On the Quasitriviality of Deformations of Bihamiltonian Structures of Hydrodynamic Type

We prove in this paper the quasitriviality of a class of deformations of the one component bihamiltonian structures of hydrodynamic type.Comment: 10 page

On Quasitriviality and Integrability of a Class of Scalar Evolutionary PDEs

For certain class of perturbations of the equation $u_t=f(u) u_x$, we prove the existence of change of coordinates, called quasi-Miura transformations, that reduce these perturbed equations to the unperturbed ones. As an application, we propose a criterion for the integrability of these equations.Comment: 23 page

Jacobi Structures of Evolutionary Partial Differential Equations

In this paper we introduce the notion of infinite dimensional Jacobi structure to describe the geometrical structure of a class of nonlocal Hamiltonian systems which appear naturally when applying reciprocal transformations to Hamiltonian evolutionary PDEs. We prove that our class of infinite dimensional Jacobi structures is invariant under reciprocal transformations. The main technical tool is in a suitable generalization of the classical Schouten-Nijenhuis bracket to the space of the so called quasi-local multi-vectors, and a simple realization of this structure in the framework of supermanifolds. These constructions are used to the computation of the Lichnerowicz-Jacobi cohomologies of Jacobi structures. We also introduce the notion of bi-Jacobi structures and consider the integrability of a system of evolutionary PDEs that possesses a bi-Jacobi structure.Comment: 59 page

Central Invariants of the Constrained KP Hierarchies

We compute the central invariants of the bihamiltonian structures of the constrained KP hierarchies, and show that these integrable hierarchies are topological deformations of their hydrodynamic limits.Comment: 20 page

Hamiltonian Structures and Reciprocal Transformations for the rr-KdV-CH Hierarchy

The $r$-KdV-CH hierarchy is a generalization of the Korteweg-de Vries and Camassa-Holm hierarchies parametrized by $r+1$ constants. In this paper we clarify some properties of its multi-Hamiltonian structures, prove the semisimplicity of the associated bihamiltonian structures and the formula for their central invariants. By introducing a class of generalized Hamiltonian structures, we give in a natural way the transformation formulae of the Hamiltonian structures of the hierarchy under certain reciprocal transformation, and prove the formulae at the level of its dispersionless limit. We also consider relations of the associated bihamiltonian structures to Frobenius manifolds.Comment: 37 page

On Hamiltonian perturbations of hyperbolic systems of conservation laws

We study the general structure of formal perturbative solutions to the Hamiltonian perturbations of spatially one-dimensional systems of hyperbolic PDEs. Under certain genericity assumptions it is proved that any bihamiltonian perturbation can be eliminated in all orders of the perturbative expansion by a change of coordinates on the infinite jet space depending rationally on the derivatives. The main tools is in constructing of the so-called quasi-Miura transformation of jet coordinates eliminating an arbitrary deformation of a semisimple bihamiltonian structure of hydrodynamic type (the quasitriviality theorem). We also describe, following \cite{LZ1}, the invariants of such bihamiltonian structures with respect to the group of Miura-type transformations depending polynomially on the derivatives.Comment: 53 page

BCFG Drinfeld-Sokolov Hierarchies and FJRW-Theory

According to the ADE Witten conjecture, which is proved by Fan, Jarvis and Ruan, the total descendant potential of the FJRW invariants of an ADE singularity is a tau function of the corresponding mirror ADE Drinfeld-Sokolov hierarchy. In the present paper, we show that there is a finite group $\Gamma$ acting on a certain ADE singularity which induces an action on the corresponding FJRW-theory, and the $\Gamma$-invariant sector also satisfies the axioms of a cohomological field theory except the gluing loop axiom. On the other hand, we show that there is also a $\Gamma$-action on the mirror Drinfeld-Sokolov hierarchy, and the $\Gamma$-invariant flows yield the BCFG Drinfeld-Sokolov hierarchy. We prove that the total descendant potential of the $\Gamma$-invariant sector of a FJRW-theory is a tau function of the corresponding BCFG Drinfeld-Sokolov hierarchy.Comment: 60 pages, final version, to appear in Inventiones Mathematicae. arXiv admin note: text overlap with arXiv:1106.6270 by other author

On the Drinfeld-Sokolov Hierarchies of D type

We extend the notion of pseudo-differential operators that are used to represent the Gelfand-Dickey hierarchies, and obtain a similar representation for the full Drinfeld-Sokolov hierarchies of $D_n$ type. By using such pseudo-differential operators we introduce the tau functions of these bi-Hamiltonian hierarchies, and prove that these hierarchies are equivalent to the integrable hierarchies defined by Date-Jimbo-Kashiware-Miwa and Kac-Wakimoto from the basic representation of the Kac-Moody algebra $D_n^{(1)}$.Comment: With 3 Figure

Hodge integrals and tau-symmetric integrable hierarchies of Hamiltonian evolutionary PDEs

For an arbitrary semisimple Frobenius manifold we construct Hodge integrable hierarchy of Hamiltonian partial differential equations. In the particular case of quantum cohomology the tau-function of a solution to the hierarchy generates the intersection numbers of the Gromov--Witten classes and their descendents along with the characteristic classes of Hodge bundles on the moduli spaces of stable maps. For the one-dimensional Frobenius manifold the Hodge hierarchy is a deformation of the Korteweg--de Vries hierarchy depending on an infinite number of parameters. Conjecturally this hierarchy is a universal object in the class of scalar Hamiltonian integrable hierarchies possessing tau-functions.Comment: 49 page

Proof of a Conjecture on the Genus Two Free Energy Associated to the A_n Singularity

In a recent paper [8], it is proved that the genus two free energy of an arbitrary semisimple Frobenius manifold can be represented as a sum of contributions associated with dual graphs of certain stable algebraic curves of genus two plus the so called genus two G-function, and for a certain class of Frobenius manifolds it is conjectured that the associated genus two G-function vanishes. In this paper, we prove this conjecture for the Frobenius manifolds associated with simple singularities of type A.Comment: 24 page