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Coupled Modified KP Hierarchy and Its Dispersionless Limit  [cached]
Takashi Takebe,Lee-Peng Teo
Symmetry, Integrability and Geometry : Methods and Applications , 2006,
Abstract: We define the coupled modified KP hierarchy and its dispersionless limit. This integrable hierarchy is a generalization of the ''half'' of the Toda lattice hierarchy as well as an extension of the mKP hierarchy. The solutions are parametrized by a fibered flag manifold. The dispersionless counterpart interpolates several versions of dispersionless mKP hierarchy.
Coupled Modified KP Hierarchy and Its Dispersionless Limit  [PDF]
Takashi Takebe,Lee-Peng Teo
Physics , 2006, DOI: 10.3842/SIGMA.2006.072
Abstract: We define the coupled modified KP hierarchy and its dispersionless limit. This integrable hierarchy is a generalization of the ''half'' of the Toda lattice hierarchy as well as an extension of the mKP hierarchy. The solutions are parametrized by a fibered flag manifold. The dispersionless counterpart interpolates several versions of dispersionless mKP hierarchy.
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.
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.
Extended discrete KP hierarchy and its dispersionless limit  [PDF]
Andrei K. Svinin
Physics , 2004,
Abstract: We exhibit the dispersionless limit of the extended discrete KP hierarchy.
Poisson Algebras associated with Constrained Dispersionless Modified KP Hierarchies  [PDF]
Jen-Hsu Chang,Ming-Hsien Tu
Physics , 2000, DOI: 10.1063/1.1322080
Abstract: We investigate the bi-Hamiltonian structures associated with constrained dispersionless modified KP hierarchies which are constructed from truncations of the Lax operator of the dispersionless modified KP hierarchy. After transforming their second Hamiltonian structures to those of Gelfand-Dickey type, we obtain the Poisson algebras of the coefficient functions of the truncated Lax operators. Then we study the conformal property and free-field realizations of these Poisson algebras. Some examples are worked out explicitly to illustrate the obtained results.
Dispersionless limit of the noncommutative potential KP hierarchy and solutions of the pseudodual chiral model in 2+1 dimensions  [PDF]
Aristophanes Dimakis,Folkert Muller-Hoissen
Physics , 2007, DOI: 10.1088/1751-8113/41/26/265205
Abstract: 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.
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.
The Moyal bracket and the dispersionless limit of the KP hierarchy  [PDF]
I. A. B. Strachan
Physics , 1994, DOI: 10.1088/0305-4470/28/7/018
Abstract: A new Lax equation is introduced for the KP hierarchy which avoids the use of pseudo-differential operators, as used in the Sato approach. This Lax equation is closer to that used in the study of the dispersionless KP hierarchy, and is obtained by replacing the Poisson bracket with the Moyal bracket. The dispersionless limit, underwhich the Moyal bracket collapses to the Poisson bracket, is particularly simple.
S-functions, reductions and hodograph solutions of the r-th dispersionless modified KP and Dym hierarchies  [PDF]
Manuel Manas
Physics , 2004, DOI: 10.1088/0305-4470/37/46/007
Abstract: 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.
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