Related papers: Darboux integrable discrete equations possessing a…
The Darboux-Halphen system of equations have common or individual additive terms depending on the matrices defining Yang-Mills gauge potential fields. Tod (Phys. Lett. A 190 (1994) 221-224), described a conserved quantity for the classical…
Here, Darboux's classical results about transformations with differential substitutions for hyperbolic equations are extended to the case of parabolic equations of the form $L u = \big(D^2_{x} + a(x,y) D_x + b(x,y) D_y + c(x,y)\big)u=0$. We…
Darboux integrability of semidiscrete and discrete 2D Toda lattices corresponding to Lie algebras of A and C series is proved.
We derive integrable discrete systems which are contiguity relations of two equations in the Painlev\'e-Gambier classification depending on some parameter. These studies extend earlier work where the contiguity relations for the six…
We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary…
We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve…
We develop the method of regularized moving frames of Fels and Olver to obtain explicit general formulas for the basis invariants that generate all the joint differential invariants, under gauge transformations, for the operators…
The Gauss-Manin equations are solved for a class of flat-metrics defined by Novikov algebras, this generalizing a result of Balinskii and Novikov who solved this problem in the case of commutative Novikov algebras (where the algebraic…
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…
We consider discrete nonlinear hyperbolic equations on quad-graphs, in particular on the square lattice. The fields are associated to the vertices and an equation Q(x_1,x_2,x_3,x_4)=0 relates four fields at one quad. Integrability of…
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…
This paper is concerned with a generalized type of Darboux transformations defined in terms of a twisted derivation $D$ satisfying $D(AB)=D(A)+\sigma(A)B$ where $\sigma$ is a homomorphism. Such twisted derivations include regular…
We provide precise formulations and proofs of two theorems from Darboux's lectures on orthogonal systems. These results provide local existence and uniqueness of solutions to certain types of first order PDE systems where each equation…
Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…
This paper is an overview of our works which are related to investigations of the integrability of natural Hamiltonian systems with homogeneous potentials and Newton's equations with homogeneous velocity independent forces. The two types of…
In this paper we propose a geometric approach to study Painlev\'e equations appearing as constrained systems of three first-order ordinary differential equations. We illustrate this approach on a system of three first-order differential…
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…
A universal system of difference equations associated with a hyperelliptic curve is derived constituting the discrete analogue of the Dubrovin equations arising in the theory of finite-gap integration. The parametrisation of the solutions…
The main purpose of this article is to show how symmetry structures in partial differential equations can be preserved in a discrete world and reflected in difference schemes. Three different structure preserving discretizations of the…
In the context of integrable partial difference equations on quad-graphs, we introduce the notion of open boundary reductions as a new means to construct discrete integrable mappings and their invariants. This represents an alternative to…