Qualitative analysis, traveling wave solutions and chaotic behavior for the perturbed Schrödinger–Hirota equation with cubic–quintic–septic law of self-phase modulation

In this paper, the dynamical behaviors, chaotic pattern and traveling wave solutions for the perturbed Schrödinger–Hirota equation with cubic–quintic–septic law have been studied by means of the qualitative analysis of planar dynamical system method. By using this method, we can not only analyze the dynamical behavior of a given equation, but also seek the corresponding traveling wave solutions. Through the traveling wave transformation, the perturbed Schrödinger–Hirota equation can easily be reduced to two-dimensional dynamical system. By selecting the relevant parameters, phase portraits are drawn via the mathematical software Maple. Furthermore, in order to analyze the chaotic behavior of the perturbed Schrödinger–Hirota equation with perturbation term, Poincaré sections and sensitivity analysis diagrams are also drawn. Finally, we also derive the periodic wave solutions, solitary wave solutions, bell-shaped wave solutions, bright solitons, kink solitary wave solutions and Jacobian elliptic function solutions for perturbed Schrödinger–Hirota equation through the planar dynamical system method.