Luis Martínez Alonso

Reductions and hodograph solutions of the dispersionless KP hierarchy

A general scheme for analysing reductions of dispersionless integrable hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order differential equations. Compatibility systems of nonlinear partial differential equations of Bourlet type characterizing both reductions and hodograph solutions of the dKP hierarchy are obtained. Wide classes of illustrative explicit examples are exhibited.

Charged free fermions, vertex operators and the classical theory of conjugate nets

We show that the quantum field theoretical formulation of the -function theory has a geometrical interpretation within the classical transformation theory of conjugate nets. In particular, we prove that (i) the partial charge transformations preserving the neutral sector are Laplace transformations, (ii) the basic vertex operators are Levy and adjoint Levy transformations and (iii) the diagonal soliton vertex operators generate fundamental transformations. We also show that the bilinear identity for the multicomponent Kadomtsev-Petviashvili hierarchy becomes, through a generalized Miwa map, a bilinear identity for the multidimensional quadrilateral lattice equations.

The Whitham hierarchies: reductions and hodograph solutions

A general scheme for analysing reductions of Whitham hierarchies is presented. It is based on a method for determining the S-function by means of a system of first-order partial differential equations. Compatibility systems of differential equations characterizing both reductions and hodograph solutions of Whitham hierarchies are obtained. The method is illustrated by exhibiting solutions of integrable models such as the dispersionless Toda equation (heavenly equation) and the generalized Benney system.

The multicomponent 2D Toda hierarchy: discrete flows and string equations

The multicomponent 2D Toda hierarchy is analyzed through a factorization problem associated to an infinite-dimensional group. A new set of discrete flows is considered and the corresponding Lax and Zakharov--Shabat equations are characterized. Reductions of block Toeplitz and Hankel bi-infinite matrix types are proposed and studied. Orlov--Schulman operators, string equations and additional symmetries (discrete and continuous) are considered. The continuous-discrete Lax equations are shown to be equivalent to a factorization problem as well as to a set of string equations. A congruence method to derive site independent equations is presented and used to derive equations in the discrete multicomponent KP sector (and also for its modification) of the theory as well as dispersive Whitham equations.

On twistor solutions of the dKP equation

The factorization problem for the group of canonical transformations close to the identity and the corresponding twistor equations for an ample family of canonical variables are considered. A method to deal with these reductions is developed for the construction of classes of nontrivial solutions of the dKP equation.

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 particularly 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 a convenient derivation of additional symmetries. Moreover, new functional symmetries of the Zakharov extension of the Bentley gas equations are given and the action of additional symmetries over the potential in terms of linear PDEs is characterized.

Dressing methods for geometric nets: I. Conjugate nets

The formalism of multicomponent KP hierarchies is applied to deriving efficient dressing methods for conjugate nets. The notion of the Cauchy propagator is used for characterizing these nets in terms of spectral data. Explicit examples in dimensions N = 2 and 3 are given. In particular, periodic nets and Cartesian nets with a Gaussian localized deformation are exhibited.

Determination of S-curves with applications to the theory of non-Hermitian orthogonal polynomials

This paper deals with the determination of the S-curves in the theory of non-hermitian orthogonal polynomials with respect to exponential weights along suitable paths in the complex plane. It is known that the corresponding complex equilibrium potential can be written as a combination of Abelian integrals on a suitable Riemann surface whose branch points can be taken as the main parameters of the problem. Equations for these branch points can be written in terms of periods of Abelian differentials and are known in several equivalent forms. We select one of these forms and use a combination of analytic an numerical methods to investigate the phase structure of asymptotic zero densities of orthogonal polynomials and of asymptotic eigenvalue densities of random matrix models. As an application we give a complete description of the phases and critical processes of the standard cubic model.

The KP hierarchy in Miwa coordinates

Abstract A systematic reformulation of the KP hierarchy by using continuous Miwa variables is presented. Basic quantities and relations are defined and determinantal expressions for Fays identities are obtained. It is shown that in terms of these variables the KP hierarchy gives rise to a Darboux system describing an infinite-dimensional conjugate net.

Dressing methods for geometric nets: II. Orthogonal and Egorov nets

The Grassmannian formalism of KP hierarchies is used to study geometric nets of orthogonal type and their subclass of Egorov nets. Efficient dressing methods for Cauchy propagators are provided which lead to wide families of explicit nets. Frobenius manifolds and solutions to the Witten-Dijkgraff-Verlinde-Verlinde associativity equations are also constructed.