Examples of G-Strand Equations

The G-strand equations for a map R×R into a Lie group G are associated to a G-invariant Lagrangian. The Lie group manifold is also the configuration space for the Lagrangian. The G-strand itself is the map g(t,s):R×R→G, where t and s are the independent variables of the G-strand equations. The Euler-Poincare´ reduction of the variational principle leads to a formulation where the dependent variables of the G-strand equations take values in the corresponding Lie algebra g and its co-algebra, g∗ with respect to the pairing provided by the variational derivatives of the Lagrangian. We review examples of G-strand constructions, including matrix Lie groups of low ranks, and the Diffeomorphism group. In some cases the arising G-strand equations are completely integrable 1+1 Hamiltonian systems that admit soliton solutions.Our presentation is aimed to illustrate the G-strand construction with several simple but instructive examples: (i) SO(3)-strand integrable equations for Lax operators, quadratic in the spectral parameter; (ii) Diff(R)-strand equations. These equations are in general non-integrable; however they admit solutions in 2+1 space-time with singular support (e.g., peakons). The one- and two-peakon equations obtained from the Diff(R)-strand equations can be solved analytically, and potentially they can be applied in the theory of image registration. Our example is with a system which is a 2+1 generalization of the Hunter-Saxton equation.