國立清華大學 數學系

Abstract: The research of nonlinear integrable systems has revolutionized the concepts of pure and applied mathematics by discovering the surprising fact that nonlinearities can be treated exactly. However, the problem of identifying integrability with a given nonlinear problem remained unsolved for a long time. Among various attempts, Lie algebraic methods provide the most elegant approach to identify and classify integrability, to construct integrable systems, to deduce and analyze almost all properties of nonlinear integrable systems. After explaining these historical backgrounds, we will apply a loop algebra factorization method to construct two new integrable twisted hierarchies with D2 symmetries in this talk. The splitting type factorization yields the generalized sinh- Gordon equation, which justifies some far-reaching generalizations of the well-known connection between the sine-Gordon equation, the Backlund transformation, and surfaces with curvature -1. The non-splitting type factorization yields the Gerdjikov-Mikhailov- Valchev equation which is an anisotropic multicomponent generalization of the classical Heisenberg ferromagnetic equation and is one of the simplest twisted integrable systems. Tea Time: 3:30pm, R707