Related papers: The Multidimensional Darboux Transformation
Algebraic Bargmann and Darboux transformations for equations of a more general form than the Schr\"odinger ones with an additional functional dependence h(r) in the right-hand side of equations are constructed. The suggested generalized…
The notion of a generalized harmonic inverse mean curvature surface in the Euclidean four-space is introduced. A backward B\"{a}cklund transform of a generalized harmonic inverse mean curvature surface is defined. A Darboux transform of a…
A detailed analysis of matrix Darboux transformations under the condition that the derivative of the superpotential be self-adjoint is given. As a onsequence, a class of the symmetries associated to Schr\"odinger matrix Hamiltonians is…
The Darbroux transformation is generalized for time-dependent Hamiltonian systems which include a term linear in momentum and a time-dependent mass. The formalism for the $N$-fold application of the transformation is also established, and…
We apply the Darboux transformation to construct new exactly-solvable cases of the two-dimensional massless Dirac equation for potential classes of Lambert-W and inverse exponential type. Both of these classes originate from the Heun…
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…
A new form of a binary Darboux transformation is used to generate analytical solutions of a nonlinear Liouville-von Neumann equation. General theory is illustrated by explicit examples.
In this paper we implement the Darboux transformation, as well as an analogue of Crum's theorem, for a discrete version of Schr\"odinger equation. The technique is based on the use of first order operators intertwining two difference…
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…
The generalized Moutard transformation of the stationary axially symmetric Schr\"odinger equation is considered. It is shown that a superposition of two Moutard transformations can provide new potentials for the eigenvalue problem. Examples…
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…
By means of certain limit technique, two kinds of generalized Darboux transformations are constructed for the derivative nonlinear Sch\"odinger equation (DNLS). These transformations are shown to lead to two solution formulas for DNLS in…
Euler-Darboux-Backlund and Laplace transformations are considered for the one- and two-dimensional Schrodinger operators. Their discrete analogs are constructed and generalized for the multidimensional lattices and two-manifolds with…
In this paper the Singular Manifold Method has allowed us to obtain the Lax pair, Darboux transformations and tau functions for a non-linear Schr\"odiger equation in 2+1 dimensions. In this way we can iteratively build different kind of…
Multivariate orthogonal polynomials in $D$ real dimensions are considered from the perspective of the Cholesky factorization of a moment matrix. The approach allows for the construction of corresponding multivariate orthogonal polynomials,…
The technique of differential intertwining operators (or Darboux transformation operators) is systematically applied to the one-dimensional Dirac equation. The following aspects are investigated: factorization of a polynomial of Dirac…
A one-fold Darboux transformation between solutions of the semi-discrete massive Thirring model is derived using the Lax pair and dressing methods. This transformation is used to find the exact expressions for soliton solutions on zero and…
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…
We study discrete isospectral symmetries for the classical acoustic spectral problem in spatial dimensions one and two, by developing a Darboux (Moutard) transformation formalism for this problem. The procedure follows the steps, similar to…
Using the method of Darboux transformations (or equivalently supersymmetric quantum mechanics) we obtain an explicit expression for the propagator for the one-dimensional Schr\"odinger equation with a multi-soliton potential.