Related papers: Darboux transformations for differential operators…

We study LU-type factorizations of the infinitesimal generator of a birth--death process on $\mathbb{N}_0$. Our goal is to characterize those factorizations whose Darboux transformations (that is, inverting the order of the factors) yield…

Infinite order differential operators appear in different fields of mathematics and physics. In the past decade they turned out to play a crucial role in the theory of superoscillations and provided new insight in the study of the evolution…

We study infinite order differential operators acting in the spaces of exponential type entire functions. We derive conditions under which such operators preserve the set of Laguerre entire functions which consists of the polynomials…

We define a class of discrete operators acting on infinite, finite or periodic sequences mimicking the standard properties of pseudo-differential operators. In particular we can define the notion of order and regularity, and we recover the…

Given a right factor and a left factor of a Linear Partial Differential Operator (LPDO), under which conditions we can refine these two-factor factorizations into one three-factor factorization? This problem is solved for LPDOs of arbitrary…

One of Darboux's seminal results is archived here

Specific global symbol classes and corresponding pseudodifferential operators of infinite order that act continuously on the space of tempered ultradistributions of Beurling and Roumieu type are constructed. For these classes, symbolic…

Interesting nonsingular parametric oscillators which are Darboux related to the classical harmonic oscillator and have periodic dissipative/gain features are identified through a modified factorization method. The same method is applied to…

The n-fold Darboux transformation (DT) is a 2\times2 matrix for the Kaup-Newell (KN) system. In this paper,each element of this matrix is expressed by a ratio of $(n+1)\times (n+1)$ determinant and $n\times n$ determinant of eigenfunctions.…

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…

We prove that second-order hyperbolic Monge-Ampere equations for one function of two variables are connected to the wave equation by a Backlund transformation if and only if they are integrable by the method of Darboux at second order. One…

In this second paper on the method of deriving linearizing transformations for nonlinear ODEs, we extend the method to a set of two coupled second order nonlinear ODEs. We show that besides the conventional point, Sundman and generalized…

In this letter, for the discrete parity-time-symmetric nonlocal nonlinear Schr\"{o}dinger equation, we construct the Darboux transformation, which provides an algebraic iterative algorithm to obtain a series of analytic solutions from a…

In this paper we discuss some results related to commuting ordinary differential operators of rank greater than one.

We study natural differential operators transforming two tensor fields into a tensor field. First, it is proved that all bilinear operators are of order one, and then we give the full classification of such operators in several concrete…

With this paper we begin an investigation of difference schemes that possess Darboux transformations and can be regarded as natural discretizations of elliptic partial differential equations. We construct, in particular, the Darboux…

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…

We give an algorithm to write down all conformally invariant differential operators acting between scalar functions on Minkowski space. All operators of order k are nonlinear and are functions on a finite family of functionally independent…

For the nonlocal Davey-Stewartson I equation, the Darboux transformation is considered and explicit expressions of the solutions are obtained. Like the nonlocal equations in 1+1 dimensions, many solutions may have singularities. However, by…

We construct a non-trivial type of 1-step exceptional Bannai-Ito polynomials which satisfy discrete orthogonality by using a generalized Darboux transformation. In this generalization, the Darboux transformed Bannai-Ito operator is directly…