Lie Symmetry Analysis, Exact Traveling Wave Solutions by \(\left(\frac{G'}{G}\right)\)-Expansion Method and Qualitative Analysis of Hirota-Schrödinger Equation - Kaur, Arshdeep; Saini, Shalu; Budhiraja, Rajeev (Communications in Mathematics and Applications, 2025)
This manuscript is based on the Hirota's equation with Kerr law nonlinearity, specifically contemplated to be a kind of nonlinear Schrödinger equation. A systematic investigation of Lie symmetry method is pertained to derive the symmetry reduction of the given equation. By utilizing these symmetry reductions, the nonlinear partial differential equation is altered into the ordinary differential equations. By utilised the \(\big(\frac{G'}{G}\big)\)-expansion technique to gain exact solutions of the Hirota-Schrödinger equation. Novel solitary wave solutions were successfully extracted, characterized by hyperbolic, rational, and trigonometric function forms. Furthermore, a qualitative analysis of the reduced system is performed by converting it into an autonomous system, allowing the investigation of the stability and behavior of critical points.  Several phase portraits are presented for various parameter values to illustrate the system's dynamics.
Keywords: Hirota-Schrödinger equation; Lie symmetry analysis; \(\big(\frac{G'}{G}\big)\)-Expansion method; Exact solution; Qualitative analysis