Hyperbolic (3+1)-Dimensional Nonlinear Schrödinger Equation: Lie Symmetry Analysis and Modulation Instability

The hyperbolic nonlinear Schrödinger equation in the (3 + 1)-dimension depicts the evolution of the elevation of the water wave surface for slowly modulated wave trains in deep water. Many researchers have studied the applicability and practicality of this model, but the analytical approach has been virtually absent from the literature. We adapted the lie symmetry analysis method to obtain a new complex solution in this work. The obtained complex solution contains bright and dark solitons. Furthermore, modulation instability is applied to this model to explain the interplay between nonlinear and dispersive effects. As a result, the modulation instability condition and the explosive rate are also discussed.