Dynamics of Abundant Wave Solutions to the Fractional Chiral Nonlinear Schrodinger’s Equation: Phase Portraits, Variational Principle and Hamiltonian, Chaotic Behavior, Bifurcation and Sensitivity Analysis - Yu Tian; Kang-Hua Yan; Shao-Hui Wang; Kang-Jia Wang; Chang Liu (Axioms)
The central objective of this study is to develop some different wave solutions and perform a qualitative analysis on the nonlinear dynamics of the time-fractional chiral nonlinear Schrodinger’s equation (NLSE) in the conformable sense. Combined with the semi-inverse method (SIM) and traveling wave transformation, we establish the variational principle (VP). Based on this, the corresponding Hamiltonian is constructed. Adopting the Galilean transformation, the planar dynamical system is derived. Then, the phase portraits are plotted and the bifurcation analysis is presented to expound the existence conditions of the various wave solutions with the different shapes. Furthermore, the chaotic phenomenon is probed and sensitivity analysis is given in detail. Finally, two powerful tools, namely the variational method (VM) which stems from the VP and Ritz method, as well as the Hamiltonian-based method (HBM) that is based on the energy conservation theory, are adopted to find the abundant wave solutions, which are the bell-shape soliton (bright soliton), W-shape soliton (double-bright solitons or double bell-shaped soliton) and periodic wave solutions. The shapes of the attained new diverse wave solutions are simulated graphically, and the impact of the fractional order <semantics>δ</semantics> on the behaviors of the extracted wave solutions are also elaborated. To the authors’ knowledge, the findings of this research have not been reported elsewhere and can enable us to gain a profound understanding of the dynamics characteristics of the investigative equation.