical Equation with Constant Coefficients and Their Applications

Abstract

We explore various analytical rational solutions with symbolic computation using the ansatz transformation functions. We gain a variety of rational solutions such as M-shaped rational solutions (MSRs), periodic cross-rationals (PCRs), multi-wave solutions, rational kink cross-solutions (RKCs), and homoclinic breather solutions (HBs), and by using the appropriate values for the relevant parameters, their dynamics are visualized in figures. Additionally, two different types of interactions between MSRs and kink waves are analyzed. Furthermore, we examine the stability of the obtained solutions and create a corresponding table. We analyze the stability of these solutions and the movement role of the wave by making graphs as two-dimensional, three-dimensional and density graphs as well as contour visual and stream plots.

Various nonlinear characteristics of breather/rogue waves and controllable interaction phenomena for a new KdV equation with variable coeffcients

“…Equation (1) can be described the ionacoustic waves in plasmas, shallow water waves in oceans, and pulse waves in large arteries [36]. If f (t), g(t), h(t), s(t) are constants, equation (1) is reduced to a (2+1)-dimensional KdV equation with constant coefficients [35], and the rational solutions, periodic cross-rational solutions, rational kink cross-solutions, M-lump solutions and homoclinic breather solutions for the KdV equation with constant coefficients are obtained [35][36][37][38][39]. But there are few researches about equation (1), some exact wave solutions, M-lump solutions and interaction phenomena of equation (1) are investigated, seen in [40,41].…”