Marianna Euler

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.

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

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.

Fourth-order recursion operators for third-order evolution equations

Abstract We report the recursion operators for a class of symmetry integrable evolution equations of third order which admit fourth-order recursion operators. Under the given assumptions we obtain the complete list of equations, one of which is the well-known Krichever-Novikov equation.

Multipotentializations and nonlocal symmetries: Kupershmidt, Kaup-Kupershmidt and Sawada-Kotera equations

In this letter we report a new invariant for the Sawada-Kotera equation that is obtained by a systematic potentialization of the Kupershmidt equation. We show that this result can be derived from nonlocal symmetries and that, conversely, a previously known invariant of the Kaup-Kupershmidt equation can be recovered using potentializations.

THE CONVERSE PROBLEM FOR THE MULTIPOTENTIALISATION OF EVOLUTION EQUATIONS AND SYSTEMS

We propose a method to identify and classify evolution equations and systems that can be multipotentialised in given target equations or target systems. We refer to this as the converse problem. Although we mainly study a method for (1 + 1)-dimensional equations/system, we do also propose an extension of the methodology to higher-dimensional evolution equations. An important point is that the proposed converse method allows one to identify certain types of auto-Bäcklund transformations for the equations/systems. In this respect we define the triangular-auto-Bäcklund transformation and derive its connections to the converse problem. Several explicit examples are given. In particular, we investigate a class of linearisable third-order evolution equations, a fifth-order symmetry-integrable evolution equation as well as linearisable systems.

A CLASS OF SEMILINEAR FIFTH-ORDER EVOLUTION EQUATIONS: RECURSION OPERATORS AND MULTIPOTENTIALISATIONS

We apply a list of criteria which leads to a class of fifth-order symmetry-integrable evolution equations. The recursion operators for this class are given explicitly. Multipotentialisations are then applied to the equations in this class in order to extend this class of integrable equations.