Parametric effects on paraxial nonlinear Schrödinger equation in Kerr media

•The mentioned techniques are applied to the paraxial NLS equation in Kerr media.•Optical soliton solutions are obtained and has parameters connected to the model.•Analyzed the nature of the obtained solutions by 3D and 2D wave profiles.•Nonlinear and free parameters effects on the paraxial NLS equation have been analyzed.•Wave profiles streamline pattern and immediate local directions have been analyzed.