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23
Normal forms of hierarchies of integrable PDEs, Frobenius manifolds and GromovWitten invariants
, 2001
"... We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their ..."
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Cited by 44 (2 self)
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We present a project of classification of a certain class of bihamiltonian 1+1 PDEs depending on a small parameter. Our aim is to embed the theory of Gromov Witten invariants of all genera into the theory of integrable systems. The project is focused at describing normal forms of the PDEs and their local bihamiltonian structures satisfying certain simple axioms. A Frobenius manifold or its degeneration is associated to every bihamiltonian structure of our type. The main result is a universal loop equation on the jet space of a semisimple Frobenius manifold that can be used for perturbative reconstruction of the integrable hierarchy. We show that first few terms of the perturbative expansion correctly reproduce the universal identities between intersection numbers of Gromov Witten classes and their descendents.
Integrable hydrodynamic chains
 J. Math. Phys
, 2003
"... A new approach for derivation of Benneylike momentum chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed, all their reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found. ..."
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Cited by 42 (4 self)
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A new approach for derivation of Benneylike momentum chains and integrable hydrodynamic type systems is presented. New integrable hydrodynamic chains are constructed, all their reductions are described and integrated. New (2+1) integrable hydrodynamic type systems are found.
Geometry and analytic theory of Frobenius manifolds
, 1998
"... Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifol ..."
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Cited by 36 (3 self)
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Main mathematical applications of Frobenius manifolds are in the theory of Gromov Witten invariants, in singularity theory, in differential geometry of the orbit spaces of reflection groups and of their extensions, in the hamiltonian theory of integrable hierarchies. The theory of Frobenius manifolds establishes remarkable relationships between these, sometimes rather distant, mathematical theories.
D.H.: Symplectic forms in the theory of solitons
 Surveys in Differential Geometry IV
, 1998
"... We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ ∗ 0 δL ..."
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Cited by 13 (3 self)
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We develop a Hamiltonian theory for 2D soliton equations. In particular, we identify the spaces of doubly periodic operators on which a full hierarchy of commuting flows can be introduced, and show that these flows are Hamiltonian with respect to a universal symplectic form ω = 1 2 Res ∞ < Ψ ∗ 0 δL ∧ δΨ0> dk. We also construct other higher order symplectic forms and compare our formalism with the case of 1D solitons. Restricted to spaces of finitegap solitons, the universal symplectic form agrees with the symplectic forms which have recently appeared in nonlinear WKB theory, topological field theory, and SeibergWitten theories. We take the opportunity to survey some developments in these areas where symplectic forms have played a major role.
ANALYTIC FUNCTIONS AND INTEGRABLE HIERARCHIESCHARACTERIZATION OF TAU FUNCTIONS
, 2003
"... We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a re ..."
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Cited by 13 (5 self)
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We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.
Quasiclassical limit of BKP hierarchy and Winfnity
"... Previous results on quasiclassical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the BakerAkhiezer function and the tau function are reformulated so as to fit into the analysis of quasiclassical limit. Two subalgebras W B 1 ..."
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Cited by 11 (2 self)
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Previous results on quasiclassical limit of the KP and Toda hierarchies are now extended to the BKP hierarchy. Basic tools such as the Lax representation, the BakerAkhiezer function and the tau function are reformulated so as to fit into the analysis of quasiclassical limit. Two subalgebras W B 1+ ∞ and wB 1+∞ of the Winfinity algebras W1+ ∞ and w1+ ∞ are introduced as fundamental Lie algebras of the BKP hierarchy and its quasiclassical limit, the dispersionless BKP hierarchy. The quantum Winfinity algebra W B 1+∞ emerges in symmetries of the BKP hierarchy. In quasiclassical limit, these W B 1+∞ symmetries are shown to be contracted into w B 1+∞ symmetries of the dispersionless BKP hierarchy
Integrable equations in nonlinear geometrical optics.
, 2004
"... Geometrical optics limit of the Maxwell equations for nonlinear media with the ColeCole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless VeselovNov ..."
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Cited by 9 (6 self)
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Geometrical optics limit of the Maxwell equations for nonlinear media with the ColeCole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless VeselovNovikov equation for the refractive index. It is demonstrated that the VeselovNovikov hierarchy is amenable to the quasiclassical ¯ ∂dressing method. Under more specific requirements for the media, one gets the dispersionless KadomtsevPetviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.