Twisted hierarchies associated with the generalized sine-Gordon equation

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 \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies are among the most important classes of twisted hierarchies. In this paper, we derive explicit interesting first and higher flows of twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies, justify that the one-dimensional systems of twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies for J = Iq, n − q(1 ⩽ q ⩽ n − 1), called the generalized sinh-Gordon equations, are the Gauss-Codazzi equations for n-dimensional timelike submanifolds with constant sectional curvature 1 and index q in pseudo-Euclidean (2n − 1)-dimensional space \documentclass[12pt]{minimal}\begin{document}$\mathbb {R}^{2n-1}_{2q-1}$\end{document}R2q−12n−1 with index 2q − 1. Furthermore, a unified treatment of the inverse scattering theory for twisted \documentclass[12pt]{minimal}\begin{document}$\frac{O(J,J)}{O(J)\times O(J)}$\end{document}O(J,J)O(J)×O(J)-hierarchies is provided.