\(1+1\) wave maps into symmetric spaces. (English) Zbl 1082.37068

Terng, Chuu-Lian; Uhlenbeck, Karen

Commun. Anal. Geom. 12, No. 1-2, 345-388 (2004).

Using Bäcklund transformations, the authors construct \(2k\)-soliton breather solutions for the -1 flow associated to a compact Lie group. They show how these give rise to \(2k\)-soliton homoclinic wave maps from the Lorentzian surface \(S^1\times \mathbb R^1\) into homogeneous spaces (where by wave maps the authors denote harmonic maps defined on a Lorentzian manifold). This result generalizes the result by J. Shatah and W. Strauss [Physica D 99, 113–133 (1996; Zbl 0890.58009)] according to which the classical breather solutions of the sine-Gordon equation give rise to periodic homoclinic wave maps into \(S^2= \text{SU}(2)/ \text{SO}(2)\).

Reviewer: Christoph Bohle (Berlin)

MSC:

37K25	Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with topology, geometry and differential geometry
35Q51	Soliton equations
35Q53	KdV equations (Korteweg-de Vries equations)
37K35	Lie-Bäcklund and other transformations for infinite-dimensional Hamiltonian and Lagrangian systems

Keywords:

wave maps; homoclinic orbit; breather; sine-Gordon equation

Citations:

Zbl 0890.58009