Related papers: Linearisable systems and the Gambier approach
The linearization of complex ordinary differential equations is studied by extending Lie's criteria for linearizability to complex functions of complex variables. It is shown that the linearization of complex ordinary differential equations…
The explicit integrability of second order ordinary differential equations invariant under time-translation and rescaling is investigated. Quadratic systems generated from the linearisable version of this class of equations are analysed to…
In this Chapter, using Riccati equation as our main example, we tried to demonstrate at least some of the ideas and notions introduced in Chapter 1 - integrability in quadratures, conservation laws, etc. Regarding transformation group and…
In the first part of the paper, we classify linear integrable (multi-dimensionally consistent) quad-equations on bipartite isoradial quad-graphs in $\mathbb C$, enjoying natural symmetries and the property that the restriction of their…
Theory of Riemann Extensions of the spaces with constant affine connection for the studying of the properties of nonlinear the first order systems of differential equations is proposed. Quadratic planar system of equations and the Lorenz…
We present the bilinear forms of the (continuous) Painlev\'e equations obtained from the continuous limit of the analogous expresssions for the discrete ones. The advantage of this method is that it leads to very symmetrical results. A new…
We present a number of second order maps, which pass the singularity confinement test commonly used to identify integrable discrete systems, but which nevertheless are non-integrable. As a more sensitive integrability test, we propose the…
We study the discretisation of the Chazy class III equation by two means: a discrete Painlev\'e test, and the preservation of a two-parameter solution to the continuous equation. We get that way a best discretisation scheme.
We introduce a method for finding general solutions of third-order nonlinear differential equations by extending the modified Prelle-Singer method. We describe a procedure to deduce all the integrals of motion associated with the given…
A method of finding general solutions of second-order nonlinear ordinary differential equations by extending the Prelle-Singer (PS) method is briefly discussed. We explore integrating factors, integrals of motion and the general solution…
We describe a set of Gaussian Process based approaches that can be used to solve non-linear Ordinary Differential Equations. We suggest an explicit probabilistic solver and two implicit methods, one analogous to Picard iteration and the…
The connection between symmetries and linearizations of discrete-time dynamical systems is being inverstigated. It is shown, that existence of semigroup structures related to the vector field and having linear representations enables…
The transformation of the Nth- order linear difference equation into a system of the first order difference equations is presented. The proposed transformation gives possibility to get new forms of the N-dimensional system of the first…
This monograph, written for educational purposes, serves as an introduction to the concept of integrability as it applies to systems of differential equations (both ordinary and partial) as well as to vector-valued fields. The general cases…
We study the discretization of Darboux integrable systems. The discretization is done using $x$-, $y$-integrals of the considered continuous systems. New examples of semi-discrete Darboux integrable systems are obtained.
The Riccati equation method is used to establish Kamenev type conditions for the existence of oscillatory solutions to third order linear ordinary differential equations. Three oscillatory theorems are proved, which generalize the Lazer's…
An approximation method is presented for probabilistic inference with continuous random variables. These problems can arise in many practical problems, in particular where there are "second order" probabilities. The approximation, based on…
In this paper, we investigate superintegrable systems which separate in parabolic coordinates and admit a third-order integral of motion. We give the corresponding determining equations and show that all such systems are multi-separable and…
In earlier work we have studied a method for discretization in time of a parabolic problem which consists in representing the exact solution as an integral in the complex plane and then applying a quadrature formula to this integral. In…
We present developments of the Hamiltonian approach to problems of the freely decay of isotropic turbulence, and also consider specific applications of the modified Prelle-Singer procedure to isotropic turbulence. It demonstrates that a…