Related papers: The nonlinear superposition principle and the Wei-…
The characterization of systems of differential equations admitting a superposition function allowing us to write the general solution in terms of any fundamental set of particular solutions is discussed. These systems are shown to be…
In this paper we develop some group theoretical methods which are shown to be very useful for a better understanding of the properties of the Riccati equation and we discuss some of its integrability conditions from a group theoretical…
By using Lie symmetry methods, we identify a class of second order nonlinear ordinary differential equations invariant under at least one dimensional subgroup of the symmetry group of the Ermakov-Pinney equation. In this context, nonlinear…
This article complements recent results of the papers [J. Math. Phys. 41 (2000), 480; 45 (2004), 336] on the symmetry classification of second-order ordinary difference equations and meshes, as well as the Lagrangian formalism and…
The theory of superposition rules for solutions of a Lie system of first-order differential equations is extended to deal with analogous systems of second-order and the theory is illustrated with the very rich example of Ermakov-like…
A superposition rule is a particular type of map that enables one to express the general solution of certain systems of first-order ordinary differential equations, the so-called Lie systems, out of generic families of particular solutions…
This paper proves a version for stochastic differential equations of the Lie-Scheffers Theorem. This result characterizes the existence of nonlinear superposition rules for the general solution of those equations in terms of the involution…
The Wei-Norman technique allows to express the solution of a system of linear non-autonomous differential equations in terms of product of exponentials. In particular it enables to find a time-ordered product of exponentials by solving a…
The Riccati equation method is used for study the behavior of solutions of the systems of two linear first order ordinary differential equations. All types of oscillation and regularity of these system are revealed. A generalization of…
The main purpose of this work is to introduce and analyse some generalizations of diverse superposition rules for first-order differential equations to the setting of second-order differential equations. As a result, we find a way to apply…
We show that the non-linear autonomus Wei-Norman equations, expressing the solution of a linear system of non-autonomous equations on a Lie algebra, can be reduced to the hierarchy of matrix Riccati equations in the case of all classical…
This paper belongs to a group of work in the intersection of symbolic computation and group analysis aiming for the symbolic analysis of differential equations. The goal is to extract important properties without finding the explicit…
Matrix Riccati equations and other nonlinear ordinary differential equations with superposition formulas are, in the case of constant coefficients, shown to have the same exact solutions as their group theoretical discretizations. Explicit…
We analyze families of non-autonomous systems of first-order ordinary differential equations admitting a common time-dependent superposition rule, i.e., a time-dependent map expressing any solution of each of these systems in terms of a…
The Riccati equation method is used to establish some global solvability criteria for some classes of second order nonlinear ordinary differential equations. Two oscillation theorems are proved. The results are applied to the Emden - Fowler…
Systems of nonlinear ordinary differential equations are constructed, for which the general solution is algebraically expressed in terms of a finite number of particular solutions. Expressions of that type are called the nonlinear…
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…
New problem is considered that is to find nonlinear differential equations with special solutions. Method is presented to construct nonlinear ordinary differential equations with exact solution. Crucial step to the method is the assumption…
A method to find exact solutions to nonlinear Schr\"odinger equation, defined on a line and on a plane, is found by connecting it with second order linear ordinary differential equation. The connection is essentially made using Riccati…
The Riccati equation method is used to establish an oscillatory and a non oscillatory criteria for nonhomogeneous linear systems of two first-order ordinary differential equations. It is shown that the obtained oscillatory criterion is a…