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Register a new userTau functions and Virasoro actions for soliton hierarchies
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There is a general method for constructing a soliton hierarchy from a splitting of a loop group as a positive and a negative sub-groups together with a commuting linearly independent sequence in the positive Lie subalgebra. Many known soliton hierarchies can be constructed this way. The formal inverse scattering associates to each f in the negative subgroup a solution u_f of the hierarchy. When there is a 2 co-cycle of the Lie algebra that vanishes on both sub-algebras, Wilson constructed a tau function tau_f for each element f in the negative subgroup. In this paper, we give integral formulas for variations of ln(tau_f) and second partials of ln(tau_f), discuss whether we can recover solutions u_f from tau_f, and give a general construction of actions of the positive half of the Virasoro algebra on tau functions. We write down formulas relating tau functions and formal inverse scattering solutions and the Virasoro vector fields for the GL(n,C)-hierarchy.
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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.
We give a survey of the following six closely related topics: (i) a general method for constructing a soliton hierarchy from a splitting of a loop algebra into positive and negative subalgebras, together with a sequence of commuting positive elements, (ii) a method---based on (i)---for constructing soliton hierarchies from a symmetric space, (iii) the dressing action of the negative loop subgroup on the space of solutions of the related soliton equation, (iv) classical Backlund, Christoffel, Lie, and Ribaucour transformations for surfaces in three-space and their relation to dressing actions, (v) methods for constructing a Lax pair for the Gauss-Codazzi Equation of certain submanifolds that admit Lie transforms, (vi) how soliton theory can be used to generalize classical soliton surfaces to submanifolds of higher dimension and co-dimension.
The solutions of a large class of hierarchies of zero-curvature equations that includes Toda and KdV type hierarchies are investigated. All these hierarchies are constructed from affine (twisted or untwisted) Kac-Moody algebras~$ggg$. Their common feature is that they have some special ``vacuum solutions corresponding to Lax operators lying in some abelian (up to the central term) subalgebra of~$ggg$; in some interesting cases such subalgebras are of the Heisenberg type. Using the dressing transformation method, the solutions in the orbit of those vacuum solutions are constructed in a uniform way. Then, the generalized tau-functions for those hierarchies are defined as an alternative set of variables corresponding to certain matrix elements evaluated in the integrable highest-weight representations of~$ggg$. Such definition of tau-functions applies for any level of the representation, and it is independent of its realization (vertex operator or not). The particular important cases of generalized mKdV and KdV hierarchies as well as the abelian and non abelian affine Toda theories are discussed in detail.
For a simple Lie algebra $mathfrak{g}$ and an irreducible faithful representation $pi$ of $mathfrak{g}$, we introduce the Schur polynomials of $(mathfrak{g},pi)$-type. We then derive the Sato-Zhou type formula for tau functions of the Drinfeld-Sokolov (DS) hierarchy of $mathfrak{g}$-type. Namely, we show that the tau functions are linear combinations of the Schur polynomials of $(mathfrak{g},pi)$-type with the coefficients being the Plucker coordinates. As an application, we provide a way of computing polynomial tau functions for the DS hierarchy. For $mathfrak{g}$ of low rank, we give several examples of polynomial tau functions, and use them to detect bilinear equations for the DS hierarchy.
We prove that for any tau-symmetric bihamiltonian deformation of the tau-cover of the Principal Hierarchy associated with a semisimple Frobenius manifold, the deformed tau-cover admits an infinite set of Virasoro symmetries.
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