Gauge transformations and symmetries of integrable systems

TL;DR

This paper explores gauge symmetries in integrable systems, deriving new equations and symmetries, and establishing connections between different integrable models through gauge transformations.

Contribution

It introduces a family of nonlinear evolution equations from gauge theory, analyzes their symmetries, and links various integrable systems via gauge transformations.

Findings

Derived a two-parameter family of equations including KdV and Harry Dym.

Identified infinite hierarchies of symmetries and conservation laws.

Established gauge equivalences among integrable systems like mKdV and Calogero KdV.

Abstract

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.