Equivariant and self-similar standing waves for a Hamiltonian hyperbolic-hyperbolic spin-field system

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 ${\mathrm{\S }}^2$ or the pseudo-sphere \H^2. This model was introduced in \cite{Martina} and it is also known as the {\it hyperbolic-hyperbolic generalized Ishimori system}. Relying on the hyperbolic coordinates introduced in \cite{KNZ11}, we prove the existence of equivariant standing waves in both regular hyperbolic coordinates as well as similarity variables, and describe their asymptotic behaviour.