Dynamic study of bifurcation, chaotic behavior and multi-soliton profiles for the system of shallow water wave equations with their stability

Abstract

The purpose of this study is to investigate the deeper characteristics of the system of shallow water wave equations that is used to model the turbulence in the atmosphere and oceans from different standpoints. The multi-soliton structures such as 1-soliton, 2-soliton, 3-soliton solutions are successfully generated with the help of simplified Hirota’s method. For the sake of physical demonstration and visual presentation, we graphically illustrate the identified solutions in 3D, 2D, and contour plots using Mathematica software. The Lie symmetry technique is used to identify the Lie group invariant transformations and symmetry reductions of the examined system, which aid in reducing the dimension by one or into an ordinary differential equation. Further, the qualitative behavior of the time-dependent dynamical system is observed using the bifurcation and chaos theory. The phase portraits of bifurcation are observed at the equilibrium point of a planar dynamical system. We discuss various tools to identify chaos (random and unpredictable behavior) in autonomous dynamic systems, such as 3D phase portraits, 2D phase portraits, time series, and Poincaré maps. At various initial conditions, the sensitivity and modulation instability analyses are also presented, and it is discovered that the investigated system is stable, as a small change in the initial conditions does not cause an abrupt change in solutions. The findings of this investigation will contribute in the overall depiction of soliton theory and nonlinear dynamical systems.