Related papers: Linearisable systems and the Gambier approach
We present a systematic construction of integrable third order systems based on the coupling of an integrable second order equation and a Riccati equation. This approach is the extension of the Gambier method that led to the equation that…
We propose a discrete form for an equation due to Gambier and which belongs to the class of the fifty second order equations that possess the Painleve property. In the continuous case, the solutions of the Gambier equation is obtained…
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 examine a family of 3-point mappings that include mappings solvable through linearization. The different origins of mappings of this type are examined: projective equations and Gambier systems. The integrable cases are obtained through…
We examine critically the Gambier equation and show that it is the generic linearisable equation containing, as reductions, all the second-order equations which are integrable through linearisation. We then introduce the general discrete…
We study the projective systems in both continuous and discrete settings. These systems are linearizable by construction and thus, obviously, integrable. We show that in the continuous case it is possible to eliminate all variables but one…
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 present a systematic study of the Gambier system, which in the continuous case is given by two Riccati equations in cascade. We derive the condition for its integrability and show that the generic Gambier system contains one free…
We examine a family of discrete second-order systems which are integrable through reduction to a linear system. These systems were previously identified using the singularity confinement criterion. Here we analyse them using the more…
This work presents a newly renovated approach to the analysis of second-order Riccati equations from the point of view of the theory of Lie systems. We show that these equations can be mapped into Lie systems through certain Legendre…
We consider the hierarchy of higher-order Riccati equations and establish their connection with the Gambier equation. Moreover we investigate the relation of equations of the Gambier family to other nonlinear differential systems. In…
This paper continues our previous work done in math.AG/0008207 and is an attempt to establish a conceptual framework which generalizes the work of Manin on the relation between non-linear second order ODEs of type Painleve VI and integrable…
Integrability conditions for Lie systems are related to reduction or transformation processes. We here analyse a geometric method to construct integrability conditions for Riccati equations following these approaches. This approach provides…
A discrete analogue of the dressing method is presented and used to derive integrable nonlinear evolution equations, including two infinite families of novel continuous and discrete coupled integrable systems of equations of nonlinear…
The asymptotic iteration method (AIM) is an iterative technique used to find exact and approximate solutions to second-order linear differential equations. In this work, we employed AIM to solve systems of two first-order linear…
Using geometric methods for linearizing systems of second order cubically semi-linear ordinary differential equations, we extend to the third order by differentiating the second order equation. This yields criteria for linearizability of a…
In this paper, we introduce an iterative numerical method to solve systems of nonlinear equations. The third-order convergence of this method is analyzed. Several examples are given to illustrate the efficiency of the proposed method.
Lie's linearizability criteria for scalar second-order ordinary differential equations had been extended to systems of second-order ordinary differential equations by using geometric methods. These methods not only yield the linearizing…
We suggest an approach for description of integrable cases of the Abel equations. It is based on increasing of the order of equations up to the second one and using equivalence transformations for the corresponding second-order ordinary…
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.