Bifurcation and Chaotic Dynamics of the Damped Nonlinear Schrödinger Equation with Soliton Solutions

Abstract

In this work, the damped nonlinear Schrödinger equation is studied within the Madelung fluid framework. Using an ansatz-based method combined with a Bernoulli-type auxiliary equation, explicit soliton solutions are derived for several nonlinear Schrödinger-type models. The dynamical behavior of the reduced system is then examined through bifurcation and analysis. It is shown that the trivial equilibrium loses stability through a supercritical pitchfork bifurcation, leading to symmetric nontrivial equilibria. Phase portraits, bifurcation diagrams, and Poincar & eacute; sections further reveal the transition to chaotic dynamics under periodic forcing. These results provide new insight into nonlinear wave propagation in dispersive media such plasmas and nonlinear optical systems.