The Anisotropic Heisenberg spin chain and the nonlinear Schrödinger equation

Abstract

It is shown that the solutions of the continuous Anisotropic Heisenberg Spin Chain (AHSC) can be obtained from the linear integral equation which was proposed in a previous paper for the solutions of the Isotropic Heisenberg Spin Chain (IHSC) and the Nonlinear Schrödinger equation (NLS). An explicit expression is obtained for the Miura transformation which maps the solutions of the AHSC on solutions of the NLS. In the second part of the paper we investigate the similarity solutions of these partial differential equations which leads to ordinary differential equations of Painlevé type. As an application we discuss some new solutions of Painlevé IV.