Soliton, stability, multistability, and diverse tools for identifying chaos in a nonlinear model with two modified methods

This research employs two analytical techniques, the modified Kudrynshov and the modified alternative <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfenced close=")" open="("> <m:mrow> <m:mfrac> <m:mrow> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mo>′</m:mo> </m:mrow> </m:msup> </m:mrow> <m:mrow> <m:mi>G</m:mi> </m:mrow> </m:mfrac> </m:mrow> </m:mfenced> </m:math> (GG) -expansion method, to investigate the soliton solutions of the well-known Zoomeron (Z) model, which arises in plasma physics, nonlinear optics, and fluid dynamics. This yields various soliton outcomes with distinct dynamic patterns, including bright solitons, localized waves, singular breather waves, kink and anti-kink patterns, and multi-breather waveforms. We attach three-dimensional, density, and two-dimensional curves to highlight the visual dynamics pattern of the outcomes. After that, we investigate equilibrium points in different scenarios to verify their stability. We also present phase portraits and various chaos assessment tools, such as return maps, Lyapunov exponents, strange attractors, and multistability, to confirm the presence of chaotic patterns in the proposed model. The results of this research will have significant implications for the future of advanced non-linear phenomena.