Related papers: Backlund Transformations for Darboux Integrable Di…
We define B\"acklund--Darboux transformations in Sato's Grassmannian. They can be regarded as Darboux transformations on maximal algebras of commuting ordinary differential operators. We describe the action of these transformations on…
B\"acklund transformations (BTs) are traditionally regarded as a tool for integrating nonlinear partial differential equations (PDEs). Their use has been recently extended, however, to problems such as the construction of recursion…
The construction of generalized Backlund transformation for the $A_n$ Affine Toda hierarchy is proposed in terms of gauge transformation acting on the zero curvature representation. Such construction is based upon the graded structure of…
We present an approach to higher dimensional Darboux transformations suitable for application to quantum integrable systems and based on the bispectral property of partial differential operators. Specifically, working with the…
General Lagrangian theory of discrete one-dimensional integrable systems is illustrated by a detailed study of B\"acklund transformations for Toda-type systems. Commutativity of B\"acklund transformations is shown to be equivalent to…
The KdV eigenfunction equation is considered: some explicit solutions are constructed. These, to the best of the authors' knowledge, new solutions represent an example of the powerfulness of the method devised. Specifically, B\"acklund…
Baecklund-Darboux transformations are closely related to the integrability and symmetry problems. For the generalized Baecklund-Darboux transformation (GBDT), we consider conservation laws, rational extensions and bispectrality. We use the…
Darboux transformation plays a key role in constructing explicit closed-form solutions of completely integrable systems. This paper provides an algebraic construction of generalized Darboux matrices with the same poles for the $2\times2$…
A new approach for obtaining the transformations of solutions of nonlinear ordinary differential equations representable as the compatibility condition of the overdetermined linear systems is proposed. The corresponding transformations of…
We construct a local action of the group of rational maps from $S^2$ to $GL(n,C)$ on local solutions of flows of the ZS-AKNS $sl(n,C)$-hierarchy. We show that the actions of simple elements (linear fractional transformations) give local…
The Darboux transformation between ordinary differential equations is a 19th century technique that has seen wide use in quantum theory for producing exactly solvable potentials for the Schr\"odinger equation with specific spectral…
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…
Starting from nonlocal symmetries related to B\"acklund transformation (BT), many interesting results can be obtained. Taking the well known potential KdV (pKdV) equation as an example, a new type of nonlocal symmetry in elegant and compact…
In this paper, we construct a Darboux transformation and the related B\"acklund transformation for the supersymmetric Sawada-Kotera (SSK) equation. The associated nonlinear superposition formula is also worked out. We demonstrate that these…
A couple of applications of B\"acklund transformations in the study of nonlinear evolution equations is here given. Specifically, we are concerned about third order nonlinear evolution equations. Our attention is focussed on one side, on…
The construction of type II Backlund transformation for the sine-Gordon and the Tzitzeica-Bullough-Dodd models are obtained from gauge transformation. An infinite number of conserved quantities are constructed from the defect matrices. This…
The note offers a proof of Darboux and Liouville theorems from a symplectic group action perspective.
The derivation of the Backlund transformations (BTs) is a standard problem of the theory of the integrable systems. Here, I discuss the equations describing the BTs for the Ablowitz-Ladik hierarchy (ALH), which have been already obtained by…
A bicomplex is a simple mathematical structure, in particular associated with completely integrable models. The conditions defining a bicomplex are a special form of a parameter-dependent zero curvature condition. We generalize the concept…
We present a non-isospectral GBDT version of B\"acklund-Darboux transformation for the gravitational and $\sigma$-model equations. New families of explicit solutions correspond to the case of GBDT with non-diagonal generalized matrix…