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1. From fully-nonlinear to semilinear evolution equations: two symmetry-integrable examples

Marianna Euler ; Norbert Euler.

In this paper we derive two examples of fully-nonlinear symmetry-integrable evolution equations with algebraic nonlinearities, namely one class of 3rd-order equations and a 5th-order equation. To achieve this we study the equations' Lie-Bäcklund symmetries and apply multipotentialisations, hodograph transformations and generalised hodograph transformations to map the equations to known semilinear integrable evolution equations. As a result of this, we also obtain interesting symmetry-integrable quasilinear equations of order five and order seven, which we display explicitly.

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2. Symmetry Approach to Integration of Ordinary Differential Equations with Retarded Argument

Vladimir Dorodnitsyn ; Roman Kozlov ; Sergey Meleshko.

We review studies on the application of Lie group methods to delay ordinary differential equations (DODEs). For first- and second-order DODEs with a single delay parameter that depends on independent and dependent variables, the group classifications are performed. Classes of invariant DODEs for each Lie subgroup are written out. The symmetries allow us to construct invariant solutions to such equations. The application of variational methods to functionals with one delay yields DODEs with two delays. The Lagrangian and Hamiltonian approaches are reviewed. The delay analog of the Legendre transformation, which relates the Lagrangian and Hamiltonian approaches, is also analysed. Noether-type operator identities relate the invariance of delay functionals with the appropriate variational equations and their conserved quantities. These identities are used to formulate Noether-type theorems that give first integrals of second-order DODEs with symmetries. Finally, several open problems are formulated in the Conclusion.

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3. Symbolic computation of optimal systems of subalgebras of three- and four-dimensional real Lie algebras

Luca Amata ; Francesco Oliveri ; Emanuele Sgroi.

The complete optimal systems of subalgebras of all nonisomorphic three- and four-dimensional real Lie algebras are analyzed by the program \symbolie running in the computer algebra system \emph{Wolfram Mathematica}\texttrademark. The approach uses the definition of $p$-families of Lie subalgebras whose set can be partitioned by introducing a binary relation (reflexive and transitive, though not necessarily symmetric) induced by inner automorphisms of the Lie algebra. The results, produced in a few minutes by \symbolie, represent a good test for the program; in fact, except for minor differences that are discussed, the results confirm those given in 1977 in a paper by Patera and Winternitz.

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4. On Riemann wave superpositions obtained from the Euler system

Łukasz Chomienia ; Alfred Michel Grundland.

The paper contains an analysis of the conditions for the existence of elastic versus non-elastic wave superpositions governed by the Euler system in (1+1)-dimensions. A review of recently obtained results is presented, including the introduction of the notion of quasi-rectifiability of vector fields and its application to both elastic and non- elastic wave superpositions. It is shown that the smallest real Lie algebra containing vector fields associated with the waves admitted by the Euler system is isomorphic to an infinite-dimensional Lie algebra which is the semi-direct sum of an Abelian ideal and the three-dimensional real Lie algebra. The maximal Lie module corresponding to the Euler system can be transformed, by an angle preserving transformation, to this algebra which is quasi-rectifiable and describes the behavior of wave superpositions. Based on these facts, we are able to find a parametrization of the region of non-elastic wave superpositions which allows for the construction of the reduced form of the Euler system.

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