Related papers: Dressing the Dressing Chain
A chain of one-dimensional Schr\"odinger operators connected by successive Darboux transformations is called the ``Darboux chain'' or ``dressing chain''. The periodic dressing chain with period $N$ has a control parameter $\alpha$. If…
The iterations are studied of the Darboux transformation for the generalized Schroedinger operator. The applications to the Dym and Camassa-Holm equations are considered.
The relation between the Darboux transformation and the solutions of the full Kostant Toda lattice is analyzed. The discrete Korteweg de Vries equation is used to obtain such solutions and the main result of [1] is extended to the case of…
This article addresses the study of the complex version of the modified Korteweg-de Vries equation using two different approaches. Firstly, the singular manifold method is applied in order to obtain the associated spectral problem, binary…
For the supersymmetric KdV equation, a proper Darboux transformation is presented. This Darboux transformation leads to the B\"{a}cklund transformation found early by Liu and Xie \cite{liu2}. The Darboux transformation and the related…
\hspace{.2in}We consider the Darboux type transformations for the spectral problems of supersymmetric KdV systems. The supersymmetric analogies of Darboux and Darboux-Levi transformations are established for the spectral problems of…
We solve the direct scattering problem for the ultradiscrete Korteweg de Vries (udKdV) equation, over $\mathbb R$ for any potential with compact (finite) support, by explicitly constructing bound state and non-bound state eigenfunctions. We…
We propose a method for construction of Darboux transformations, which is a new development of the dressing method for Lax operators invariant under a reduction group. We apply the method to the vector sine-Gordon equation and derive its…
In this paper we present a vectorial Darboux transformation, in terms of ordinary determinants, for the supersymmetric extension of the Korteweg-de Vries equation proposed by Manin and Radul. It is shown how this transformation reduces to…
We present the hierarchy and soliton solutions associated to a multi-component generalisation of the modified Korteweg-de Vries equation. A recursive formula for obtaining the Lax operators associated to the higher flows of the hierarchy is…
In this paper, we consider a supersymmetric AKNS spectral problem. Two elementary and a binary Darboux transformations are constructed. By means of reductions, Darboux and B\"acklund transformations are given for the supersymmetric modified…
The dressing chain equations for the second order Sturm-Liouville differential operators is integrated. The simplest closure at the third step is linked to the spectral curve of the genus 1 and the explicit solution in elliptic Weierstrass…
To describe two-place events, Alice-Bob systems have been established by means of the shifted parity and delayed time reversal in Ref. [1]. In this paper, we mainly study exact solutions of the integrable Alice-Bob modified Korteweg…
By introducing a Miura transformation, we derive a generalized super modified Korteweg-de Vries (gsmKdV) equation from the generalized super KdV (gsKdV) equation. It is demonstrated that, while the gsKdV equation takes Kupershmidt's super…
We identify the self-similarity limit of the second flow of $sl(N)$ mKdV hierarchy with the periodic dressing chain thus establishing % a connection to $A^{(1)}_{N-1}$ invariant Painlev\'e equations. The $A^{(1)}_{N-1}$ B\"acklund…
In the KdV context we put forward a continuous version of the binary Darboux transformation (aka the double commutation method). Our approach is based on the Riemann-Hilbert problem and yields a new explicit formula for perturbation of the…
The $N=2 \;a=-2$ supersymmetric KdV equation is studied. A Darboux transformation and the corresponding B\"acklund transformation are constructed for this equation. Also, a nonlinear superposition formula is worked out for the associated…
We apply the method of dressing chains to reproduction of Toda lattice in the case of D=1 and D=2. On the example of modified equations $m_0TL$ and $m_1TL$ it is shown that combination of the Darboux and Schlesinger transformations results…
The action of a B\"acklund-Darboux transformation on a spectral problem associated with a known integrable system can define a new discrete spectral problem. In this paper, we interpret a slightly generalized version of the binary…
In this paper, we develop discrete versions of Darboux transformations and Crum's theorems for two second order difference equations. The difference equations are discretised versions (using Darboux transformations) of the spectral problems…