Bifurcation, Stability, and Nonlinear Parametric Effects on the Solitary Wave Profile of the Riemann Wave Equation

Abstract

The enduring stability exhibited by solitons is strikingly demonstrated as a soliton pulse traverses an ideal lossless optical fibre, thereby highlighting a compelling attribute for their integration into optical communication systems. In this study, we employ the improved Bernoulli sub-equation function method to systematically derive stable and functionally robust soliton solutions for the Riemann wave equation. The stability of the soliton solutions is demonstrated through their composition involving hyperbolic and exponential functions, among others. The physical significance of these solutions is meticulously analyzed by presenting 2D and 3D graphs that illustrate the behaviour of the solutions for specific parameter values. Additionally, a comprehensive investigation into the influence of the nonlinear parameter on the wave velocity and solution curve is conducted. The study further explores local stability through bifurcation and phase plane analysis. Our findings affirm the reliability of the improved Bernoulli sub-equation function method and suggest its potential application in future endeavours to uncover diverse and novel soliton solutions for other nonlinear evolution equations encountered in the realms of mathematical physics and engineering.