Related papers: Transformations of ordinary differential equations…
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…
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…
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…
A generalization of the already studied transformations of the linear differential equation into a system of the first order equations is given. The proposed transformation gives possibility to get new forms of the N-dimensional system of…
We develop a transformer-based sequence-to-sequence model that recovers scalar ordinary differential equations (ODEs) in symbolic form from irregularly sampled and noisy observations of a single solution trajectory. We demonstrate in…
A version of the iterated B\"acklund-Darboux transformation, where Darboux matrix takes a form of the transfer matrix function from the system theory, is constructed for the discrete canonical system and Non-Abelian Toda lattice. Results on…
We construct a Darboux transformation of a general $su(3)$-valued spin system called the $\Gamma$-spin system. Using this Darboux transformation we derive a recursive formula for the soliton solutions of this spin system. Then using these…
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…
We study discretization of Darboux integrable systems. The discretization is done by using $x$- or $y$-integrals of the considered systems. New examples of semi-discrete Darboux integrable systems are obtained.
A matricial Darboux operator intertwining two one-dimensional stationary Dirac Hamiltonians is constructed. This operator is such that the potential of the second Dirac Hamiltonian as well as the corresponding eigenfunctions are determined…
The nonlocal Darboux transformation for the stationary axially symmetric Schr\"odinger and Helmholtz equations is considered. Formulae for the nonlocal Darboux transformation are obtained and its relation to the generalized Moutard…
Singular Darboux transformations, in contrast to the conventional ones, have a singular matrix as a coefficient before the derivative. We incorporated such transformations into a chain of conventional transformations and presented…
Meromorphic solutions of autonomous nonlinear ordinary differential equations are studied. An algorithm for constructing meromorphic solutions in explicit form is presented. General expressions for meromorphic solutions (including rational,…
General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find although powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases…
A new strategy, using Darboux transformations, of finding self-switching solutions of $i\dot{\rho} = [H, f({\rho})]$ is introduced. Unlike the previous ones, working for any f but for Hamiltonians whose spectrum contains at least three…
Differential equations with constant and variable coefficients over octonions are investigated. It is found that different types of differential equations over octonions can be resolved. For this purpose non-commutative line integration is…
New types of irreducible second order Darboux transformations for the one dimensional Schroedinger equation are described. The main feature of such transformations is that the transformation functions have the eigenvalues grater then the…
We adapt the notion of the Darboux transformation to the context of polynomial Sturm-Liouville problems. As an application, we characterize the recently described $X_m$ Laguerre polynomials in terms of an isospectral Darboux transformation.…
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…
We present here the solution of the problem on linearization of fourth-order equations by means of point transformations. We show that all fourth-order equations that are linearizable by point transformations are contained in the class of…