1+1 Wave Maps into Symmetric Spaces

Abstract

We explain how to apply techniques from integrable systems to construct $2k$-soliton homoclinic wave maps from the periodic Minkowski space $S^1\times R^1$ to a compact Lie group, and more generally to a compact symmetric space. We give a correspondence between solutions of the -1 flow equation associated to a compact Lie group $G$ and wave maps into $G$. We use B\acklund transformations to construct explicit $2k$-soliton breather solutions for the -1 flow equation and show that the corresponding wave maps are periodic and homoclinic. The compact symmetric space $G/K$ can be embedded as a totally geodesic submanifold of $G$ via the Cartan embedding. We prescribe the constraint condition for the -1 flow equation associated to $G$ which insures that the corresponding wave map into $G$ actually lies in $G/K$. For example, when $G/K=SU(2)/SO(2)=S^2$, the constrained -1-flow equation associated to SU(2) has the sine-Gordon equation (SGE) as a subequation and classical breather solutions of the SGE are 2-soliton breathers. Thus our result generalizes the result of Shatah and Strauss that a classical breather solution of the SGE gives rise to a periodic homoclinic wave map to $S^2$. When the group $G$ is non-compact, the bi-invariant metric on $G$ is pseudo-Riemannian and B\acklund transformations of a smooth solution often are singular. We use B\acklund transformations to show that there exist smooth initial data with constant boundary conditions and finite energy such that the Cauchy problem for wave maps from $R^{1,1}$ to the pseudo-Riemannian manifold $SL(2,R)$ develops singularities in finite time.