Wave solitons of hyper-elliptic function in anisotropic Heisenberg spin chain

Abstract

There are various nonlinear solutions in the anisotropic Heisenberg spin chain model (AHSCM), such as soliton solutions. In consideration of high-order nonlinear terms, a good modified nonlinear analytical solution can be obtained under reasonable simplification conditions. The purpose of this paper is to find the nonlinear solutions other than soliton of AHSCM. We use Holstein-Primakoff representation to study the AHSCM. Under the semi-classical approximation, considering the high order nonlinear term and the periodic boundary condition, an improved nonlinear Schrodinger equation and its wave solutions of the hyper-elliptic function expressed by the combination of the inverse function of Jacobi elliptic function are obtained through using the coherent state. These solutions can be expressed by the combination of the inverse functions of the first kind of elliptic functions. In the limit case, these solutions are reduced to wave solutions of sinusoidal (or cosine) functions, or wave solutions that can be represented by hyperbolic tangent functions. The energy levels of these nonlinear solutions can be obtained theoretically by the normalized conditions, but even by using hyper-elliptic functions, it is difficult to express them as analytic expressions.