Non-local symmetries of first-order equations

It is shown how one can transform scalar first-order ordinarydifferential equations which admit non-local symmetries of theexponential type to integrable equations admitting canonicalexponential non-local symmetries. As examples we invoke theAbel equation of the second kind, the Riccati equation and naturalgeneralizations of these. Moreover, our method describes howa double reduction of order for a second-order ordinary differentialequation which admits a two-dimensional Lie algebra of generatorsof point symmetries can be affected if the second-order equationis first reduced in order once by a symmetry which does notspan an ideal of the two-dimensional Lie algebra.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras I

We study coupled systems of nonlinear wave equations from the point of view of their formal Darboux integrability. By making use of Vessiot's geometric theory of differential equations, it is possible to associate to each system of nonlinear wave equations a module of vector fields on the second-order jet bundle — the Vessiot distribution. By imposing certain conditions of the structure of the Vessiot distributions, we identify the so-called separable Vessiot distributions. By expressing the separable Vessiot distributions in a basis of singular vector fields, we show that there are, at most, 27 equivalence classes of such distributions. Of these, 14 classes are associated with Darboux integrable nonlinear systems. We take one of these Darboux integrable classes and show that it is in correspondence with the class of six-dimensional simply transitive Lie algebras. Finally, this later result is used to reduce the problem of constructing exact general solutions of the nonlinear wave equations understudy to the integration of Lie systems. These systems were first discovered by Sophus Lie as the most general class of ordinary differential equations which admit nonlinear superposition principles.     

The method of noncommutative integration for linear differential equations. Functional algebras and noncommutative dimensional reduction

The method of noncommutative integration for linear partial differential equations [1] is extended to the case of the so-called functional algebras for which the commutators of their generators are nonlinear functions of the same generators. The linear functions correspond to Lie algebras, whereas the quadratics are associated with the so-called quadratic algebras having wide applications in quantum field theory. A nontrivial example of integration of the Klein-Gordon equation in a curved space not allowing separation of variables is considered. A classification of four- and five-dimensional quadratic algebras is performed.A method of dimensional reduction for noncommutatively integrable many-dimensional partial differential equations is suggested. Generally, the reduced equation has a complicated functional symmetry algebra. The method permits integration of the reduced equation without the use of the explicit form of its functional algebra.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 106, No. 1, pp. 3–15, January, 1996.     

Group classification of quasilinear elliptic-type equations. II. Invariance under solvable Lie algebras

We study the problem of group classification of quasilinear elliptic equations in a two-dimensional space. The list of all equations of this type admitting solvable Lie algebras of symmetry operators is obtained. Together with the results obtained earlier by the authors, these results give a complete solution of the problem of group classification of quasilinear elliptic equations.     

An infinite set of conservation laws for infinite symmetries

We consider partial differential equations of a variational problem admitting infinite-dimensional Lie symmetry algebras parameterized by arbitrary functions of dependent variables and their derivatives. We show that unlike differential systems with symmetry algebras parameterized by arbitrary functions of independent variables, these equations have infinite sets of essential conservation laws. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 3, pp. 518–528, June, 2007.     

Symmetries of shallow water equations on a rotating plane

We consider a system of nonlinear differential equations which describes the spatial motion of an ideal incompressible fluid on a rotating plane in the shallow water approximation and a more general system of the theory of long waves which takes into account the specifics of shear flows. Using the group analysis methods, we calculate the 9-dimensional Lie algebras of infinitesimal operators admissible by the models. We establish an isomorphism of these Lie algebras with a known Lie algebra of operators admissible by the system of equations for the two-dimensional isentropic motions of a polytropic gas with the adiabatic exponent γ = 2. The nontrivial symmetries of the models under consideration enable us to carry out the group generation of the solutions. The class of stationary solutions to the equations of rotating shallow water transforms into a new class of periodic solutions.     

Abelian Lie symmetry algebras of two-dimensional quasilinear evolution equations

We carry out the classification of abelian Lie symmetry algebras of two-dimensional second-order nondegenerate quasilinear evolution equations. It is shown that such an equation is linearizable if it admits an abelian Lie symmetry algebra that is of dimension greater than or equal to 5 or of dimension greater than or equal to 3 with rank-one.     

Investigation of new solutions for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif equation

We investigate and concentrate on new infinitesimal generators of Lie symmetries for an extended (2+ 1)-dimensional Calogero-Bogoyavlenskii-Schif (eCBS) equation using the commutator table which results in a system of nonlinear ordinary differential equations (ODEs) which can be manually solved. Through two stages of Lie symmetry reductions, the eCBS equation is reduced to non-solvable nonlinear ODEs using different combinations of optimal Lie vectors. Using the integration method and the Riccati and Bernoulli equation methods, we investigate new analytical solutions to those ODEs. Back substituting to the original variables generates new solutions to the eCBS equation. These results are simulated through three- and two-dimensional plots.     

On the geometry of the Clairin theory of conditional symmetries for higher-order systems of PDEs with applications

This work presents a geometrical formulation of the Clairin theory of conditional symmetries for higher-order systems of partial differential equations (PDEs). We devise methods for obtaining Lie algebras of conditional symmetries from known conditional symmetries, and unnecessary previous assumptions of the theory are removed. As a consequence, new insights into other types of conditional symmetries arise. We then apply the so-called PDE Lie systems to the derivation and analysis of Lie algebras of conditional symmetries. In particular, we develop a method for obtaining solutions of a higher-order system of PDEs via the solutions and geometric properties of a PDE Lie system, whose form gives a Lie algebra of conditional symmetries of the Clairin type. Our methods are illustrated with physically relevant examples such as nonlinear wave equations, the Gauss–Codazzi equations for minimal soliton surfaces, and generalised Liouville equations.     

Three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras

Dynamical systems attract much attention due to their wide applications. Many significant results have been obtained in this field from various points of view. The present paper is devoted to an algebraic method of integration of three-dimensional nonlinear time dependent dynamical systems admitting nonlinear superposition with four-dimensional Vessiot-Guldberg-Lie algebras $L_4.$ The invariance of the relation between a dynamical system admitting nonlinear superposition and its Vessiot-Guldberg-Lie algebra is the core of the integration method. It allows to simplify the dynamical systems in question by reducing them to \textit{standard forms}. We reduce the three-dimensional dynamical systems with four-dimensional Vessiot-Guldberg-Lie algebras to 98 standard types and show that 86 of them are integrable by quadratures.     

Laplace invariants for a fourth-order equation with two independent variables

We construct the Laplace invariants for an equation with the leading partial derivative. We write defining equations in terms of the Laplace invariants. We obtain classes of equations admitting four-dimensional Lie algebras.     

Narrow Positively Graded Lie Algebras

The present paper is devoted to the classification of infinite-dimensional naturally graded Lie algebras that are narrow in the sense of Zelmanov and Shalev [9]. Such Lie algebras are Lie algebras of slow linear growth. In the theory of nonlinear hyperbolic partial differential equations the notion of the characteristic Lie algebra of equation is introduced [3]. Two graded Lie algebras n₁ and n₂ from our list, that are positive parts of the affine Kac–Moody algebras A₁⁽¹⁾ and A₂⁽²⁾, respectively, are isomophic to the characteristic Lie algebras of the sinh-Gordon and Tzitzeika equations [6]. We also note that questions relating to narrow and slowly growing Lie algebras have been extensively studied in the case of a field of positive characteristic [2].

     

On some classes of fourth-order Bianchi equations with constant ratios of Laplace invariants

On the basis of defining equations written out in terms of Laplace invariants, we single out some classes of fourth-order Bianchi equations similar to the well-known classes of hyperbolic equations with two independent variables admitting Lie algebras of maximum dimension.     

Coupled systems of nonlinear wave equations and finite-dimensional lie algebras II: A nonlinear system arising from the group G 6.1 and its exact integration

In the first paper of this series a correspondence was established between coupled systems of two-dimensional nonlinear wave equations and the six-dimensional simply transitive Lie algebras. In the present paper we make use of this result to construct a Darboux integrable and exactly integrable nonlinear system associated with the six-parameter nilpotent Lie group G _6,1 and we give its exact general solution in terms of four arbitrary functions. The procedure is shown to be an exact linearization of the nonlinear problem.     

On a certain class of locally conformal symplectic structures of the second kind

We study locally conformal symplectic (LCS) structures of the second kind on a Lie algebra. We show a method to construct new examples of Lie algebras admitting LCS structures of the second kind starting with a lower dimensional Lie algebra endowed with a LCS structure and a suitable extension. Moreover, we characterize all LCS Lie algebras obtained with our construction. Finally, we study the existence of lattices in the associated simply connected Lie groups in order to obtain compact examples of manifolds admitting this kind of structure.     

The exotic conformal Galilei algebra and nonlinear partial differential equations

The conformal Galilei algebra (cga) and the exotic conformal Galilei algebra (ecga) are applied to construct partial differential equations (PDEs) and systems of PDEs, which admit these algebras. We show that there are no single second-order PDEs invariant under the cga but systems of PDEs can admit this algebra. Moreover, a wide class of nonlinear PDEs exists, which are conditionally invariant under cga. It is further shown that there are systems of nonlinear PDEs admitting ecga with the realisation obtained very recently in [D. Martelli, Y. Tachikawa, Comments on Galilei conformal field theories and their geometric realisation, preprint, arXiv:0903.5184v2 [hep-th], 2009]. Moreover, wide classes of nonlinear systems, invariant under two different 10-dimensional subalgebras of ecga are explicitly constructed and an example with possible physical interpretation is presented.     

The Grunewald–O'Halloran Conjecture for Nilpotent Lie Algebras of Rank ≥1

Grunewald and O'Halloran conjectured in 1993 that every complex nilpotent Lie algebra is the degeneration of another, nonisomorphic, Lie algebra. We prove the conjecture for the class of nilpotent Lie algebras admitting a semisimple derivation, and also for 7-dimensional nilpotent Lie algebras. The conjecture remains open for characteristically nilpotent Lie algebras of dimension grater than or equal to 8.     

Classical Lie point symmetry analysis of nonlinear diffusion equations describing thermal energy storage

In this paper, we employed the linear transformation group approach to time dependent nonlinear diffusion equations describing thermal energy storage problem. Symmetry analysis of the governing equation resulted in admitted large Lie symmetry algebras for some special cases of the arbitrary constants and the source term. Some transformations that lead to equations with fewer arbitrary parameters are applied and classical Lie point symmetry methods are employed to analyze the transformed equations. Some symmetry reductions are performed and wherever possible the reduced ordinary differential equations are completely solved subject to realistic boundary conditions.