Bifurcation Analysis and Dynamical Solitary Wave Solutions of the Lonngren‐Wave Equation With Fractional Derivative

ABSTRACT The study of soliton theory is crucial for understanding nonlinear phenomena in nonlinear science and engineering. This work analyzes bifurcation analysis and soliton solutions analytically and numerically. The study investigates a beta‐fractional Lonngren‐wave model to understand nonlinear electrical signal behavior in telegraph lines containing tunnel diodes. Initially, a new technique known as bifurcation analysis is applied to study critical points or phase portraits, where systems transition to novel behaviors, such as stability shifts or the emergence of chaos. Then, the chaotic nature of the dynamical system is analyzed by adding a trigonometric perturbation term. The shockwave, quasi‐periodic wave, super‐periodic wave, and sensitivity analysis are illustrated through 3D and 2D phase portraits with initial conditions. Finally, the modified F ‐expansion technique is also employed to analytically obtain soliton solutions of the beta fractional Lonngren‐wave model. The solutions are presented using transcendental functions under specific conditions, and their physical interpretation, including bright bell, dark bell shape soliton solutions, bright and dark periodic soliton solutions, periodic lump wave solution, and linked lump wave solutions for several free parameters. It also demonstrates the impact of beta fractional parameters on these solutions. Bifurcation and chaotic nature analysis, and soliton solutions through the modified F ‐expansion technique are explored in this work for the first time. The findings provide insight into the physical characteristics of waves propagating in a dispersive medium, potentially enhancing our understanding of these phenomena.