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     

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.     

Quasiconformal Mappings and Solutions of the Dispersionless KP Hierarchy

A 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.     

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.     

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.     

Contact integrable extensions and differential coverings for the generalized (2 + 1)-dimensional dispersionless Dym equation

The method of contact integrable extensions is used to find new differential coverings for the generalized (2 + 1)-dimensional dispersionless Dym equation and corresponding Bäcklund transformations.     

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.     

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.     

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.     

Darboux transformation and N-soliton solutions for a more general set of coupled integrable dispersionless system

The Darboux transformation and Lax pair of a more general set of coupled integrable dispersionless system are derived. By Darboux transformation, N-soliton solutions for the coupled integrable dispersionless system are obtained. In particular, the multi-soliton solutions are shown through some figures.     

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.     

On the isospectral problem of the dispersionless Camassa–Holm equation

We discuss direct and inverse spectral theory for the isospectral problem of the dispersionless Camassa–Holm equation, where the weight is allowed to be a finite signed measure. In particular, we prove that this weight is uniquely determined by the spectral data and solve the inverse spectral problem for the class of measures which are sign definite. The results are applied to deduce several facts for the dispersionless Camassa–Holm equation. In particular, we show that initial conditions with integrable momentum asymptotically split into a sum of peakons as conjectured by McKean.     

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.     

Semiclassical $$\bar \partial $$ -Method: Generating Equations for Dispersionless Integrable Hierarchies

We use the semiclassical -dressing method to derive compact generating equations for dispersionless hierarchies. The considered illustrative examples are the dispersionless Kadomtsev–Petviashvili and two-dimensional Toda lattice hierarchies.     

Elliptic parameterization of Pfaff integrable hierarchies in the zero-dispersion limit

We show that the dispersionless limits of the Pfaff–KP (also known as the DKP or Pfaff lattice) and the Pfaff–Toda hierarchies admit a reformulation in terms of elliptic functions. In the elliptic form, they look like natural elliptic deformations of the respective dispersionless KP and two-dimensional Toda hierarchies.     

Transformations of integrable hydrodynamic chains and their hydrodynamic reductions

Hydrodynamic reductions of the hydrodynamic chain associated with the dispersionless limit of the 2+1-dimensional Harry-Dym equation are found by using the Miura type and reciprocal transformations applied to the Benney hydrodynamic chain. __________ Translated from Fundamentalnaya i Prikladnaya Matematika (Fundamental and Applied Mathematics), Vol. 12, No. 7, pp. 167–175, 2006.     

Dbar-穿衣法和一个耦合无色散方程

利用基于2×2矩阵（e）（Dbar）-问题的推广穿衣法,研究了一个耦合无色散方程,进而利用Cauchy矩阵的性质导出其孤立子解.此外,还讨论了N-孤立子解的渐近行为.     

Light Propagation in a Cole-Cole Nonlinear Medium via the Burgers-Hopf Equation

A new model of light propagation through a so-called weakly three-dimensional Cole-Cole nonlinear medium with short-range nonlocality was recently proposed. In particular, it was shown that in the geometric optics limit, the model is integrable and is governed by the dispersionless Veselov-Novikov (dVN) equation. The Burgers-Hopf equation can be obtained as a (1+1)-dimensional reduction of the dVN equation. We discuss its properties in the specific context of nonlinear geometric optics and consider an illustrative explicit example.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 1, pp. 102–109, July, 2005.     

Nonlinear Dynamics on the Plane and Integrable Hierarchies of Infinitesimal Deformations

A class of nonlinear problems on the plane, described by nonlinear inhomogeneous     -equations, is considered. It is shown that the corresponding dynamics, generated by deformations of inhomogeneous terms (sources), is described by Hamilton–Jacobi-type equations associated with hierarchies of dispersionless integrable systems. These hierarchies are constructed by applying the quasiclassical     -dressing method.