Universal Whitham Hierarchy Research Articles

Integrable structures of dispersionless systems and differential geometry

We develop the theory of Whitham type hierarchies integrable by hydrodynamic reductions as a theory of certain differential-geometric objects. As an application we construct Gibbons-Tsarev systems associated to moduli space of algebraic curves of arbitrary genus and prove that the universal Whitham hierarchy is integrable by hydrodynamic reductions.

Kernel formula approach to the universal Whitham hierarchy

Based on the kernel formula proposed by Carroll and Kodama, we derive the dispersionless Hirota equations of the universal Whitham hierarchy. We also verify the associativity equations in this hierarchy from the dispersionless Hirota equations and give a realization of the associative algebra with the structure constants expressed in terms of residue formulas.

Non-degenerate solutions of the universal Whitham hierarchy

The notion of non-degenerate solutions for the dispersionless Toda hierarchy is generalized to the universal Whitham hierarchy of genus zero with M + 1 marked points. These solutions are characterized by a Riemann–Hilbert problem (generalized string equations) with respect to two-dimensional canonical transformations and may be thought of as a kind of general solutions of the hierarchy. The Riemann–Hilbert problem contains M arbitrary functions Ha(z0, za), a = 1, …, M, which play the role of generating functions of two-dimensional canonical transformations. The solution of the Riemann–Hilbert problem is described by period maps on the space of (M + 1)-tuples (zα(p): α = 0, 1, …, M) of conformal maps from M disks of the Riemann sphere and their complements to the Riemann sphere. The period maps are defined by an infinite number of contour integrals that generalize the notion of harmonic moments. The F-function (free energy) of these solutions is also shown to have a contour integral representation.

Löwner equations, Hirota equations and reductions of the universal Whitham hierarchy

This paper reconsiders finite variable reductions of the universal Whitham hierarchy of genus zero in the perspective of dispersionless Hirota equations. In the case of one-variable reduction, dispersionless Hirota equations turn out to be a powerful tool for understanding the mechanism of reduction. All relevant equations describing the reduction (Löwner-type equations and diagonal hydrodynamic equations) can be thereby derived and justified in a unified manner. The case of multi-variable reductions is not so straightforward. Nevertheless, the reduction procedure can be formulated in a general form, and justified with the aid of dispersionless Hirota equations. As an application, previous results of Guil, Mañas and Martínez Alonso are reconfirmed in this formulation.

Dispersionful analog of the Whitham hierarchy

The transition from the zero-genus universal Whitham hierarchy to its dispersionful counterpart, making use only of the Lax representations, is presented. This is an alternative approach to that of Takasaki who has recently shown that the dispersionless limit of the charged multicomponent Kadomtsev-Petviashvili (KP) hierarchy is the Whitham hierarchy. The advantage of the presented approach is the construction of finite-field reductions, which are the main concern in this paper. The theory is illustrated by several significant examples.

Universal Whitham hierarchy, dispersionless Hirota equations and multicomponent KP hierarchy

The goal of this paper is to identify the universal Whitham hierarchy of genus zero with a dispersionless limit of the multicomponent KP hierarchy. To this end, the multicomponent KP hierarchy is (re)formulated to depend on several discrete variables called “charges”. These discrete variables play the role of lattice coordinates in underlying Toda field equations. A multicomponent version of the so called differential Fay identity are derived from the Hirota equations of the τ -function of this “charged” multicomponent KP hierarchy. These multicomponent differential Fay identities have a well-defined dispersionless limit (the dispersionless Hirota equations). The dispersionless Hirota equations turn out to be equivalent to the Hamilton–Jacobi equations for the S -functions of the universal Whitham hierarchy. The differential Fay identities themselves are shown to be a generating functional expression of auxiliary linear equations for scalar-valued wave functions of the multicomponent KP hierarchy.

Coisotropic Deformations of Associative Algebras and Dispersionless Integrable Hierarchies

The paper is an inquiry of the algebraic foundations of the theory of dispersionless integrable hierarchies, like the dispersionless KP and modified KP hierarchies and the universal Whitham hierarchy of genus zero. It stands out for the idea of interpreting these hierarchies as equations of coisotropic deformations for the structure constants of certain associative algebras. It discusses the link between the structure constants and Hirota’s tau function, and shows that the dispersionless Hirota bilinear equations are, within this approach, a way of writing the associativity conditions for the structure constants in terms of the tau function. It also suggests a simple interpretation of the algebro-geometric construction of the universal Whitham equations of genus zero due to Krichever.

Two-matrix model with semiclassical potentials and extended Whitham hierarchy

We consider the two-matrix model with potentials whose derivative are arbitrary rational function of fixed pole structure and the support of the spectra of the matrices are union of intervals (hard-edges). We derive an explicit formula for the planar limit of the free energy and we derive a calculus which allows to compute derivatives of arbitrarily high order by extending classical Rauch's variational formulae. The four-points correlation functions are explicitly worked out. The formalism extends naturally to the computation of residue formulae for the tau function of the so-called universal Whitham hierarchy studied mainly by I. Krichever: our setting extends that moduli space in that there are certain extra data.

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

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.

The twistor theory of Whitham hierarchy

We have generalized the approach of M. Dunajski, L. Mason and P. Tod [Einstein–Weyl geometry, the dKP equation and twistor theory, J. Geom. Phys. 37 (2001) 63–93] and established a 1–1 correspondence between a solution of the Universal Whitham hierarchy [I.M. Krichever, The τ -function of the universal Whitham hierarchy, matrix models and topological field theories, Comm. Pure Appl. Math. 47 (1994) 437–475] and a twistor space. The twistor space consists of a complex surface and a family of complex curves together with a meromorphic 2-form. The solution of the Universal Whitham hierarchy is given by deforming the curve in the surface. By treating the family of algebraic curves in C P 1 × C P 1 as a twistor space, we were able to express the deformations of the isomonodromic spectral curve in terms of the deformations generated by the Universal Whitham hierarchy.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Whitham hierarchy in growth problems

We discuss the recently established equivalence between the Laplacian growth in the limit of zero surface tension and the universal Whitham hierarchy known in soliton theory. This equivalence allows distinguishing a class of exact solutions of the Laplacian growth problem in the multiply connected case. These solutions correspond to finite-dimensional reductions of the Whitham hierarchy representable as equations of hydrodynamic type, which are solvable by the generalized hodograph method.

Dispersionless scalar integrable hierarchies, Whitham hierarchy, and the quasiclassical ∂̄-dressing method

The quasiclassical limit of the scalar nonlocal ∂̄-problem is derived and a quasiclassical version of the ∂̄-dressing method is presented. Dispersionless Kadomtsev–Petviashvili (KP), modified KP, and dispersionless two-dimensional Toda lattice (2DTL) hierarchies are discussed as illustrative examples. It is shown that the universal Whitham hierarchy is nothing but the ring of symmetries for the quasiclassical ∂̄-problem. The reduction problem is discussed and, in particular, the d2DTL equation of B type is derived.

The $\bar{\partial}$-approach to the dispersionless KP hierarchy

The dispersionless limit of the scalar nonlocal ∂-problem is derived. It is given by a special class of nonlinear first-order equations. A quasiclassical version of the ∂-dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami-type equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasiclassical ∂-problem. PACS number: 02.30.Uk

Formula omitted]-equations, integrable deformations of quasiconformal mappings and Whitham hierarchy

It is shown that the dispersionless scalar integrable hierarchies and, in general, the universal Whitham hierarchy are nothing but classes of integrable deformations of quasiconformal mappings on the plane. Examples of deformations of quasiconformal mappings associated with explicit solutions of the dispersionless KP hierarchy are presented.

The τ‐function of the universal whitham hierarchy, matrix models and topological field theories

AbstractThe universal Whitham hierarchy is considered from the viewpoint of topological field theories. The τ‐function is defined for this hierarchy. It is proved that the algebraic orbits of Whitham hierarchy can be identified with various topological matter models coupled with topological gravity. © 1994 John Wiley & Sons, Inc.

The universal Whitham hierarchy and the geometry of the moduli space of pointed Riemann surfaces

We show that certain structures and constructions of the Whitham theory, an essential part of the perturbation theory of soliton equations, can be instrumental in understanding the geometry of the moduli spaces of Riemann surfaces with marked points. We use the ideas of the Whitham theory to define local coordinates and construct foliations on the moduli spaces. We use these constructions to give a new proof of the Diaz’ bound on the dimension of complete subvarieties of the moduli spaces. Geometrically, we study the properties of meromorphic differentials with real periods and their degenerations.