Invariant Solutions and Conservation Laws of the Variable-Coefficient Heisenberg Ferromagnetic Spin Chain Equation

Na Liu

doi:10.3389/fphy.2020.00260

Abstract

The variable-coefficient Heisenberg ferromagnetic spin chain (vcHFSC) equation is investigated using the Lie group method. The infinitesimal generators and Lie point symmetries are reported. Four types of similarity reductions are acquired by virtue of the optimal system of one-dimensional subalgebras. Several invariant solutions are derived, including trigonometric and hyperbolic function solutions. Furthermore, conservation laws for the vcHFSC equation are obtained with the help of Lagrangian and nonlinear self-adjointness.

Highlights

The investigation of physical phenomenon modeled by non-linear partial differential equations (NLPDEs) and searching for their underlying dynamics remain the hot issue of research for applied and theoretical sciences
We study the variable-coefficient Heisenberg ferromagnetic spin chain (vcHFSC) equation (1) via the Lie group method and obtain new invariant solutions, including the trigonometric and hyperbolic function solutions
The Lie group method has been successfully used to establish the invariant solutions for the vcHFSC equation

Summary

Introduction

The investigation of physical phenomenon modeled by non-linear partial differential equations (NLPDEs) and searching for their underlying dynamics remain the hot issue of research for applied and theoretical sciences. A great many powerful methods have been proposed to construct the explicit solutions of NLPDEs, such as the inverse scattering method [1], the Lie group method [2,3,4,5], the Hirota bilinear method [6, 7], the extended tanh method [8,9,10], the homoclinc test method [11,12,13], the F-expansion technique [14], and so on [15,16,17,18]. Invariant solutions of a class of constant and variable coefficient NLPDEs have been obtained by virtue of this method, such as Keller-Segel models [19], generalized fifth-order non-linear integrable equation [20], KdV equation [21], and Davey-Stewartson equation [22]

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Results

Conclusion