Related papers: Invertible Darboux Transformations
A well-known theorem factors a scalar coefficient differential operator given a linearly independent set of functions in its kernel. The goal of this paper is to generalize this useful result to other types of operators. In place of the…
We construct arbitrary-order Darboux transformations for Schroedinger equations with energy-dependent potential and position-dependent mass within the Dunkl formalism. Our construction is based on a point transformation that interrelates…
The purpose of this paper is to study Toda-Darboux transforms, i.e., Darboux transforms for operators L(t) flowing according to the Toda lattice. Each element of the null-space $L(t)-z$ specifies a factorization for all t and thus a…
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 prove a Darboux theorem for formal deformations of Hamiltonian operators of hydrodynamic type (Dubrovin-Novikov). Not all deformations are equivalent to the original operator: there is a moduli 2-stack of normal forms. The paper utilizes…
We develop a theory of Darboux transformations for CMV matrices, canonical representations of the unitary operators. In perfect analogy with their self-adjoint version -- the Darboux transformations of Jacobi matrices -- they are equivalent…
Differential-difference equation $\frac{d}{dx}t(n+1,x)=f(x,t(n,x),t(n+1,x),\frac{d}{dx}t(n,x))$ with unknown $t(n,x)$ depending on continuous and discrete variables $x$ and $n$ is studied. We call an equation of such kind Darboux…
Using bidifferential calculus, we derive a vectorial binary Darboux transformation for an integrable matrix version of the first negative flow of the Kaup-Newell hierarchy. A reduction from the latter system to an integrable matrix version…
We construct the supersymmetric extensions of the Darboux-Backlund transformations (DBTs) for the Manin-Radul super KdV hierarchy using the super-pseudo-differential operators. The elementary DBTs are triggered by the gauge operators…
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…
Our purpose in this paper is to study when a planar differential system polynomial in one variable linearizes in the sense that it has an inverse integrating factor which can be constructed by means of the solutions of linear differential…
Darboux transformations are employed in construction and analysis of Dirac Hamiltonians with pseudoscalar potentials. By this method, we build a four parameter class of reflectionless systems. Their potentials correspond to composition of…
We construct explicit Darboux transformations for a generalized, two-dimensional Dirac equation. Our results contain former findings for the one-dimensional, stationary Dirac equation, as well as for the fully time-dependent case in (1+1)…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
In this paper we utilize the covariance of Ricatti equation with respect to linear fractional transformations to define classes of conformally equivalent second order differential equations. This motivates then the introduction of…
We analyze the parabolic Dirac operator $D \pm i\partial_t$ in a biquaternionic setting, characterizing its kernel via generalized div-curl systems and Cauchy-Riemann-type relations between the real and imaginary parts. Using the machinery…
We propose a generalization of Laplace transformations to the case of linear partial differential operators (LPDOs) of arbitrary order in R^n. Practically all previously proposed differential transformations of LPDOs are particular cases of…
This paper considers Darboux transformations of a bispectral operator which preserve its bispectrality. A sufficient condition for this to occur is given, and applied to the case of generalized Airy operators of arbitrary order $r>1$. As a…
We study invariants under gauge transformations of linear partial differential operators on two variables. Using results of BK-factorization, we construct hierarchy of general invariants for operators of an arbitrary order. Properties of…
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,…