Instability modulation for the (2+1)-dimension paraxial wave equation and its new optical soliton solutions in Kerr media

Abstract In this paper, we use the modified auxiliary expansion method to seek some new solutions of the paraxial nonlinear Schrödinger equation. The solutions have a hyperbolic function, trigonometric function, exponential function, and rational function forms The linear stability analysis of paraxial NLSE is also presented to study the modulational instability (MI). Two cases when the instability modulation becomes to occur are investigated. Depending on MI cases, the MI gain spectrum are also investigated and presented graphically. All solutions are new and verified the main equation of the paraxial wave equation. Moreover, the constraint conditions for the existence of soliton solutions are also showed.