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Differential Fay identities and auxiliary linear problem of integrable hiearchies  [PDF]
Kanehisa Takasaki
Physics , 2007,
Abstract: We review the notion of differential Fay identities and demonstrate, through case studies, its new role in integrable hierarchies of the KP type. These identities are known to be a convenient tool for deriving dispersionless Hirota equations. We show that differential (or, in the case of the Toda hierarchy, difference) Fay identities play a more fundamental role. Namely, they are nothing but a generating functional expression of the full set of auxiliary linear equations, hence substantially equivalent to the integrable hierarchies themselves. These results are illustrated for the KP, Toda, BKP and DKP hierarchies. As a byproduct, we point out some new features of the DKP hierarchy and its dispersionless limit.
On a generalization of the Fay-Sato identity for KP Baker functions and its application to constrained hierarchies  [PDF]
L. A. Dickey,W. Strampp
Physics , 1996,
Abstract: Some new formulas for the KP hierarchy are derived from the differential Fay identity. They proved to be useful for the $k$-constrained hierarchies providing a series of determinant identities for them. A differential equation is introduced which is called ``universal" since it plays an important role for all the $k$-constrained hierarchies. In the cases $k=1,2$ and 3 explicit formulas are presented, in all the others recurrence relations are given which enable one to obtain the identities.
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy
Kanehisa Takasaki
Symmetry, Integrability and Geometry : Methods and Applications , 2009,
Abstract: Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as ''the coupled KP hierarchy'' and ''the Pfaff lattice''). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called ''the Pfaff-Toda hierarchy''). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.
Auxiliary Linear Problem, Difference Fay Identities and Dispersionless Limit of Pfaff-Toda Hierarchy  [PDF]
Kanehisa Takasaki
Physics , 2009, DOI: 10.3842/SIGMA.2009.109
Abstract: Recently the study of Fay-type identities revealed some new features of the DKP hierarchy (also known as "the coupled KP hierarchy" and "the Pfaff lattice"). Those results are now extended to a Toda version of the DKP hierarchy (tentatively called "the Pfaff-Toda hierarchy"). Firstly, an auxiliary linear problem of this hierarchy is constructed. Unlike the case of the DKP hierarchy, building blocks of the auxiliary linear problem are difference operators. A set of evolution equations for dressing operators of the wave functions are also obtained. Secondly, a system of Fay-like identities (difference Fay identities) are derived. They give a generating functional expression of auxiliary linear equations. Thirdly, these difference Fay identities have well defined dispersionless limit (dispersionless Hirota equations). As in the case of the DKP hierarchy, an elliptic curve is hidden in these dispersionless Hirota equations. This curve is a kind of spectral curve, whose defining equation is identified with the characteristic equation of a subset of all auxiliary linear equations. The other auxiliary linear equations are related to quasi-classical deformations of this elliptic spectral curve.
Fay-like identities of the Toda Lattice Hierarchy and its dispersionless limit  [PDF]
Lee-Peng Teo
Physics , 2006, DOI: 10.1142/S0129055X06002838
Abstract: In this paper, we derive the Fay-like identities of tau function for the Toda lattice hierarchy from the bilinear identity. We prove that the Fay-like identities are equivalent to the hierarchy. We also show that the dispersionless limit of the Fay-like identities are the dispersionless Hirota equations of the dispersionless Toda hierarchy.
On rational solutions of multicomponent and matrix KP hierarchies  [PDF]
Alberto Tacchella
Physics , 2010, DOI: 10.1016/j.geomphys.2011.02.007
Abstract: We derive some rational solutions for the multicomponent and matrix KP hierarchies generalising an approach by Wilson. Connections with the multicomponent version of the KP/CM correspondence are discussed.
The KP Hierarchy in Miwa Coordinates  [PDF]
Boris Konopelchenko,Luis Martinez Alonso
Physics , 1999, DOI: 10.1016/S0375-9601(99)00373-4
Abstract: A systematic reformulation of the KP hierarchy by using continuous Miwa variables is presented. Basic quantities and relations are defined and determinantal expressions for Fay's identities are obtained. It is shown that in terms of these variables the KP hierarchy gives rise to a Darboux system describing an infinite-dimensional conjugate net.
Soliton Fay identities. I. Dark soliton case  [PDF]
V. E. Vekslerchik
Physics , 2014, DOI: 10.1088/1751-8113/47/41/415202
Abstract: We derive a set of bilinear identities for the determinants of the matrices that have been used to construct the dark soliton solutions for various models. To give examples of the application of the obtained identities we present soliton solutions for the equations describing multidimensional quadrilateral lattices, Darboux equations and multidimensional multicomponent systems of the nonlinear Schrodinger type.
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System  [cached]
Aristophanes Dimakis,Folkert Müller-Hoissen
Symmetry, Integrability and Geometry : Methods and Applications , 2009,
Abstract: Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more general relation between the multicomponent linear heat hierarchy and the multicomponent KP hierarchy. From this results a construction of exact solutions of the latter via a matrix linear system.
Multicomponent Burgers and KP Hierarchies, and Solutions from a Matrix Linear System  [PDF]
Aristophanes Dimakis,Folkert Müller-Hoissen
Physics , 2008, DOI: 10.3842/SIGMA.2009.002
Abstract: Via a Cole-Hopf transformation, the multicomponent linear heat hierarchy leads to a multicomponent Burgers hierarchy. We show in particular that any solution of the latter also solves a corresponding multicomponent (potential) KP hierarchy. A generalization of the Cole-Hopf transformation leads to a more general relation between the multicomponent linear heat hierarchy and the multicomponent KP hierarchy. From this results a construction of exact solutions of the latter via a matrix linear system.
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