Physics

Nonlinear system

Invariant (physics)

Pure mathematics

Mathematics

Mathematical physics

Partial differential equation

Mathematical analysis

Trigonometric functions

Lie group

Geometry

Quantum mechanics

Differential equation

Conservation law

Symmetry group

First-order partial differential equation

Journal of Taibah University for Science

In this study, we consider the third order nonlinear Schrödinger equation (TONSE) that models the wave pulse transmission in a time period less than one-trillionth of a second. With the help of the extended modified method, we obtain numerous exact travelling wave solutions containing sets of generalized hyperbolic, trigonometric and rational solutions that are more general than classical ones. Secondly, we construct the transformation groups which left the equations invariant and vector fields with the Lie symmetry groups approach. With the help of these vector fields, we obtain the symmetry reductions and exact solutions of the equation. The obtained group-invariant solutions are Jacobi elliptic function and exponential type. We discuss the dynamic behaviour and structure of the exact solutions for distinct solutions of arbitrary constants. Lastly, we obtain conservation laws of the considered equation by construing the complex equation as a system of two real partial differential equations (PDEs).

A third-order nonlinear Schrödinger equation: the exact solutions, group-invariant solutions and conservation laws