Related papers: Darboux formulae for linear hyperbolic equations i…
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…
Differential operators commuting with integral operators were discovered in the work of C. Tracy and H. Widom [37, 38] and used to derive asymptotic expansions of the Fredholm determinants of integral operators arising in random matrix…
In this work, we continue the development of methods for constructing Lax pairs and recursion operators for nonlinear integrable hyperbolic equations of soliton type, previously proposed in the work of Habibullin et al. (2016 {\it J. Phys.…
A method of G. Wilson for generating commutative algebras of ordinary differential operators is extended to higher dimensions. Our construction, based on the theory of D-modules, leads to a new class of examples of commutative rings of…
The article contributes to the theory of infinitesimal bendings of smooth surfaces in Euclidean 3-space. We derive a linear differential equation of the first order, which previously did not appear in the literature and which is satisfied…
In this work we investigate the existence of solutions, their uniqueness and finally dependence on parameters for solutions of second order neutral nonlinear difference equations. The main tool which we apply is Darbo fixed point theorem.
This paper establishes the symmetries of Darboux's equations (1882) on tori. We extend Ince's work (1940) by developing new infinite series expansions in terms of Jacobi elliptic functions around each of the four regular singular points of…
We investigate and derive second solutions to linear homogeneous second-order difference equations using a variety of methods, in each case going beyond the purely formal solution and giving explicit expressions for the second solution. We…
Highly accurate simulations of problems including second derivatives on complex geometries are of primary interest in academia and industry. Consider for example the Navier-Stokes equations or wave propagation problems of acoustic or…
We propose a discrete Darboux-Lax scheme for deriving auto-B\"acklund transformations and constructing solutions to quad-graph equations that do not necessarily possess the 3D consistency property. As an illustrative example we use the…
We exploit some relations which exist when (rigid) special geometry is formulated in real symplectic special coordinates $P^I=(p^\Lambda,q_\Lambda), I=1,...,2n$. The central role of the real $2n\times 2n$ matrix $M(\Re \mathcal{F},\Im…
The use of effective Darboux transformations for general classes Lax pairs is discussed. The general construction of ``binary'' Darboux transformations preserving certain properties of the operator, such as self-adjointness, is given. The…
We introduce a method for constructing Darboux (or supersymmetric) pairs of pseudoscalar and scalar Dirac potentials that are associated with exceptional orthogonal polynomials. Properties of the transformed potentials and regularity…
In $L_2({\mathbb R}^d;{\mathbb C}^n)$, a selfadjoint strongly elliptic second order differential operator ${\mathcal A}_\varepsilon$ is considered. It is assumed that the coefficients of the operator ${\mathcal A}_\varepsilon$ are periodic…
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…
First-order systems of hyperbolic partial differential equations (PDEs) occur ubiquitously throughout computational physics, commonly used in simulations of fluid turbulence, shock waves, electromagnetic interactions, and even general…
We present a framework which enables the analysis of dynamic inverse problems for wave phenomena that are modeled through second-order hyperbolic PDEs. This includes well-posedness and regularity results for the forward operator in an…
In this article we describe the novel method to construct fundamental solutions for operators with variable coefficients. That method was introduced in "A note on the fundamental solution for the Tricomi-type equation in the hyperbolic…
In this paper, we propose a complex short pulse equation of both focusing and defocusing types, which governs the propagation of ultra-short pulses in nonlinear optical fibers. It can be viewed as an analogue of the nonlinear Schr\"odinger…
There are many methods for finding a particular solution to a nonhomogeneous linear ordinary differential equation (ODE) with constant coefficients. The method of undetermined coefficients, Laplace transform method and differential operator…