Bifurcation and Traveling Wave Solutions for a Generalized Nonlinear Schrödinger Equation

This paper considers special traveling wave solutions for a generalized nonlinear Schrödinger equation. Through a combined transformation of both dependent and independent variables, the equation is reduced to a planar dynamic system. By employing dynamical systems theory and a real root discrimination method for cubic polynomials, the bifurcation of this planar system is thoroughly analyzed. The analysis reveals that the system exhibits 12 distinct phase portrait structures. Subsequently, via direct integration, parametric representations for typical bounded traveling waves are derived, including periodic, homoclinic, and heteroclinic solutions. Finally, some typical solutions are plotted to illustrate their characteristics.