1,354 research outputs found
On Darboux transformations for the derivative nonlinear Schr\"odinger equation
We consider Darboux transformations for the derivative nonlinear
Schr\"odinger equation. A new theorem for Darboux transformations of operators
with no derivative term are presented and proved. The solution is expressed in
quasideterminant forms. Additionally, the parabolic and soliton solutions of
the derivative nonlinear Schr\"odinger equation are given as explicit examples.Comment: 14 page
On solutions to the non-Abelian Hirota-Miwa equation and its continuum limits
In this paper, we construct grammian-like quasideterminant solutions of a
non-Abelian Hirota-Miwa equation. Through continuum limits of this non-Abelian
Hirota-Miwa equation and its quasideterminant solutions, we construct a cascade
of noncommutative differential-difference equations ending with the
noncommutative KP equation. For each of these systems the quasideterminant
solutions are constructed as well.Comment: 9 pages, 1 figur
Quasideterminant solutions of a non-Abelian Hirota-Miwa equation
A non-Abelian version of the Hirota-Miwa equation is considered. In an
earlier paper [Nimmo (2006) J. Phys. A: Math. Gen. \textbf{39}, 5053-5065] it
was shown how solutions expressed as quasideterminants could be constructed for
this system by means of Darboux transformations. In this paper we discuss these
solutions from a different perspective and show that the solutions are
quasi-Pl\"{u}cker coordinates and that the non-Abelian Hirota-Miwa equation may
be written as a quasi-Pl\"{u}cker relation. The special case of the matrix
Hirota-Miwa equation is also considered using a more traditional, bilinear
approach and the techniques are compared
Yang-Baxter Maps from the Discrete BKP Equation
We construct rational and piecewise-linear Yang-Baxter maps for a general
N-reduction of the discrete BKP equation
Darboux dressing and undressing for the ultradiscrete KdV equation
We solve the direct scattering problem for the ultradiscrete Korteweg de
Vries (udKdV) equation, over for any potential with compact
(finite) support, by explicitly constructing bound state and non-bound state
eigenfunctions. We then show how to reconstruct the potential in the scattering
problem at any time, using an ultradiscrete analogue of a Darboux
transformation. This is achieved by obtaining data uniquely characterising the
soliton content and the `background' from the initial potential by Darboux
transformation.Comment: 41 pages, 5 figures // Full, unabridged version, including two
appendice
Darboux and binary Darboux transformations for discrete integrable systems 1. Discrete potential KdV equation
The Hirota-Miwa equation can be written in `nonlinear' form in two ways: the
discrete KP equation and, by using a compatible continuous variable, the
discrete potential KP equation. For both systems, we consider the Darboux and
binary Darboux transformations, expressed in terms of the continuous variable,
and obtain exact solutions in Wronskian and Grammian form. We discuss
reductions of both systems to the discrete KdV and discrete potential KdV
equations, respectively, and exploit this connection to find the Darboux and
binary Darboux transformations and exact solutions of these equations
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