Stationary Solutions to Nonlinear Partial Differential Equations for the Classical Continuous Isotropic Heisenberg Model

Abstract

Stationary solutions to nonlinear partial differential equations obeyed by two angles of rotation, e and cp, of spins for the classical continuous isotropic Heisenberg model are studied. By paying attention to particular solutions for which cp satisfies the Laplace equation, three types of solutions of physical interest describing nonlinear excitations in the system are studied. These are cylindrical symmetric or vortex string type solutions, axisymmetric solutions and uniform·flow solutions, for each of which an equation obeyed by e takes the form of the sine· Gordon equation having the corresponding symmetry. For the first type multi-vortex string solutions are obtained in terms of the Jacobi elliptic functions with argument given by a stream function for vortices. The second type of solutions is written in terms of the Painleve trans­ cendents of the third kind. Here solutions of another type are also obtained in close similarity to solutions to the Ernst equation for the axisymmetric gravitational field problem. The third type of solutions is given as solutions to the conventional static two-dimensional sine-Gordon equation for which multi-kink solutions exist.