Twisted hierarchies associated with the generalized sine-Gordon equation

TL;DR

This paper explores twisted hierarchies related to the generalized sine-Gordon equation, deriving new flows, analyzing their geometric properties, and providing a unified inverse scattering framework.

Contribution

It explicitly derives new flows of twisted (J,J)/((J) imes (J)) hierarchies, investigates their geometric aspects, and unifies the inverse scattering theory approach.

Findings

Derived interesting first and higher flows of twisted (J,J)/((J) imes (J))-hierarchies.

Investigated the associated submanifold geometry.

Provided a unified treatment of the inverse scattering theory.

Abstract

Twisted $U$- and twisted $U/K$-hierarchies are soliton hierarchies introduced by Terng to find higher flows of the generalized sine-Gordon equation. Twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are among the most important classes of twisted hierarchies. In this paper, interesting first and higher flows of twisted $\frac {O(J,J)}{O(J)\times O(J)}$-hierarchies are explicitly derived, the associated submanifold geometry is investigated and a unified treatment of the inverse scattering theory is provided.