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| The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ 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 $g(t,s): \mathbb{R}\times \mathbb{R}\to G$, where $t$ and $s$ are the independent variables of the $G$-strand equations. The Euler-Poincar\'e 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 $\mathfrak{g}$ and its co-algebra, $\mathfrak{g}^*$ 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) $SO(3)$ and $SO(4)$-strand integrable equations for Lax operators, quadratic in the spectral parameter;
(ii) ${\rm Diff}(\mathbb{R})$-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 ${\rm Diff}(\mathbb{R})$-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. |
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