Qualitative analysis and wave propagation for a class of nonlinear partial differential equation

Abstract

This study presents a qualitative analysis of a class of nonlinear partial differential equations, including notable examples such as the Davey–Stewartson equation, the generalized Zakharov equation, and the nonlinear Schrödinger equation. Using a specific transformation, we reformulate this class as a one-dimensional Hamiltonian system. By applying the qualitative theory of planar dynamical systems, we construct phase portraits and provide detailed descriptions. Through bifurcation analysis of the system parameters, we identify novel solutions, including periodic, solitary, and kink (or anti-kink) solutions. The validity of these solutions is confirmed by examining the degeneration of phase plane orbit families into limiting orbits. Additionally, we graphically illustrate some of the obtained solutions.