Related papers: B\"acklund--Darboux transformations in Sato's Gras…
Painleve analysis and the singular manifold method are the tools used in this paper to perform a complete study of an equation in 2+1 dimensions. This procedure has allowed us to obtain the Lax pair, Darboux transformation and tau functions…
Generalized Euler-Arnold-von Neumann density matrix equations can be solved by a binary Darboux transformation given here in a new form: $\rho[1]=e^{P\ln(\mu/\nu)}\rho e^{-P\ln(\mu/\nu)}$ where $P=P^2$ is explicitly constructed in terms of…
We construct linear and quadratic Darboux matrices compatible with the reduction group of the Lax operator for each of the seven known non-Abelian derivative nonlinear Schr\"odinger equations that admit Lax representations. The…
We construct the Darboux-Backlund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix…
The technique of Darboux transformation is applied to nonlocal partner of two-dimensional periodic A_{n-1} Toda lattice. This system is shown to admit a representation as the compatibility conditions of direct and dual overdetermined linear…
The plane wave solutions of the three-wave resonant interaction in the plane are considered. It is shown that rank-one constraints over the right derivatives of invertible operators on an arbitrary linear space gives solutions of the…
We construct so called Darboux matrices and fundamental solutions in the important case of the generalised Hamiltonian (or canonical) systems depending rationally on the spectral parameter. A wide class of explicit solutions is obtained in…
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…
The Darboux process, also known by many other names, played a very important role in some extremely enjoyable joint work that Hans and I did 25 years ago. I revisit a version of this problem in a case when scalars are replaced by matrices,…
Using Cartan's Method of Equivalence, we prove an upper bound for the generality of generic rank-1 B\"acklund transformations relating two hyperbolic Monge-Amp\`ere systems. In cases when the B\"acklund transformation admits a symmetry…
Elementary Darboux--Laplace transformations for semidiscrete and discrete second order hyperbolic operators are classified. It is proved that in the (semi)-discrete case there are two types of elementary Darboux--Laplace transformations as…
The discrete Schr\"odinger equation on a half-line lattice with the Dirichlet boundary condition is considered when the potential is real valued, is summable, and has a finite first moment. The Darboux transformation formulas are derived…
We describe a class of algebraically solvable SUSY models by considering the deformation of invariant polynomial flags by means of the Darboux transformation. The algebraic deformations corresponding to the addition of a bound state to a…
For a class of Schrodinger Hamiltonians the supersymmetry transformations can degenerate to simple coordinate displacements. We examine this phenomenon and show that it distinguishes the Weierstrass potentials including the one-soliton…
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…
The Darboux transformations for the two dimensional elliptic affine Toda equations corresponding to all seven infinite series of affine Kac-Moody algebras, including $A_l^{(1)}$, $A_{2l}^{(2)}$, $A_{2l-1}^{(2)}$, $B_l^{(1)}$, $C_l^{(1)}$,…
The Darboux transformation on matrix solutions to the generalized coupled dispersionless integrable system based on some non-abelian Lie group, is studied and the solutions are shown to be expressed in terms of quasideterminants. As an…
In the first part of the paper we present the dressing method which generates multi-soliton solutions to integrable systems of nonlinear partial differential equations. We compare the approach of Neugebauer with that of Zakharov, Shabat and…
In this work we show that, under certain conditions, parametric Backlund transformations (BTs) for a finite dimensional integrable system can be interpreted as solutions to the equations of motion defined by an associated non-autonomous…
For the n-dimensional integrable system with a twisted so(p,q) reduction, Darboux transformations given by Darboux matrices of degree 2 are constructed explicitly. These Darboux transformations are applied to the local isometric immersion…