666 research outputs found
A Class of Coupled KdV systems and Their Bi-Hamiltonian Formulations
A Hamiltonian pair with arbitrary constants is proposed and thus a sort of
hereditary operators is resulted. All the corresponding systems of evolution
equations possess local bi-Hamiltonian formulation and a special choice of the
systems leads to the KdV hierarchy. Illustrative examples are given.Comment: 8 pages, late
The constrained modified KP hierarchy and the generalized Miura transformations
In this letter, we consider the second Hamiltonian structure of the
constrained modified KP hierarchy. After mapping the Lax operator to a pure
differential operator the second structure becomes the sum of the second and
the third Gelfand-Dickey brackets defined by this differential operator. We
simplify this Hamiltonian structure by factorizing the Lax operator into linear
terms.Comment: 8 pages, latex, no figure
Why the general Zakharov-Shabat equations form a hierarchy?
The totality of all Zakharov-Shabat equations (ZS), i.e., zero-curvature
equations with rational dependence on a spectral parameter, if properly
defined, can be considered as a hierarchy. The latter means a collection of
commuting vector fields in the same phase space. Further properties of the
hierarchy are discussed, such as additional symmetries, an analogue to the
string equation, a Grassmannian related to the ZS hierarchy, and a Grassmannian
definition of soliton solutions.Comment: 13p
On the constrained KP hierarchy
An explanation for the so-called constrained hierarhies is presented by
linking them with the symmetries of the KP hierarchy. While the existence of
ordinary symmetries (belonging to the hierarchy) allows one to reduce the KP
hierarchy to the KdV hierarchies, the existence of additional symmetries allows
to reduce KP to the constrained KP.Comment: 7pp, LaTe
Hamiltonian Structures for the Generalized Dispersionless KdV Hierarchy
We study from a Hamiltonian point of view the generalized dispersionless KdV
hierarchy of equations. From the so called dispersionless Lax representation of
these equations we obtain three compatible Hamiltonian structures. The second
and third Hamiltonian structures are calculated directly from the r-matrix
approach. Since the third structure is not related recursively with the first
two ones the generalized dispersionless KdV hierarchy can be characterized as a
truly tri-Hamiltonian system.Comment: 16 pages, plain Te
Reduction of bihamiltonian systems and separation of variables: an example from the Boussinesq hierarchy
We discuss the Boussinesq system with stationary, within a general
framework for the analysis of stationary flows of n-Gel'fand-Dickey
hierarchies. We show how a careful use of its bihamiltonian structure can be
used to provide a set of separation coordinates for the corresponding
Hamilton--Jacobi equations.Comment: 20 pages, LaTeX2e, report to NEEDS in Leeds (1998), to be published
in Theor. Math. Phy
-analogue of modified KP hierarchy and its quasi-classical limit
A -analogue of the tau function of the modified KP hierarchy is defined by
a change of independent variables. This tau function satisfies a system of
bilinear -difference equations. These bilinear equations are translated to
the language of wave functions, which turn out to satisfy a system of linear
-difference equations. These linear -difference equations are used to
formulate the Lax formalism and the description of quasi-classical limit. These
results can be generalized to a -analogue of the Toda hierarchy. The results
on the -analogue of the Toda hierarchy might have an application to the
random partition calculus in gauge theories and topological strings.Comment: latex2e, a4 paper 15 pages, no figure; (v2) a few references are
adde
Dressing Technique for Intermediate Hierarchies
A generalized AKNS systems introduced and discussed recently in \cite{dGHM}
are considered. It was shown that the dressing technique both in matrix
pseudo-differential operators and formal series with respect to the spectral
parameter can be developed for these hierarchies.Comment: 16 pages, LaTeX Report/no: DFTUZ/94/2
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