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Register a new userGeometric transformations and soliton equations
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Chuu-Lian Terng
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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.
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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.
In these lecture notes we discuss the solution theory of geometric wave equations as they arise in Lorentzian geometry: for a normally hyperbolic differential operator the existence and uniqueness properties of Green functions and Green operators is discussed including a detailed treatment of the Cauchy problem on a globally hyperbolic manifold both for the smooth and finite order setting. As application, the classical Poisson algebra of polynomial functions on the initial values and the dynamical Poisson algebra coming from the wave equation are related. The text contains an introduction to the theory of distributions on manifolds as well as detailed proofs.
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.
We prove that conformally parametrized surfaces in Euclidean space $Rcubec$ of curvature $c$ admit a symmetry reduction of their Gauss-Codazzi equations whose general solution is expressed with the sixth Painleve function. Moreover, it is shown that the two known solutions of this type (Bonnet 1867, Bobenko, Eitner and Kitaev 1997) can be recovered by such a reduction.
We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painleve equations: difference Painleve equation d-$Pleft({A}_{2}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painleve equation d-$Pleft({A}_{1}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.
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