Soliton Hierarchies Constructed from Involutions

TL;DR

This paper constructs new soliton hierarchies related to symmetric spaces, providing integrable systems for the generalized sine-Gordon equation and exploring their symmetries.

Contribution

It introduces two families of soliton hierarchies with Laurent polynomial Lax pairs, linking them to geometric equations like the GSGE and twisted Grassmannian systems.

Findings

Established hierarchies for the GSGE as commuting flows.

Connected soliton hierarchies to symmetric space geometry.

Analyzed symmetries of the generalized sine-Gordon equation.

Abstract

We introduce two families of soliton hierarchies: the twisted hierarchies associated to symmetric spaces. The Lax pairs of these two hierarchies are Laurent polynomials in the spectral variable. Our constructions gives a hierarchy of commuting flows for the generalized sine-Gordon equation (GSGE), which is the Gauss-Codazzi equation for n-dimensional submanifolds in Euclidean (2n-1)-space with constant sectional curvature -1. In fact, the GSGE is the first order system associated to a twisted Grassmannian system. We also study symmetries for the GSGE.