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27
Extended affine Weyl groups and Frobenius manifolds
 Compositio Math
, 1998
"... Abstract. For the root system of type Bl and Cl, we generalize the result of [5] by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of [5]. 1. ..."
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Cited by 41 (8 self)
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Abstract. For the root system of type Bl and Cl, we generalize the result of [5] by showing the existence of a Frobenius manifold structure on the orbit space of the extended affine Weyl group that corresponds to any vertex of the Dynkin diagram instead of a particular choice of [5]. 1.
The extended bigraded Toda hierarchy
 J. Phys. A: Math. Gen
"... Abstract. We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are ǫseries of differential polynomials in the dependent variables, and we use them to provi ..."
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Cited by 32 (4 self)
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Abstract. We generalize the Toda lattice hierarchy by considering N + M dependent variables. We construct roots and logarithms of the Lax operator which are uniquely defined operators with coefficients that are ǫseries of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy, generalizing [4]. Using Rmatrix theory we give the bihamiltonian formulation of this hierarchy and we prove the existence of a tau function for its solutions. Finally we study the dispersionless limit and its connection with a class of Frobenius manifolds on the orbit space of the extended affine Weyl groups ˜ W (N) (AN+M−1) of the A series, defined in [9].
On universality of critical behaviour in Hamiltonian PDEs, in Geometry
 Topology, and Mathematical Physics, Amer. Math. Soc. Transl. Ser
"... on the occasion of his 70th birthday. Abstract. Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimensi ..."
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Cited by 21 (4 self)
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on the occasion of his 70th birthday. Abstract. Our main goal is the comparative study of singularities of solutions to the systems of first order quasilinear PDEs and their perturbations containing higher derivatives. The study is focused on the subclass of Hamiltonian PDEs with one spatial dimension. For the systems of order one or two we describe the local structure of singularities of a generic solution to the unperturbed system near the point of “gradient catastrophe ” in terms of standard objects of the classical singularity theory; we argue that their perturbed companions must be given by certain special solutions of Painlevé equations and their generalizations. Contents
Virasoro Symmetries of the Extended Toda Hierarchy
, 2008
"... We prove that the extended Toda hierarchy of [1] admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the ..."
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Cited by 20 (1 self)
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We prove that the extended Toda hierarchy of [1] admits nonabelian Lie algebra of infinitesimal symmetries isomorphic to the half of the Virasoro algebra. The generators Lm, m ≥ −1 of the Lie algebra act by linear differential operators onto the tau function of the hierarchy. We also prove that the tau function of a generic solution to the extended Toda hierarchy is annihilated by a combination of the Virasoro operators and the flows of the hierarchy. As an application we show that the validity of the Virasoro constraints for the CP 1 GromovWitten invariants and their descendents implies that their generating function is the logarithm of a particular tau function of the extended Toda hierarchy.
quadratic equations for the extended Toda hierarchy”, arXiv:math/0501336; “The equivariant GromovWitten theory of CP 1 and integrable hierarchies”, arXiv:mathph/0508054; “GromovWitten theory of CP 1 and integrable hierarchies
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Bihamiltonian Systems of Hydrodynamic Type and Reciprocal Transformations
, 2005
"... We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian. Mathematics Subject Classification (2000). 37K25; 37K10. Key words. bihamiltonian sys ..."
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We prove that under certain linear reciprocal transformation, an evolutionary PDE of hydrodynamic type that admits a bihamiltonian structure is transformed to a system of the same type which is still bihamiltonian. Mathematics Subject Classification (2000). 37K25; 37K10. Key words. bihamiltonian system, reciprocal transformation
Rmatrices and Hamiltonian structures for certain Lax equations
 Rev. Math. Phys
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GromovWitten invariants of CP 1 and integrable hierarchies, available at arXiv:math.AG/0501336
"... Abstract. The paper [CDZ] gives a Lax type presentation of the flows of the Extended Toda Hierarchy (shortly ETH). Our first result is a description of the ETH in terms of Hirota Quadratic Equations (shortly HQEs), which can be viewed as flows on a certain infinite dimensional manifold of functions, ..."
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Abstract. The paper [CDZ] gives a Lax type presentation of the flows of the Extended Toda Hierarchy (shortly ETH). Our first result is a description of the ETH in terms of Hirota Quadratic Equations (shortly HQEs), which can be viewed as flows on a certain infinite dimensional manifold of functions, called taufunctions of the ETH. A new feature here is that the Hirota equations are given in terms of vertex operators taking values in the algebra of differential operators on the affine line. On the other hand, in [G1], the author constructs vertex operators in terms of period mappings associated with isolated singularities of holomorphic functions. In the second part of this paper we apply the methods from [G1] to the mirror of CP 1 to construct certain deformations of the HQEs of the ETH, equivalent to the original ones and parameterized by the semisimple points in the cohomology algebra H ∗ (CP 1; C). In particular, applying Givental’s formula which describes the total descendent potential of CP 1 in terms of taufunctions of the KdV hierarchy, we obtain a new proof of the Toda conjecture: the Gromov–Witten invariants of CP 1 are governed by the flows of the ETH. 1.