4,672 research outputs found
Darboux transformations for a twisted derivation and quasideterminant solutions to the super KdV equation
This paper is concerned with a generalized type of Darboux transformations
defined in terms of a twisted derivation satisfying
where is a homomorphism. Such twisted derivations include regular
derivations, difference and -difference operators and superderivatives as
special cases. Remarkably, the formulae for the iteration of Darboux
transformations are identical with those in the standard case of a regular
derivation and are expressed in terms of quasideterminants. As an example, we
revisit the Darboux transformations for the Manin-Radul super KdV equation,
studied in Q.P. Liu and M. Ma\~nas, Physics Letters B \textbf{396} 133--140,
(1997). The new approach we take enables us to derive a unified expression for
solution formulae in terms of quasideterminants, covering all cases at once,
rather than using several subcases. Then, by using a known relationship between
quasideterminants and superdeterminants, we obtain expressions for these
solutions as ratios of superdeterminants. This coincides with the results of
Liu and Ma\~nas in all the cases they considered but also deals with the one
subcase in which they did not obtain such an expression. Finally, we obtain
another type of quasideterminant solutions to the Main-Radul super KdV equation
constructed from its binary Darboux transformations. These can also be
expressed as ratios of superdeterminants and are a substantial generalization
of the solutions constructed using binary Darboux transformations in earlier
work on this topic
Integrable discrete autonomous quad-equations admitting, as generalized symmetries, known five-point differential-difference equations
In this paper we construct the autonomous quad-equations which admit as
symmetries the five-point differential-difference equations belonging to known
lists found by Garifullin, Yamilov and Levi. The obtained equations are
classified up to autonomous point transformations and some simple
non-autonomous transformations. We discuss our results in the framework of the
known literature. There are among them a few new examples of both sine-Gordon
and Liouville type equations.Comment: 27 page
Darboux Transformation for Supersymmetric KP Hierarchies
We construct Darboux transformations for the super-symmetric KP hierarchies
of Manin--Radul and Jacobian types. We also consider the binary Darboux
transformation for the hierarchies. The iterations of both type of Darboux
transformations are briefly discussed.Comment: 14 pages, LaTeX2e with amsmath,amssymb,amsthm and geometry packages.
In this new version we consider both the Manin-Radul and the Jacobian SKP
hierachies and we show how the elementary Darboux transformation composed
with a reversion of signs in the fermionic times constitute a proper
transformation of these hierarchie
Generalized Darboux transformations for the KP equation with self-consistent sources
The KP equation with self-consistent sources (KPESCS) is treated in the
framework of the constrained KP equation. This offers a natural way to obtain
the Lax representation for the KPESCS. Based on the conjugate Lax pairs, we
construct the generalized binary Darboux transformation with arbitrary
functions in time for the KPESCS which, in contrast with the binary Darboux
transformation of the KP equation, provides a non-auto-B\"{a}cklund
transformation between two KPESCSs with different degrees. The formula for
N-times repeated generalized binary Darboux transformation is proposed and
enables us to find the N-soliton solution and lump solution as well as some
other solutions of the KPESCS.Comment: 20 pages, no figure
Two binary Darboux transformations for the KdV hierarchy with self-consistent sources
Two binary (integral type) Darboux transformations for the KdV hierarchy with
self-consistent sources are proposed. In contrast with the Darboux
transformation for the KdV hierarchy, one of the two binary Darboux
transformations provides non auto-B\"{a}cklund transformation between two n-th
KdV equations with self-consistent sources with different degrees. The formula
for the m-times repeated binary Darboux transformations are presented. This
enables us to construct the N-soliton solution for the KdV hierarchy with
self-consistent sources.Comment: 19 pages, LaTeX, no figures, to be published in Journal of
Mathematical Physic
Darboux transformations for 5-point and 7-point self-adjoint schemes and an integrable discretization of the 2D Schrodinger operator
With this paper we begin an investigation of difference schemes that possess
Darboux transformations and can be regarded as natural discretizations of
elliptic partial differential equations. We construct, in particular, the
Darboux transformations for the general self adjoint schemes with five and
seven neighbouring points. We also introduce a distinguished discretization of
the two-dimensional stationary Schrodinger equation, described by a 5-point
difference scheme involving two potentials, which admits a Darboux
transformation.Comment: 15 pages, 1 figur
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