Dispersionless KP Hierarchy Research Articles

Integrable hierarchies associated to infinite families of Frobenius manifolds

We propose a new construction of an integrable hierarchy associated to any infinite series of Frobenius manifolds satisfying a certain stabilization condition. We study these hierarchies for Frobenius manifolds associated to A N , D N and B N singularities. In the case of A N Frobenius manifolds our hierarchy turns out to coincide with the dispersionless KP hierarchy; for B N Frobenius manifolds it coincides with the dispersionless BKP hierarchy; and for D N hierarchy it is a certain reduction of the dispersionless 2-component BKP hierarchy. As a side product to these results we illustrate the enumerative meaning of certain coefficients of A N , D N and B N Frobenius potentials.

LÖwner evolution and finite-dimensional reductions of integrable systems

The Loewner equation is known as a one-dimensional reduction of the Benney chain as well as the dispersionless KP hierarchy. We propose a reverse process showing that time splitting in the Loewner or the Loewner-Kufarev equation leads to some known integrable systems.

Old and New Reductions of Dispersionless Toda Hierarchy

This paper is focused on geometric aspects of two particular types of finite-variable reductions in the dispersionless Toda hierarchy. The reductions are formulated in terms of "Landau-Ginzburg potentials" that play the role of reduced Lax functions. One of them is a generalization of Dubrovin and Zhang's trigonometric polynomial. The other is a transcendental function, the logarithm of which resembles the waterbag models of the dispersionless KP hierarchy. They both satisfy a radial version of the L\"owner equations. Consistency of these L\"owner equations yields a radial version of the Gibbons-Tsarev equations. These equations are used to formulate hodograph solutions of the reduced hierarchy. Geometric aspects of the Gibbons-Tsarev equations are explained in the language of classical differential geometry (Darboux equations, Egorov metrics and Combescure transformations). Flat coordinates of the underlying Egorov metrics are presented.

Recursion Operators for Dispersionless KP Hierarchy

Based on the corresponding theorem between dispersionless KP (dKP) hierarchy and ħ-dependent KP (ħKP) hierarchy, a general formal representation of the recursion operators for dKP hierarchy under n-reduction is given in a systematical way from the corresponding ħKP hierarchy. To illustrate this method, the recursion operators for dKP hierarchy under 2-reduction and 3-reduction are calculated in detail.

A new extended dispersionless mKP hierarchy and hodograph solution

In this article, a new extended dispersionless mKP hierarchy (exdmKPH) is constructed to obtain two types of dispersionless mKP equations with self-consistent sources (dmKPSCS) and their associated conservation equations. Two reductions of this hierarchy are used to get two types of the corresponding dispersionless mKdV equations with self-consistent sources (dmKdVSCS). A hodograph solution for the first type of dmKdVSCS and Bäcklund transformation between the extended dispersionless KP hierarchy (exdKPH) and exdmKPH are also given.

Growth of fat slits and dispersionless KP hierarchy

A ‘fat slit’ is a compact domain in the upper half-plane bounded by a curve with endpoints on the real axis and a segment of the real axis between them. We consider conformal maps of the upper half-plane to the exterior of a fat slit parameterized by harmonic moments of the latter and show that they obey an infinite set of Lax equations for the dispersionless KP hierarchy. Deformation of a fat slit under changing a particular harmonic moment can be treated as a growth process similar to the Laplacian growth of domains in the whole plane. This construction extends the well-known link between solutions to the dispersionless KP hierarchy and conformal maps of slit domains in the upper half-plane and provides a new, large family of solutions.

Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2 + 1 dimensions

The usual dispersionless limit of the KP hierarchy does not work in the case where the dependent variable has values in a noncommutative (e.g. matrix) algebra. Passing over to the potential KP hierarchy, there is a corresponding scaling limit in the noncommutative case, which turns out to be the hierarchy of a ‘pseudodual chiral model’ in 2 + 1 dimensions (‘pseudodual’ to a hierarchy extending Ward's (modified) integrable chiral model). Applying the scaling procedure to a method generating exact solutions of a matrix (potential) KP hierarchy from solutions of a matrix linear heat hierarchy, leads to a corresponding method that generates exact solutions of the matrix dispersionless potential KP hierarchy, i.e. the pseudodual chiral model hierarchy. We use this result to construct classes of exact solutions of the su(m) pseudodual chiral model in 2 + 1 dimensions, including various multiple lump configurations.

Logarithmic Deformations of the Rational Superpotential/Landau-Ginzburg Construction of Solutions of the WDVV Equations

The superpotential in the Landau-Ginzburg construction of solutions to the Witten-Dijkgraaf-Verlinde-Verlinde (or WDVV) equations is modified to include logarithmic terms. This results in deformations - quadratic in the deformation parameters- of the normal prepotential solutions of the WDVV equations. Such solutions satisfy various pseudo-quasi-homogeneity conditions, on assigning a notional weight to the deformation parameters. These solutions originate in the so-called ‘water-bag’ reductions of the dispersionless KP hierarchy. This construction includes, as a special case, deformations which are polynomial in the flat coordinates, resulting in a new class of polynomial solutions of the WDVV equations.

The initial-value problem for the dKP hierarchy

The solution corresponding to each initial condition for the dispersionless KP hierarchy can be found from the integration of a Hamilton–Jacobi equation by means of a transformation of coordinates. The solution is explicitly determined in a parametric form as well as a power series in a deformation parameter. In addition, a twistor formulation of the solution is given.

Bäcklund transformations for the constrained dispersionless hierarchies and dispersionless hierarchies with self-consistent sources

The Bäcklund transformations between the constrained dispersionless KP hierarchy (cdKPH) and the constrained dispersionless mKP hierarchy (cdmKPH) and between the dispersionless KP hierarchy with self-consistent sources (dKPHSCS) and the dispersionless mKP hierarchy with self-consistent sources (dmKPHSCS) are constructed. The auto-Bäcklund transformations for the cdmKPH and for the dmKPHSCS are also formulated.

The quasiclassical limit of the symmetry constraint of the KP hierarchy and the dispersionless KP hierarchy with self-consistent sources

For the first time we show that the quasiclassical limit of the symmetry constraint of the Sato operator for the KP hierarchy leads to the generalized Zakharov reduction of the Sato function for the dispersionless KP (dKP) hierarchy which has been proved to be result of symmetry constraint of the dKP hierarchy recently. By either regarding the symmetry constrained dKP hierarchy as its stationary case or taking the dispersionless limit of the KP hierarchy with self-consistent sources directly, we construct a new integrable dispersionless hierarchy, i.e., the dKP hierarchy with self-consistent sources and find its associated conservation equations (or equations of Hamilton-Jacobi type). Some solutions of the dKP equation with self-consistent sources are also obtained by hodograph transformations.

Dressing operator approach to the toroidal model of higher-dimensional dispersionless KP hierarchy

Based upon the dressing operator approach we investigate the twistor theoretical construction, additional symmetries, hodograph solutions, and Miura transformation for the toroidal model of higher-dimensional dispersionless KP hierarchy introduced by Takasaki.

Classification of Integrable (2+1)-Dimensional Quasilinear Hierarchies

We investigate the (2+1)-dimensional hierarchies associated with the integrable PDEs of the form Δ tt = F(Δxx, Δxt, Δxy), which generalize the dispersionless KP hierarchy. Integrability is understood as the existence of infinitely many hydrodynamic reductions.

S-functions, reductions and hodograph solutions of the rth dispersionless modified KP and Dym hierarchies

We introduce an S-function formulation for the recently found r-th dispersionless modified KP and r-th dispersionless Dym hierarchies, giving also a connection of these $S$-functions with the Orlov functions of the hierarchies. Then, we discuss a reduction scheme for the hierarchies that together with the $S$-function formulation leads to hodograph systems for the associated solutions. We consider also the connection of these reductions with those of the dispersionless KP hierarchy and with hydrodynamic type systems. In particular, for the 1-component and 2-component reduction we derive, for both hierarchies, ample sets of examples of explicit solutions.

Symmetry constraints for dispersionless integrable equations and systems of hydrodynamic type

Symmetry constraints for ( 2 + 1 ) -dimensional dispersionless integrable equations are considered. It is demonstrated that they naturally lead to certain systems of hydrodynamic type which arise within the reduction method. One also easily obtains an associated complex curve (Sato function) and corresponding generating equations. Dispersionless KP and 2DTL hierarchy are considered as illustrative examples.

Hydrodynamic reductions of multidimensional dispersionless PDEs: The test for integrability

A (d+1)-dimensional dispersionless PDE is said to be integrable if its n-component hydrodynamic reductions are locally parametrized by (d−1)n arbitrary functions of one variable. The most important examples include the four-dimensional heavenly equation descriptive of self-dual Ricci-flat metrics and its six-dimensional generalization arising in the context of sdiff(Σ2) self-dual Yang–Mills equations. Given a multidimensional PDE which does not pass the integrability test, the method of hydrodynamic reductions allows one to effectively reconstruct additional differential constraints which, when added to the equation, make it an integrable system in fewer dimensions. As an example of this phenomenon we discuss the second commuting flow of the dispersionless KP hierarchy. Considered separately, this is a four-dimensional PDE which does not pass the integrability test. However, the method of hydrodynamic reductions generates additional differential constraints which reconstruct the full (2+1)-dimensional dispersionless KP hierarchy.

Functional symmetries and solutions of the dKP hierarchy

It is shown that the dispersionless KP hierarchy possesses an infinite collection of symmetries depending on arbitrary functions of the time variables. The twistor formulation of a factorization problem determines the family of symmetries and serves for the characterization of solutions of the hierarchy.

Nonlinear Beltrami equation and τ-function for dispersionless hierarchies

It is proved that the action for nonlinear Beltrami equation (quasi-classical ∂ ̄ -problem) evaluated on its solution gives a τ-function for dispersionless KP hierarchy. Infinitesimal transformations of τ-function corresponding to variations of ∂ ̄ -data are found. Determinant equations for the function generating these transformations are derived. They represent a dispersionless analogue of singular manifold (Schwarzian) KP equations. Dispersionless 2DTL hierarchy is also considered.

Additional symmetries and solutions of the dispersionless KP hierarchy

The dispersionless KP hierarchy is considered from the point of view of the twistor formalism. A set of explicit additional symmetries is characterized and its action on the solutions of the twistor equations is studied. A method for dealing with the twistor equations by taking advantage of hodograph type equations is proposed. This method is applied for determining the orbits of solutions satisfying reduction constraints of Gelfand–Dikii type under the action of additional symmetries.

Analytic Functions and Integrable Hierarchies–Characterization of Tau Functions

We prove the dispersionless Hirota equations for the dispersionless Toda, dispersionless coupled modified KP and dispersionless KP hierarchies using an idea from classical complex analysis. We also prove that the Hirota equations characterize the tau functions for each of these hierarchies. As a result, we establish the links between the hierarchies.