The Dynamics on Soliton Molecules and Soliton Bifurcation for an Extended Generalization of Vakhnenko Equation - Ma, Yu-Lan; Li, Bang-Qing (Qualitative theory of dynamical systems, 2024)
Vakhnenko-type equations play a critical role in nonlinear electromagnetic and optical fiber applications. In this article, we present a new advancement in high-frequency wave propagation in electromagnetic and optical fiber applications by investigating an extended generalization of the Vakhnenko equation. In our study, we employ the bilinear method and introduce an auxiliary function that combines exponential and cosine functions. By utilizing this approach, we are able to derive two distinct sets of analytical solutions for the equation. By choosing suitable parameters involved in the solutions, we identify the presence of soliton molecules and soliton bifurcation in the equation. Furthermore, for the soliton molecules, we note the existence of two distinct types of phase transitions: (i) a transition from soliton molecules to loop-like breathers, and (ii) a transition from soliton molecules to intersection solitons. Regarding soliton bifurcation, we observe phase transitions among loop-like, cusp-like, and peak-like solitons. Moreover, the results of this study illuminate that the structures of all solitons, within the context of soliton molecules and soliton bifurcation, remain stable throughout the phase transitions. This stability carries considerable significance for practical applications.