Dispersionless Integrable Hierarchies Research Articles

Dispersionless integrable systems and the Bogomolny equations on an Einstein–Weyl geometry background

We derive a dispersionless integrable system describing a local form of a general three-dimensional Einstein-Weyl geometry with an Euclidean (positive) signature, construct its matrix extension and demonstrate that it leads to the Bogomolny equations for a non-abelian monopole on an Einstein-Weyl geometry background. The corresponding dispersionless integrable hierarchy, its matrix extension and the dressing scheme are also considered.

Dispersionless Integrable Hierarchy via Kodaira–Spencer Gravity

We explain how dispersionless integrable hierarchy in 2d topological field theory arises from the Kodaira-Spencer gravity (BCOV theory). The infinitely many commuting Hamiltonians are given by the current observables associated to the infinite abelian symmetries of the Kodaira-Spencer gravity. We describe a BV framework of effective field theories that leads to the B-model interpretation of dispersionless integrable hierarchy.

Dispersionless integrable hierarchies andGL(2, ℝ) geometry

AbstractParaconformal orGL(2, ℝ) geometry on ann-dimensional manifoldMis defined by a field of rational normal curves of degreen– 1 in the projectivised cotangent bundle ℙT*M. Such geometry is known to arise on solution spaces of ODEs with vanishing Wünschmann (Doubrov–Wilczynski) invariants. In this paper we discuss yet another natural source ofGL(2, ℝ) structures, namely dispersionless integrable hierarchies of PDEs such as the dispersionless Kadomtsev–Petviashvili (dKP) hierarchy. In the latter context,GL(2, ℝ) structures coincide with the characteristic variety (principal symbol) of the hierarchy.Dispersionless hierarchies provide explicit examples of particularly interesting classes of involutiveGL(2, ℝ) structures studied in the literature. Thus, we obtain torsion-freeGL(2, ℝ) structures of Bryant [5] that appeared in the context of exotic holonomy in dimension four, as well as totally geodesicGL(2, ℝ) structures of Krynski [33]. The latter possess a compatible affine connection (with torsion) and a two-parameter family of totally geodesicα-manifolds (coming from the dispersionless Lax equations), which makes them a natural generalisation of the Einstein–Weyl geometry.Our main result states that involutiveGL(2, ℝ) structures are governed by a dispersionless integrable system whose general local solution depends on 2n– 4 arbitrary functions of 3 variables. This establishes integrability of the system of Wünschmann conditions.

Projective differential geometry of multidimensional dispersionless integrable hierarchies

We introduce a general setting for multidimensional dispersionless integrable hierarchy in terms of differential m-form Ωm with the coefficients satisfying the Plücker relations, which is gauge-invariantly closed and its gauge-invariant coordinates (ratios of coefficients) are (locally) holomorphic with respect to one of the variables (the spectral variable). We demonstrate that this form defines a hierarchy of dispersionless integrable equations in terms of commuting vector fields locally holomorphic in the spectral variable. The equations of the hierarchy are given by the gauge-invariant closedness equations.

On dispersionless Lax representations for two-primary models

The equivalence between bi-Hamiltonian formulation and Lax representation of dispersionless integrable hierarchies associated with two-primary models of Frobenius manifolds is explicitly verified.

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.

Dispersionless integrable equations as coisotropic deformations: Extensions and reductions

Interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systInterpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain algebras and other algebraic structures like Jordan triple systems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level. ems is discussed. Several generalizations are considered. Stationary reductions of the dispersionless integrable equations are shown to be connected with the dynamical systems on the plane completely integrable on a fixed energy level.

BIDIFFERENTIAL CALCULI, BICOMPLEX STRUCTURE AND ITS APPLICATION TO BIHAMILTONIAN SYSTEMS

In this exposition, we study the relationship between the bihamiltonian formalism of completely integrable systems using the bidifferential calculi introduced by Dimakis and Müller-Hoissen in [1] and the bihamiltonian formulation of integrable systems with a finite number of degrees of freedom via the Frölicher–Nijenhuis geometry. This pair of bidifferetial operators are used to construct alternative Lie algebroids as shown by Camacaro and Carinena. We find its connection to Finsler geometry. We also find the dispersionless integrable hierarchies using the bidifferential ideals. Finally, we lay out its connection to Gelfand–Zakharevich bihamiltonian geometry.

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.

Quasiconformal Mappings and Solutions of the Dispersionless KP hierarchy

A \(\overline \partial \) formalism for studying dispersionless integrable hierarchies is applied to the dispersionless KP (dKP) hierarchy. We relate this formalism to the theory of quasiconformal mappings on the plane and present some classes of explicit solutions of the dKP hierarchy.

Reductions of the Dispersionless KP Hierarchy

We present a method for constructing the S-function based on a system of first-order differential equations and use it to analyze reductions of dispersionless integrable hierarchies.

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.