The method of Darboux transformation matrix for solving the Landau-Lifschitz equation for a spin chain with an easy plane

Abstract

The Landau-Lifschitz equation for a spin chain with an easy plane is solved by the method of the Darboux transformation matrix. In terms of a particular parameter k, Jost solutions and Darboux matrices are generated in a recursive manner. The Jost solutions are shown to satisfy the corresponding Lax equations by a suitable choice of the constants involved in the Darboux matrices. A system of linear equations is derived and can yield the expressions for multi-soliton solutions. Asymptotic behaviour in the limits as t to +or- infinity is derived. An expression for the one-soliton solution is given in terms of elementary functions of x and t, as an example.