Seminar on Integrable Systems
主講者: 1.馬文秀教授 (University of South Florida) 2.張仁煦教授 (National Defence University)
講題: 1.Variational Identities and Hamiltonian Structures of Soliton Equations 2.Integrable Model in Nonlinear Geometrical Optics
時間: 2008-12-08 (Mon.)  13:30 - 16:30
Abstract: 1. We will discuss variational identities associated with continuous and discrete spectral problems involving both classical variables and super variables. The variational identities provide a systematic way to construct Hamiltonian structures for soliton equations generated from semi-direct sums of Lie algebras. Our illustrative examples are the AKNS hierarchy, the Volterra lattice hierarchy, and their integrable couplings. The resulting Hamiltonian structures yield infinitely many symmetries and conservation laws. 2. The geometrical optics limit of the Maxwell equations for the Cole-Cole nonlinear media with slow variation along one axis gives rise to the dispersionless Veselov-Novikov equation(dVN) for the refractive index. We investigate the dispersionless Veselov-Novikov equation based on the framework of dispersionless two-component BKP hierarchy. Symmetry constraints for real dVN system are considered. It is shown that under symmetry reductions, the conserved densities are therefore related to the associated Faber polynomials and can be solved recursively. Moreover, the method of hodograph transformation as well as the expressions of Faber polynomials are used to find exact real solutions of the dVN hierarchy.
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