Cubic-quintic nonlinear Helmholtz model: novel exact solutions, stability and phase portraits

Abstract

The nonlinear Helmholtz equation, which is a Schrödinger-type wave equation, is studied. This model accurately simulates strong fields by accounting for nonparaxial effects that are ignored in the nonlinear Schrödinger equation. We derive novel optical soliton solutions demonstrated by hyperbolic functions for the cubic-quintic nonlinear Helmholtz equation using the effective sine-Gordon equation expansion method. Various wave structures, such as dark, bright, kink, and anti-kink optical solitons, are obtained. Moreover, the equation is reduced to a planar dynamical system to investigate the stability and phase portraits. Because of the singularity in the system, a topological classification using the Separatrix method is presented. To enhance the understanding of the behaviour of these solutions, their physıcal features are illustrated using two- and three-dimensional graphs. Furthermore, the behaviour of the equilibrium points, phase portraits, and the physical features of the studied equation are presented graphically.