New investigation on soliton solutions of two nonlinear PDEs in mathematical physics with a dynamical property: Bifurcation analysis

Abstract To comprehend nonlinear dynamics, one must have access to soliton solutions, which faithfully portray the actions of numerous physical systems and nonlinear equations. Notable nonlinear equations in relativistic physics, quantum field theory, nonlinear optics, dispersive wave phenomena, contemporary industrial applications, and plasma physics include the Klein–Gordon and Sharma–Tasso–Olver equations, which shed light on wave behavior and interactions. This study introduces a powerful approach to uncovering some novel soliton solutions for these equations, namely, the new generalized <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">′</m:mo> </m:mrow> </m:msup> <m:mo>/</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:math> ({G}^{^{\prime} }\left/G) -expansion method. The derived soliton solutions are articulated in terms of rational, trigonometric, and hyperbolic functions, each embodying the physical implications of the equations through meticulously specified parameters. The resulting solutions encompass several waveforms, including sharp solitons, singular periodic solitons, flat kink solitons, and singular kink solitons. The results indicate that the employed method is both robust and very effective for the analysis of nonlinear evolution equations (NLEEs). It is compatible with computer algebra systems, facilitating the generation of more generalized wave solutions. The strength and versatility of the new generalized <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>(</m:mo> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo accent="true">′</m:mo> </m:mrow> </m:msup> <m:mo>/</m:mo> <m:mi>G</m:mi> <m:mo>)</m:mo> </m:math> ({G}^{^{\prime} }\left/G) -expansion method suggest its potential for further research, particularly in exploring exact solutions for other NLEEs. The approach represents a significant expansion in the methodologies available for handling nonlinear wave equations, opening new avenues for theoretical and applied investigations in nonlinear science. Furthermore, the bifurcation analysis is carried out, which reveals the comprehension and precise representation of the dynamics of these two nonlinear partial differential equations. It offers the information required to build a comprehensive and significant phase portrait, including insights into solution behaviors, stability changes, and parameter dependencies.