Related papers: Integral transformation and Darboux transformation
A brief sketch of the formulation of the monodromy transform approach and corresponding integral equation methods as well as of various applications of this approach for solution of integrable symmetry reductions of Einstein's field…
In this work, we give some criteria that allow us to decide when two sequences of matrix-valued orthogonal polynomials are related via a Darboux transformation and to build explicitly such transformation. In particular, they allow us to see…
Classical Bianchi-Lie, Backlund and Darboux transformations are considered. Their generalizations for the dynamical systems are discussed. For the transformation being the generalization of the normal shift the special class of dynamical…
Baecklund-Darboux transformations are closely related to the integrability and symmetry problems. For the generalized Baecklund-Darboux transformation (GBDT), we consider conservation laws, rational extensions and bispectrality. We use the…
Hypergeometric equations with a dihedral monodromy group can be solved in terms of elementary functions. This paper gives explicit general expressions for quadratic monodromy invariants for these hypergeometric equations, using a…
In this paper we construct Darboux transformations for the supersymmetric Two-boson equation. Two Darboux transformations and associated B\"acklund transformations are presented. For one of them, we also obtain the corresponding the…
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…
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.
In this work we establish new forms of Heun-to-Heun transformations and Heun-to-Hypergeometric transformations. The transformations are realised by changing the independent variable in a non-linear way. Using these we also point out some…
As a generalization of Riemann-Liouville integral, we introduce integral transformations of convergent power series which can be applied to hypergeometric functions with several variables.
We construct the Darboux-Backlund transformation for the sigma model describing static configurations of the 2-dimensional classical continuum Heisenberg chain. The transformation is characterized by a non-trivial normalization matrix…
We consider a classical problem of Computer Algebra: symbolic solution of PDEs. We transform the famous Darboux theorems on differential transformations of hyperbolic operator into the space of invariants. We introduce a new idea -- $X$-…
The standard binary Darboux transformation is investigated and is used to obtain quasi-Grammian multisoliton solutions of the generalized coupled dispersionless integrable system.
The purpose of this manuscript is to derive two distinct integral representations of the generalized polylogarithm using two different techniques. The first approach involves the Dirichlet series and its Laplace representation, which leads…
The integration procedure based on the the generalized Darboux transform is suggested for the Ishimori magnet model. Exact solutions are constructed for the model of background of spiral structures. The possibility of phase transition in…
This paper presents the basic ideas and properties of elliptic functions and elliptic integrals as an expository essay. It explores some of their numerous consequences and includes applications to some problems such as the simple pendulum,…
In the paper we find solution representations in the compact integral form to the Cauchy problem for a general form of the Euler--Poisson--Darboux equation with Bessel operators via generalized translation and spherical mean operators for…
The Darboux transformations of Krawtchouk polynomials are investigated and all possible exceptional Krawtchouk polynomials obtainable from a single-step Darboux transformation are considered. The properties of these exceptional Krawtchouk…
Darboux Wronskian formulas allow to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one cannot be represented this way. It has been a long standing problem on what are other…
The reductions of the Heun equation to the hypergeometric equation by polynomial transformations of its independent variable are enumerated and classified. Heun-to-hypergeometric reductions are similar to classical hypergeometric…