Gauge transformations and symmetries of integrable systems

Details Gauge transformations and symmetries of integrable systems Gauge transformations and symmetries of integrable systems

2007-05-24

arxiv.org/abs/0705.3530v2

J.Phys.A40:12227-12242,2007

Physics

High Energy Physics

High Energy Physics - Theory

18pages, no figure Journal version

Scientific paper

10.1088/1751-8113/40/40/014

We analyze several integrable systems in zero-curvature form within the framework of $SL(2,\R)$ invariant gauge theory. In the Drienfeld-Sokolov gauge we derive a two-parameter family of nonlinear evolution equations which as special cases include the Kortweg-de Vries (KdV) and Harry Dym equations. We find residual gauge transformations which lead to infinintesimal symmetries of this family of equations. For KdV and Harry Dym equations we find an infinite hierarchy of such symmetry transformations, and we investigate their relation with local conservation laws, constants of the motion and the bi-Hamiltonian structure of the equations. Applying successive gauge transformatinos of Miura type we obtain a sequence of gauge equivalent integrable systems, among them the modified KdV and Calogero KdV equations.

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