Norbert Euler

Nonlinear evolution equations and Painlevé test

This book is an edited version of lectures given by the authors at a seminar at the Rand Afrikaans University. It gives a survey on the Painleve test, Painleve property and integrability. Both ordinary differential equations and partial differential equations are considered.

The Riccati and Ermakov-Pinney hierarchies

Abstract Rota-Baxter operators or relations were introduced to solve certain analytic and combinatorial problems and then applied to many fields in mathematics and mathematical physics. In this paper, we commence to study the Rota-Baxter operators of weight zero on pre-Lie algebras. Such operators satisfy (the operator form of) the classical Yang-Baxter equation on the sub-adjacent Lie algebras of the pre-Lie algebras. We not only study the invertible Rota-Baxter operators on pre-Lie algebras, but also give some interesting construction of Rota-Baxter operators. Furthermore, we give all Rota-Baxter operators on 2-dimensional complex pre-Lie algebras and some examples in higher dimensions.

Linearisable Third-Order Ordinary Differential Equations and Generalised Sundman Transformations: The Case X′′′=0

We calculate in detail the conditions which allow the most general third-order ordinary differential equation to be linearised in X′′′(T)=0 under the transformation X(T)=F(x,t), dT=G(x,t) dt.

Sundman Symmetries of Nonlinear Second-Order and Third-Order Ordinary Differential Equations

Abstract We investigate the Sundman symmetries of second-order and third-order nonlinear ordinary differential equations. These symmetries, which are in general nonlocal transformations, arise from generalised Sundman transformations of autonomous equations. We show that these transformations and symmetries can be calculated systematically and can be used to find first integrals of the equations.

Approximate symmetries and approximate solutions for a multidimensional Landau-Ginzburg equation

The authors give the approximate symmetries for the multidimensional Landau-Ginzburg equation delta 2u/ delta x2i+ delta u/ delta x4=a1+a2u+ in un where n in R and 0( in <<1. They also construct approximate solutions for this nonlinear equation using the approximate symmetries.

On the construction of approximate solutions for a multidimensional nonlinear heat equation

We study three methods, based on continuous symmetries, to find approximate solutions for the multidimensional nonlinear heat equation delta u/ delta x0+ Delta u=aun+ epsilon f(u), where a and n are arbitrary real constants, f is a smooth function, and 0< epsilon <<1.

A tree of linearisable second-order evolution equations by generalised hodograph transformations

Abstract We present a list of (1+1)-dimensional second-order evolution equations all connected via a proposed generalised hodograph transformation, resulting in a tree of equations transformable to the linear second-order autonomous evolution equation. The list includes autonomous and nonautonomous equations. To the memory of Wilhelm Fushchych

A note on Nambu mechanics

SummaryIn Nambu mechanics an autonomous system of first-order ordinary differential equations dui/dt=Fi(u) (i=1,…,n) is constructed with the help of (n−1) smooth functionsIi. These smooth functions are first integrals of this dynamical system. If the functionsIi are polynomials, then the system is algebraic completely integrable. We discuss the question whether the first integrals determine uniquely the autonomous system of first-order differential equations. Then we give a generalization of Nambu mechanics.

Multipotentialisations and iterating-solution formulae : the Krichever-Novikov equation

We derive solution-formulae for the Krichever–Novikov equation by a systematic multipotentialisation of the equation. The formulae are achieved due to the connections of the Krichever–Novikov equations to certain symmetry-integrable 3rd-order evolution equations which admit autopotentialisations.

Q-symmetry generators and exact solutions for nonlinear heat conduction

We investigate conditional invariance by considering Q-symmetry generators of the nonlinear heat equation ∂u/∂x0 – λ∂2u/∂x12 = f(u), where λ is a real constant and f an arbitrary differentiable function. With the obtained Q-generators we construct exact solutions by the use of similarity ansatze and reductions to ordinary differential equations. A generalization to m-space dimensions is performed.