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Hydrodynamic Reductions of Dispersionless Harry Dym Hierarchy  [PDF]
Jen-Hsu Chang
Physics , 2005, DOI: 10.1088/0305-4470/38/29/006
Abstract: We investigate the reductions of dispersionless Harry Dym hierarchy to systems of finitely many partial differential equations. These equations must satisfy the compatibility condition and they are diagonalizable and semi-Hamiltonian. By imposing a further constraint, the compatibility is reduced to a system of algebraic equations, whose solutions are described.
On the r-th dispersionless Toda hierarchy I: Factorization problem, symmetries and some solutions  [PDF]
M. Manas
Physics , 2004, DOI: 10.1088/0305-4470/37/39/010
Abstract: For a family of Poisson algebras, parametrized by by an integer number r, and an associated Lie algebraic splitting, we consider the factorization of given canonical transformations. In this context we rederive the recently found r-th dispersionless modified KP hierachies and r-th dispersionless Dym hierarchies, giving a new Miura map among them. We also found a new integrable hierarchy which we call the r-th dispersionless Toda hierarchy. Moreover, additional symmetries for these hierarchies are studied in detail and new symmetries depending on arbitrary functions are explicitly constructed for the r-th dispersionless KP, r-th dispersionless Dym and r-th dispersionless Toda equations. Some solutions are derived by examining the imposition of a time invariance to the potential r-th dispersionless Dym equation, for which a complete integral is presented and therefore an appropriate envelope leads to a general solution. Symmetries and Miura maps are applied to get new solutions and solutions of the r-th dispesionless modified KP equation.
On dispersionless coupled modified KP hierarchy  [PDF]
Lee-Peng Teo
Physics , 2003,
Abstract: We define and study dispersionless coupled modified KP hierarchy, which incorporates two different versions of dispersionless modified KP hierarchies.
On the Miura map between the dispersionless KP and dispersionless modified KP hierarchies  [PDF]
Jen-Hsu Chang,Ming-Hsien Tu
Physics , 1999, DOI: 10.1063/1.533416
Abstract: We investigate the Miura map between the dispersionless KP and dispersionless modified KP hierarchies. We show that the Miura map is canonical with respect to their bi-Hamiltonian structures. Moreover, inspired by the works of Takasaki and Takebe, the twistor construction of solution structure for the dispersionless modified KP hierarchy is given.
On the Whitham Hierarchies: Reductions and Hodograph Solutions  [PDF]
Francisco Guil,Manuel Manas,Luis Martinez Alonso
Physics , 2002, DOI: 10.1088/0305-4470/36/14/309
Abstract: A general scheme for analyzing 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.
A note on the modified KP hierarchy and its (yet another) dispersionless limit  [PDF]
Takashi Takebe
Physics , 2001,
Abstract: The modified KP hierarchies of Kashiwara and Miwa is formulated in Lax formalism by Dickey. Their solutions are parametrised by flag varieties. Its dispersionless limit is considered.
dbar-approach to the dispersionless KP hierarchy  [PDF]
B. Konopelchenko,L. Martinez Alonso,O. Ragnisco
Physics , 2001,
Abstract: The dispersionless limit of the scalar nonlocal dbar-problem is derived. It is given by a special class of nonlinear first-order equations. A quasi-classical version of the dbar-dressing method is presented. It is shown that the algebraic formulation of dispersionless hierarchies can be expressed in terms of properties of Beltrami tupe equations. The universal Whitham hierarchy and, in particular, the dispersionless KP hierarchy turn out to be rings of symmetries for the quasi-classical dbar-problem.
Additional symmetries and solutions of the dispersionless KP hierarchy  [PDF]
Luis Martinez Alonso,Manuel Manas
Physics , 2003,
Abstract: 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.
Hodograph solutions of the dispersionless coupled KdV hierarchies, critical points and the Euler-Poisson-Darboux equation  [PDF]
B. Konopelchenko,L. Martinez Alonso,E. Medina
Physics , 2010, DOI: 10.1088/1751-8113/43/43/434020
Abstract: It is shown that the hodograph solutions of the dispersionless coupled KdV (dcKdV) hierarchies describe critical and degenerate critical points of a scalar function which obeys the Euler-Poisson-Darboux equation. Singular sectors of each dcKdV hierarchy are found to be described by solutions of higher genus dcKdV hierarchies. Concrete solutions exhibiting shock type singularities are presented.
Recursion operators for KP, mKP and Harry-Dym Hierarchies  [PDF]
Jipeng Cheng,Lihong Wang,Jingsong He
Physics , 2010, DOI: 10.1142/S1402925111001234
Abstract: In this paper, we give a unified construction of the recursion operators from the Lax representation for three integrable hierarchies: Kadomtsev-Petviashvili (KP), modified Kadomtsev-Petviashvili (mKP) and Harry-Dym under $n$-reduction. This shows a new inherent relationship between them. To illustrate our construction, the recursion operator are calculated explicitly for $2$-reduction and $3$-reduction.
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