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

Abstract

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.

Time-fractional (2+1)-dimensional navier-stokes equations: similarity reduction and exact solutions for one-parameter lie group of rotations

“…It is easy to verify that transformation T δ forms an infinite dimensional Lie Group of transformation with the identity element T 0 . Further, expanding the Lie Group of transformation (9) with the help of the Taylor series in δ about the δ = 0 with the initial conditions…”

“…admitted by A i are indexed within the table 1. These subgroups are obtained with the help of the Lie Group of Transformation (9). It is worth mentioning here that these subgroups can modulate the complex physical phenomena generated by the two-dimensional time-fractional Navier-Stokes equation.…”

“…As we know Lie symmetry analysis [1][2][3][4][5][6][7][8][9][10][11][12][13] is a powerful and flexible tool for developing algorithms for analyzing nonlinear differential equations. R E Boisvert et al [14] applied this powerful tool to the classical integer order Navier-Stokes equations.…”