Related papers: On generating relative and absolute invariants of …
In this paper, we present a method to identify integrable complex nonlinear oscillator systems and construct their solutions. For this purpose, we introduce two types of nonlocal transformations which relate specific classes of nonlinear…
Properties of partial integrals such as real and complex-valued polynomial, multiple polynomial, exponential, and conditional for ordinary differential systems are studied. The possibilities of constructing first integrals and last…
We establish Liouville type theorems for degenerate conformally invariant equations.
Borel summation techniques are developed to obtain exact invariants from formal adiabatic invariants (given as divergent series in a small parameter) for a class of differential equations, under assumptions of analyticity of the…
We prove the meromorphy of solutions for a wide class of ordinary differential equations. These equations are given by invariant manifolds of non-linear partial differential equations integrable by the inverse scattering method. Some higher…
Fractional calculus generalizes the derivative and antiderivative operations of differential and integral calculus from integer orders to the entire complex plane. Methods are presented for using this generalized calculus with Laplace…
We establish a consistency result by comparing two independent notions of generalised solutions to a large class of linear hyperbolic first order PDE systems with constant coefficients, showing that they eventually coincide. The first is…
Given a linear ordinary differential equation (ODE) on $\RE$ and a set of interface conditions at a finite set of points $I \subset \RE$, we consider the problem of determining another differential equation whose {\it global} solutions…
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…
Here we present a method to find elementary first integrals of rational second order ordinary differential equations (SOODEs) based on a Darboux type procedure \cite{ManMac,firsTHEOps1,secondTHEOps1}. Apart from practical computational…
We revisit the results on admissible transformations between normal linear systems of second-order ordinary differential equations with an arbitrary number of dependent variables under several appropriate gauges of the arbitrary elements…
This paper gives out the general solutions of variable coefficients ODE and Riccati equation by way of integral series E(X) and F(X). Such kinds of integral series are the generalized form of exponential function, and keep the properties of…
A large family of linear, usually overdetermined, systems of partial differential equations that admit a multiplication of solutions, i.e, a bi-linear and commutative mapping on the solution space, is studied. This family of PDE's contains…
Consider complex semisimple Lie algebras of a given dimension specified by their structure constants. We describe a finite collection of rational functions in the structure constants that form a complete set of invariants: two sets of…
We discuss the general properties of the theory of joint invariants of a smooth Lie group action in a manifold. Many of the known results about differential invariants, including Lie's finiteness theorem, have simpler versions in the…
Below, the explicit solution to a certain finite-difference equation is given and the required steps for derivation of these results are outlined. Everything is included as Mathematica formulae, so the notebook itself can be used for…
A comprehensive symmetry analysis of the N=1 supersymmetric sine-Gordon equation is performed. Two different forms of the supersymmetric system are considered. We begin by studying a system of partial differential equations corresponding to…
Classification theorems for linear differential equations in two real variables, possessing eigenfunctions in the form of the polynomials (the generalized Bochner problem) are given. The main result is based on the consideration of the…
For a nonlinear ordinary differential equation solved with respect to the highest order derivative and rational in the other derivatives and in the independent variable, we devise two algorithms to check if the equation can be reduced to a…
We provide an explicit algorithm to calculate invariant tensors for the adjoint representation of the simple Lie algebra $sl(n)$, as well as arbitrary representation in terms of roots. We also obtain explicit formulae for the adjoint…