Equivariante and Self-similar Standing Waves for a Hamiltonian Hyperbolic-hyperbolic Spin-field System

Abstract

In this paper we study the existence of special symmetric solutions to a Hamiltonian hyperbolic-hyperbolic coupled spin-field system, where the spins are maps from $\mathbb R^{2+1}$ into the sphere ${\mathbb S}^2$ or the pseudo sphere ${\mathbb H}^2$. This model was introduced by Martina et al. in [Phys. Rev. B, 49 (1994), pp. 12915--12922] from the hyperbolic-hyperbolic generalized Ishimori systems. Relying on the hyperbolic coordinates introduced in [P. Kevrekidis, A. Nahmod, and C. Zeng, Nonlinearity, 24 (2011), pp. 1523--1538], we prove the existence of equivariant standing waves both in regular hyperbolic coordinates and in similarity variables and describe their asymptotic behavior.