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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 93 (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.
On a class of threedimensional integrable Lagrangians
"... We characterize nondegenerate Lagrangians of the form f(ux, uy, ut) dx dy dt such that the corresponding EulerLagrange equations (fux)x +(fuy)y +(fut)t = 0 are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an overdetermined system of fourth order PDE ..."
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Cited by 4 (1 self)
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We characterize nondegenerate Lagrangians of the form f(ux, uy, ut) dx dy dt such that the corresponding EulerLagrange equations (fux)x +(fuy)y +(fut)t = 0 are integrable by the method of hydrodynamic reductions. The integrability conditions constitute an overdetermined system of fourth order PDEs for the Lagrangian density f, which is in involution and possess interesting differentialgeometric properties. The moduli space of integrable Lagrangians, factorized by the action of a natural equivalence group, is threedimensional. Familiar examples include the dispersionless KadomtsevPetviashvili (dKP) and the BoyerFinley Lagrangians, f = u 3 x/3 + u 2 y − uxut and f = u 2 x + u 2 y − 2e ut, respectively. A complete description of integrable cubic and quartic Lagrangians is obtained. Up to the equivalence transformations, the list of integrable cubic Lagrangians reduces to three examples, f = uxuyut, f = u 2 xuy + uyut and f = u 3 x/3 + u 2 y − uxut (dKP). There exists a unique integrable quartic Lagrangian, f = u 4 x + 2u 2 xut − uxuy − u 2 t. We conjecture that these examples exhaust the list of integrable polynomial Lagrangians which are essentially threedimensional (it was verified that there exist no polynomial integrable Lagrangians of degree five). We prove that the EulerLagrange equations are integrable by the method of hydrodynamic reductions if and only if they possess a scalar pseudopotential playing the role of a dispersionless ‘Lax pair’.
COMPATIBLE METRICS ON A MANIFOLD AND NONLOCAL BIHAMILTONIAN STRUCTURES
, 2004
"... Abstract. Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local biHamiltonian structure, and with additional quasihomogeneity conditions one obtains the structure of a Fr ..."
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Cited by 4 (2 self)
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Abstract. Given a flat metric one may generate a local Hamiltonian structure via the fundamental result of Dubrovin and Novikov. More generally, a flat pencil of metrics will generate a local biHamiltonian structure, and with additional quasihomogeneity conditions one obtains the structure of a Frobenius manifold. With appropriate curvature conditions one may define a curved pencil of compatible metrics and these give rise to an associated nonlocal biHamiltonian structure. Specific examples include the Fmanifolds of Hertling and Manin equipped with an invariant metric. In this paper the geometry supporting such compatible metrics is studied and interpreted in terms of a multiplication on the cotangent bundle. With additional quasihomogeneity assumptions one arrives at a socalled weak Fmanifold a curved version of a Frobenius manifold (which is not, in general, an Fmanifold). A submanifold theory is also developed. Contents
Geometric Structures on the Target Space of Hamiltonian Evolution Equations
"... This thesis is concerned with the relationship between integrable Hamiltonian partial differential equations and geometric structures on the manifold in which the dependent variables take their values. Chapters 1 and 2 are introductory chapters, and as such contains no original material. Chapter 1 c ..."
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This thesis is concerned with the relationship between integrable Hamiltonian partial differential equations and geometric structures on the manifold in which the dependent variables take their values. Chapters 1 and 2 are introductory chapters, and as such contains no original material. Chapter 1 covers some basic material from the theory of integrable systems, including the Hamiltonian formalism for PDE’s, the concept of a biHamiltonian system, and the dispersionless Lax equation. Chapter 2 is about Frobenius manifolds. It explains their relationship to the WDVV equations of topological quantum field theory, and how they form part of the theory of integrable systems via both the deformed LeviCivita connection and a flat pencil of metrics. Chapter 3 is based on [39], which is to appear in the Journal of Geometry and Physics. It is original, except for the background material in Section 3.1. In it we explain the (almost) symplectic geometry associated to Hamiltonian operators of degree 2, and use it to formulate the geometric conditions for two such operators to constitute a biHamiltonian structure. In the case that these operators are associated to symplectic forms, these
(1)
, 709
"... Abstract. We construct the free energy associated with the waterbag model of dToda. Also, the relations of conserved densities are investigated. Key Words: waterbag model, WDVV equation, conserved densities MSC (2000): 35Q58, 37K10, 37K35 ..."
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Abstract. We construct the free energy associated with the waterbag model of dToda. Also, the relations of conserved densities are investigated. Key Words: waterbag model, WDVV equation, conserved densities MSC (2000): 35Q58, 37K10, 37K35
TEICHMÜLLER SPACES AS DEGENERATED SYMPLECTIC LEAVES IN DUBROVIN–UGAGLIA POISSON MANIFOLDS
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