Novel soliton solutions and phase plane analysis in nonlinear Schrödinger equations with logarithmic nonlinearities.

This paper investigates a generalized form of the nonlinear Schrödinger equation characterized by a logarithmic nonlinearity. The nonlinear Schrödinger equation, a fundamental equation in nonlinear wave theory, is applied across various physical systems including nonlinear optics, Bose-Einstein condensates, and fluid dynamics. We specifically explore a logarithmic variant of the nonlinear Schrödinger equation to model complex wave phenomena that conventional polynomial nonlinearities fail to capture. We derive four distinct forms of the nonlinear Schrödinger equation with logarithmic nonlinearity and provide exact solutions for each, encompassing bright, dark, and kink-type solitons, as well as a range of periodic solitary waves. Analytical techniques are employed to construct bounded and unbounded traveling wave solutions, and the dynamics of these solutions are analyzed through phase portraits of the associated dynamical systems. These findings extend the scope of the nonlinear Schrödinger equation to more accurately describe wave behaviors in complex media and open avenues for future research into non-standard nonlinear wave equations.