String equations in Whitham hierarchies: tau-functions and Virasoro constraints

A scheme for solving Whitham hierarchies satisfying a special class of string equations is presented. The tau-function of the corresponding solutions is obtained and the differential expressions of the underlying Virasoro constraints are characterized. Illustrative examples of exact solutions of Whitham hierarchies are derived and applications to conformal maps dynamics are indicated.Comment: 26 pages, 2 figure

Dispersionless scalar integrable hierarchies, Whitham hierarchy and the quasi-classical dbar-dressing method

The quasi-classical limit of the scalar nonlocal dbar-problem is derived and a quasi-classical version of the dbar-dressing method is presented. Dispersionless KP, mKP and 2DTL hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy it is nothing but the ring of symmetries for the quasi-classical dbar-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.Comment: LaTex file,19 page

Bi-Hamiltonian structures for integrable systems on regular time scales

A construction of the bi-Hamiltonian structures for integrable systems on regular time scales is presented. The trace functional on an algebra of $\delta$-pseudo-differential operators, valid on an arbitrary regular time scale, is introduced. The linear Poisson tensors and the related Hamiltonians are derived. The quadratic Poisson tensors is given by the use of the recursion operators of the Lax hierarchies. The theory is illustrated by $\Delta$-differential counterparts of Ablowitz-Kaup-Newell-Segur and Kaup-Broer hierarchies.Comment: 18 page

On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized

On the Whitham hierarchy: dressing scheme, string equations and additional symmetrie

A new description of the universal Whitham hierarchy in terms of a factorization problem in the Lie group of canonical transformations is provided. This scheme allows us to give a natural description of dressing transformations, string equations and additional symmetries for the Whitham hierarchy. We show how to dress any given solution and prove that any solution of the hierarchy may be undressed, and therefore comes from a factorization of a canonical transformation. A particulary important function, related to the $\tau$-function, appears as a potential of the hierarchy. We introduce a class of string equations which extends and contains previous classes of string equations considered by Krichever and by Takasaki and Takebe. The scheme is also applied for an convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Benney gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized

Classification of integrable hydrodynamic chains and generating functions of conservation laws

New approach to classification of integrable hydrodynamic chains is established. Generating functions of conservation laws are classified by the method of hydrodynamic reductions. N parametric family of explicit hydrodynamic reductions allows to reconstruct corresponding hydrodynamic chains. Plenty new hydrodynamic chains are found

The Extended Bigraded Toda hierarchy

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 $\epsilon$-series of differential polynomials in the dependent variables, and we use them to provide a Lax pair definition of the extended bigraded Toda hierarchy. Using R-matrix 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 of the $A$ series.Comment: 32 pages, corrected typo

The algebraic and Hamiltonian structure of the dispersionless Benney and Toda hierarchies

The algebraic and Hamiltonian structures of the multicomponent dispersionless Benney and Toda hierarchies are studied. This is achieved by using a modified set of variables for which there is a symmetry between the basic fields. This symmetry enables formulae normally given implicitly in terms of residues, such as conserved charges and fluxes, to be calculated explicitly. As a corollary of these results the equivalence of the Benney and Toda hierarchies is established. It is further shown that such quantities may be expressed in terms of generalized hypergeometric functions, the simplest example involving Legendre polynomials. These results are then extended to systems derived from a rational Lax function and a logarithmic function. Various reductions are also studied.Comment: 29 pages, LaTe

The dispersive self-dual Einstein equations and the Toda lattice

The Boyer-Finley equation, or $SU(\infty)$-Toda equation is both a reduction of the self-dual Einstein equations and the dispersionlesslimit of the $2d$-Toda lattice equation. This suggests that there should be a dispersive version of the self-dual Einstein equation which both contains the Toda lattice equation and whose dispersionless limit is the familiar self-dual Einstein equation. Such a system is studied in this paper. The results are achieved by using a deformation, based on an associative $\star$-product, of the algebra $sdiff(\Sigma^2)$ used in the study of the undeformed, or dispersionless, equations.Comment: 11 pages, LaTeX. To appear: J. Phys.