A note on the extended dispersionless Toda hierarchy

We derive dispersionless Hirota equations for the extended dispersionless Toda hierarchy. We show that the dispersionless Hirota equations are just a direct consequence of the genus-zero topological recurrence relation for the topological ?P¹ model. Using the dispersionless Hirota equations, we compute the twopoint functions and express the result in terms of Catalan numbers     

Dispersionless integrable equations as coisotropic deformations: Extensions and reductions

We discuss the interpretation of dispersionless integrable hierarchies as equations of coisotropic deformations for certain associative algebras and other algebraic structures. We show that with this approach, the dispersionless Hirota equations for the dKP hierarchy are just the associativity conditions in a certain parameterization. We consider several generalizations and demonstrate that B-type dispersionless integrable hierarchies, such as the dBKP and the dVN hierarchies, are coisotropic deformations of the Jordan triple systems. We show that stationary reductions of the dispersionless integrable equations are connected with dynamical systems on the plane that are completely integrable on a fixed energy level. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 439–457, June, 2007.     

Dispersionless Limit of Hirota Equations in Some Problems of Complex Analysis

We study the integrable structure recently revealed in some classical problems in the theory of functions in one complex variable. Given a simply connected domain bounded by a simple analytic curve in the complex plane, we consider the conformal mapping problem, the Dirichlet boundary problem, and the 2D inverse potential problem associated with the domain. A remarkable family of real-valued functionals on the space of such domains is constructed. Regarded as a function of infinitely many variables, which are properly defined moments of the domain, any functional in the family gives a formal solution of the above problems. These functions satisfy an infinite set of dispersionless Hirota equations and are therefore tau-functions of an integrable hierarchy. The hierarchy is identified with the dispersionless limit of the 2D Toda chain. In addition to our previous studies, we show that within a more general definition of the moments, this connection pertains not to a particular solution of the Hirota equations but to the hierarchy itself.     

On a new generalization of dispersionless Kadomtsev–Petviashvili hierarchy and its reductions

The dispersionless Kadomtsev–Petviashvili hierarchy is generalized by introducing two new time series γ_n and σ_k with two parameters η_n and λ_k. By this hierarchy, we obtain the first type, the second type as well as mixed type of dispersionless Kadomtsev–Petviashvili equation with self‐consistent sources and their related conservation equations. In addition, the reduction and constrained flow of this new hierarchy are studied. The first type, the second type and the mixed type of dispersionless Korteweg–de Vries equation with self‐consistent sources and of dispersionless Boussinesq equation with self‐consistent sources are obtained. Copyright © 2013 John Wiley & Sons, Ltd.     

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.     

Classifying Integrable Egoroff Hydrodynamic Chains

We introduce the notion of Egoroff hydrodynamic chains. We show how they are related to integrable (2+1)-dimensional equations of hydrodynamic type. We classify these equations in the simplest case. We find (2+1)-dimensional equations that are not just generalizations of the already known Khokhlov–Zabolotskaya and Boyer–Finley equations but are much more involved. These equations are parameterized by theta functions and by solutions of the Chazy equations. We obtain analogues of the dispersionless Hirota equations.     

N=(1|1) Supersymmetric Dispersionless Toda Lattice Hierarchy

Generalizing the graded commutator in superalgebras, we propose a new bracket operation on the space of graded operators with an involution. We study properties of this operation and show that the Lax representation of the two-dimensional N=(1|1) supersymmetric Toda lattice hierarchy can be realized via the generalized bracket operation; this is important in constructing the semiclassical (continuum) limit of this hierarchy. We construct the continuum limit of the N=(1|1) Toda lattice hierarchy, the dispersionless N=(1|1) Toda hierarchy. In this limit, we obtain the Lax representation, with the generalized graded bracket becoming the corresponding Poisson bracket on the graded phase superspace. We find bosonic symmetries of the dispersionless N=(1|1) supersymmetric Toda equation.     

Additional symmetries and Bäcklund transformations for the dispersionless Harry Dym hierarchy

We give a dispersionless Toda-like extension to the dispersionless Harry Dym (dDym) hierarchy. The extended dDym (EdDym) hierarchy has a dressing formulation, and its underlying solution structure can be investigated through the twistor construction. We show that additional symmetries of the solution space generate Backlund transformations of the EdDym hierarchy. We present some examples of constructing new solutions of the (2+1)-dimensional dDym and EdDym equations via Bäcklund transformations.     

Bilinear identities for an extended B-type Kadomtsev–Petviashvili hierarchy

We construct bilinear identities for wave functions of an extended B-type Kadomtsev–Petviashvili (BKP) hierarchy containing two types of (2+1)-dimensional Sawada–Kotera equations with a self-consistent source. Introducing an auxiliary variable corresponding to the extended flow for the BKP hierarchy, we find the τ -function and bilinear identities for this extended BKP hierarchy. The bilinear identities generate all the Hirota bilinear equations for the zero-curvature forms of this extended BKP hierarchy. As examples, we obtain the Hirota bilinear equations for the two types of (2+1)-dimensional Sawada–Kotera equations in explicit form.     

On Associativity Equations

We consider the associativity or Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations and discuss their solution class based on the existence of the residue formulas, which is most relevant for nonperturbative physics. We demonstrate that for this case, proving the associativity equations reduces to solving a system of linear algebraic equations. Particular examples of solutions related to Landau–Ginzburg topological theories, Seiberg–Witten theories, and the tau functions of semiclassical hierarchies are discussed in detail. We also discuss related questions including the covariance of associativity equations, their relation to dispersionless Hirota relations, and the auxiliary linear problem for the WDVV equations.     

From the Real and Complex Coupled Dispersionless Equations to the Real and Complex Short Pulse Equations

In the present paper, we study the real and complex coupled dispersionless (CD) equations, the real and complex short pulse (SP) equations geometrically and algebraically. From the geometric point of view, we first establish the link of the motions of space curves to the real and complex CD equations, then to the real and complex SP equations via hodograph transformations. The integrability of these equations are confirmed by constructing their Lax pairs geometrically. In the second part of the paper, it is made clear for the connection between the real and complex CD and SP equations and the two‐component extended Kadomtsew‐Petviashvili (KP) hierarchy. As a by‐product, the N‐soliton solutions in the form of determinants for these equations are provided.     

Integrable Equations in Nonlinear Geometrical Optics

Geometrical optics limit of the Maxwell equations for nonlinear media with the Cole–Cole dependence of dielectric function and magnetic permeability on the frequency is considered. It is shown that for media with slow variation along one axis such a limit gives rise to the dispersionless Veselov–Novikov equation for the refractive index. It is demonstrated that the Veselov–Novikov hierarchy is amenable to the quasiclassical     -dressing method. Under more specific requirements for the media, one gets the dispersionless Kadomtsev–Petviashvili equation. Geometrical optics interpretation of some solutions of the above equations is discussed.     

A generalization of the coupled integrable dispersionless equations

The paper investigates an extension of the coupled integrable dispersionless equations, which describe the current‐fed string within an external magnetic field. By using the relation among the coupled integrable dispersionless equations, the sine‐Gordon equation and the two‐dimensional Toda lattice equation, we propose a generalized coupled integrable dispersionless system. N‐soliton solutions to the generalized system are presented in the Casorati determinant form with arbitrary parameters. By choosing real or complex parameters in the Casorati determinant, the properties of one‐soliton and two‐soliton solutions are investigated. It is shown that we can obtain solutions in soliton profile and breather profile. Copyright © 2016 John Wiley & Sons, Ltd.     

Infinite-dimensional Frobenius manifolds for 2 + 1 integrable systems

We introduce a structure of an infinite-dimensional Frobenius manifold on a subspace in the space of pairs of functions analytic inside/outside the unit circle with simple poles at 0/∞, respectively. The dispersionless 2D Toda equations are embedded into a bigger integrable hierarchy associated with this Frobenius manifold.     

Equivalence of formulations of the MKP hierarchy and its polynomial tau-functions

We show that a system of Hirota bilinear equations introduced by Jimbo and Miwa defines tau-functions of the modified KP (MKP) hierarchy of evolution equations introduced by Dickey. Some other equivalent definitions of the MKP hierarchy are established. All polynomial tau-functions of the KP and the MKP hierarchies are found. Similar results are obtained for the reduced KP and MKP hierarchies.     

The master T-operator for vertex models with trigonometric R-matrices as a classical τ-function

We apply the recently proposed construction of the master T-operator to integrable vertex models and the associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. It also turns out to be the τ-function of an integrable hierarchy of classical soliton equations in the sense that it satisfies the same bilinear Hirota equations. We characterize the class of solutions of the Hirota equations that correspond to eigenvalues of the master T-operator and discuss its relation to the classical Ruijsenaars-Schneider system of particles.     

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.