Nonlinear differential Galois theory

We show that if a holomorphic Hamiltonian system is holomorphically integrable in the non-commutative sense in a neighbourhood of a non-equilibrium phase curve which is located at a regular level of the first integrals, then the identity component of the differential Galois group of the variational equations along the phase curve is Abelian. Thus necessary conditions for the commutative and non-commutative integrability given by the differential Galois approach are the same.

The main motivation of our work is to create an efficient algorithm that decides hypertranscendence of solutions of linear differential equations, via the parameterized differential and Galois theories. To achieve this, we expand the representation theory of linear differential algebraic groups and develop new algorithms that calculate unipotent radicals of parameterized differential Galois groups for differential equations whose coefficients are rational functions. P. Berman...

In this paper we study the general group classification of systems of linear second-order ordinary differential equations inspired from earlier works and recent results on the group classification of such systems. Some interesting results and subsequent Theorem arising from this particular study are discussed here. This paper considers the study of irreducible systems of second-order ordinary differential equations.

A Galois theory of differential fields with parameters is developed in a manner that generalizes Kolchin's theory. It is shown that all connected differential algebraic groups are Galois groups of some appropriate differential field extension.

The aim of this article is to provide a method to prove the irreducibility of non-linear ordinary differential equations by means of the differential Galois group of their variational equations along algebraic solutions. We show that if the dimension of the Galois group of a variational equation is large enough then the equation must be irreducible. We propose a method to compute this dimension via reduced forms. As an application, we reprove the irreducibility of the second ...

An algebraic technique is presented that does not use results of model theory and makes it possible to construct a general Galois theory of arbitrary nonlinear systems of partial differential equations. The algebraic technique is based on the search for prime differential ideals of special form in tensor products of differential rings. The main results demonstrating the work of the technique obtained are the theorem on the constructedness of the differential closure and the g...

n this paper we formulate necessary conditions for the integrability in the Jacobi sense of Newton equations $\ddot \vq=-\vF(\vq)$, where $\vq\in\C^n$ and all components of $\vF$ are polynomial and homogeneous of the same degree $l$. These conditions are derived from an analysis of the differential Galois group of the variational equations along special particular solutions of the Newton equations. We show that, taking all admissible particular solutions, we restrict consider...

This paper is centred on solving differential equations by symmetry groups for first order ODEs and is in response to Starrett (2007). It also explores the possibility of averting the assumptions by Olver (2000) that, in practice finding the solutions of the linearized symmetry condition is usually a much more difficult problem than solving the original ODE but, by inspired guesswork or geometric intuition, it is possible to ascertain a particular solution of the linearized s...

In this paper we give the global conditions for an ordinary differential equation to admit a superposition law of solutions in the classical sense. This completes the well-known Lie superposition theorem. We introduce rigorous notions of pretransitive Lie group action and Lie-Vessiot systems. We proof that pretransitive Lie group actions are transitive. We proof that an ordinary differential equation admit a superposition law if and only if it is a pretransitive Lie-Vessiot s...

We present a non-linear difference-differential equation whoes Galois group is a non-commutative and non-co-comutative Hopf algebra.