Optimal Systems and Group Classification of (1+2)-Dimensional Heat Equation

The optimal systems and symmetry breaking interactions for the (1+2)-dimensional heat equation are systematically studied. The equation is invariant under the nine-dimensional symmetry group H ₂. The details of the construction for an one-dimensional optimal system is presented. The optimality of one- and two-dimensional systems is established by finding some algebraic invariants under the adjoint actions of the group H ₂. A list of representatives of all Lie subalgebras of the Lie algebra h ₂ of the Lie group H ₂ is given in the form of tables and many of their properties are established. We derive the most general interactions F(t,x,y,u,u _x ,u _y ) such that the equation u _t =u _xx +u _yy +F(t,x,y,u,u _x ,u _y ) is invariant under each subgroup.     

Symmetries of the Discrete Nonlinear Schrödinger Equation

The Lie algebra L(h) of point symmetries of a discrete analogue of the nonlinear Schrödinger equation (NLS) is described. In the continuous limit, the discrete equation is transformed into the NLS, while the structure of the Lie algebra changes: a contraction occurs with the lattice spacing h as the contraction parameter. A five-dimensional subspace of L(h), generated by both point and generalized symmetries, transforms into the five-dimensional point symmetry algebra of the NLS.     

Nonlocally related systems, linearization and nonlocal symmetries for the nonlinear wave equation

The nonlinear wave equation utt=(c²x(u)ux) arises in various physical applications. Ames et al. [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] did the complete group classification for its admitted point symmetries with respect to the wave speed function c(u) and as a consequence constructed explicit invariant solutions for some specific cases. By considering conservation laws for arbitrary c(u), we find a tree of nonlocally related systems and subsystems which include related linear systems through hodograph transformations. We use existing work on such related linear systems to extend the known symmetry classification in [W.F. Ames, R.J. Lohner, E. Adams, Group properties of utt=x[f(u)ux], Int. J. Nonlin. Mech. 16 (1981) 439-447] to include nonlocal symmetries. Moreover, we find sets of c(u) for which such nonlinear wave equations admit further nonlocal symmetries and hence significantly further extend the group classification of the nonlinear wave equation.     

Lie group classification of the generalized Lane–Emden equation

We carry out the Lie group classification of the generalized Lane–Emden equation xu^″+nu^′+xH(u)=0, which has many applications in mathematical physics and astrophysics. We show that the equation admits a three-dimensional equivalence Lie algebra. It is also shown that the principal Lie algebra, which in this case is trivial, has seven possible extensions. Three new cases arise for which the Lie point symmetry algebra is non-trivial. Comparison is then made of these cases with the Noether symmetry cases as well as the partial Noether operators.     

Special conformal groups of a Riemannian manifold and Lie point symmetries of the nonlinear Poisson equation

We obtain a complete group classification of the Lie point symmetries of nonlinear Poisson equations on generic (pseudo) Riemannian manifolds M. Using this result we study their Noether symmetries and establish the respective conservation laws. It is shown that the projection of the Lie point symmetries on M are special subgroups of the conformal group of M. In particular, if the scalar curvature of M vanishes, the projection on M of the Lie point symmetry group of the Poisson equation with critical nonlinearity is the conformal group of the manifold. We illustrate our results by applying them to the Thurston geometries.     

Group classification of differential-difference equations: Low-dimensional Lie algebras

Differential-difference equations of the form u? _n = F _n (t, u_n?1, u _n , u_n+1, u?_n?1, u? _n , u?_n+1) are classified according to their intrinsic Lie point symmetries, equivalence group and some low-dimensional Lie algebras including the Abelian symmetry algebras, nilpotent nonAbelian symmetry algebras, solvable symmetry algebras with nonAbelian nilradicals, solvable symmetry algebras with Abelian nilradicals and nonsolvable symmetry algebras. Here F _n is a nonlinear function of its arguments and the dot over u denotes differentiation with respect to t.     

Group classification of the generalized Emden–Fowler-type equation

We perform the Lie group classification of the Emden–Fowler-type equation xu^″+nu^′+x^νF(u)=0, which arises in several applications. These include the theory of stellar structure, the thermal behaviour of a spherical cloud of gas, isothermal gas spheres and the theory of thermionic currents. Seven cases arise for the possible extension of the principal Lie algebra, which in this case is trivial. Three new cases occur for which we have non-trivial Lie point symmetry algebra. We compare these cases with the Noether symmetry cases. Moreover, we also make comparisons with the partial Noether operators. Finally for three cases we reduce the Emden–Fowler-type equation to quadratures.     

P ∞ algebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operatorsalgebra of symmetries of the kadomtsev-petviashvili equation, free fermions, and 2-cocycles in the lie algebra of pseudo-differential operators

The symmetry algebraP _∞=W _∞ ⊕P ⊕I _∞ of integrable systems is defined. As an example, the classical Lie point symmetries of all higher Kadomtsev-Petviashvili equations are obtained. It is shown that of the point symmetries, the (positive) ones belong to theW _∞ symmetries, while the other (negative) ones belong toI _∞ symmetries. The corresponding action on the τ-function is obtained for the positive symmetries. The negative symmetries cannot be obtained from the free fermion algebra. A new embedding of the Virasoro algebra intogl(∞) describes the conformal transformations of the KP time variables. A free fermion algebra cocycle is described as a PDO Lie algebra cocycle. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 113, No. 2, pp. 231–260, November, 1997.     

Invariance properties of a general bond-pricing equation

We perform the group classification of a bond-pricing partial differential equation of mathematical finance to discover the combinations of arbitrary parameters that allow the partial differential equation to admit a nontrivial symmetry Lie algebra. As a result of the group classification we propose “natural” values for the arbitrary parameters in the partial differential equation, some of which validate the choices of parameters in such classical models as that of Vasicek and Cox-Ingersoll-Ross. For each set of these natural parameter values we compute the admitted Lie point symmetries, identify the corresponding symmetry Lie algebra and solve the partial differential equation.     

Exceptional symmetry reductions of Burgers' equation in two and three spatial dimensions

Lie point symmetry analysis of the general class of nonlinear diffusion-convection equations in two and three dimensions has shown that only for Burgers' equation (that isD(u)=const,K(u)=quadratic) is a full symmetry reduction to an ordinary differential equation possible. The optimal system of symmetry operators is determined to ensure that a minimal complete set of reductions is obtained. For each reduced partial differential equation, classical Lie group analysis has been performed and further reductions obtained. In this manner, all possible reductions to an ordinary differential equation are found, leading to exact solutions to both the two and three dimensional Burgers' equation.     

Similarity variables and reduction of the heat equation on torus

The problem of symmetry classification for the heat equation on torus is studied by means of classical Lie group theory. The Lie point symmetries are constructed and Lie algebra is formed for equation under consideration. Then these algebras are used to classify its subalgebras up to conjugacy classes. In general the heat equation on torus admits one-, two-, three- and four-dimensional algebras. For one-dimensional algebra £₁ and £₂ the heat equation on torus is reduced in independent variables whereas in two-dimensional algebras £₃ and £₄ the considered heat equation is investigated by quadrature. While three- and four-dimensional algebras lead to a trivial solution.     

Group-Theoretical Analysis of Variable Coefficient Nonlinear Telegraph Equations

Given a class \(\mathcal{F(\theta)}\) of differential equations with arbitrary element θ, the problems of symmetry group, nonclassical symmetry and conservation law classifications are to determine for each member \(f\in\mathcal{F(\theta)}\) the structure of its Lie symmetry group G _f , conditional symmetry Q _f and conservation law \(\mathop {\rm CL}\nolimits _{f}\) under some proper equivalence transformations groups.In this paper, an extensive investigation of these three aspects is carried out for the class of variable coefficient (1+1)-dimensional nonlinear telegraph equations with coefficients depending on the space variable f(x)u _tt =(g(x)H(u)u _x ) _x +h(x)K(u)u _x . The usual equivalence group and the extended one including transformations which are nonlocal with respect to arbitrary elements are first constructed. Then using the technique of variable gauges of arbitrary elements under equivalence transformations, we restrict ourselves to the symmetry group classifications for the equations with two different gauges g=1 and g=h. In order to get the ultimate classification, the method of furcate split is also used and consequently a number of new interesting nonlinear invariant models which have non-trivial invariance algebra are obtained. As an application, exact solutions for some equations which are singled out from the classification results are constructed by the classical method of Lie reduction.The classification of nonclassical symmetries for the classes of differential equations with gauge g=1 is discussed within the framework of singular reduction operator. This enabled to obtain some exact solutions of the nonlinear telegraph equation which are invariant under certain conditional symmetries.Using the direct method, we also carry out two classifications of local conservation laws up to equivalence relations generated by both usual and extended equivalence groups. Equivalence with respect to these groups and correct choice of gauge coefficients of equations play the major role for simple and clear formulation of the final results.     

Classification of Similarity Solutions for Inviscid Burgers’ Equation

Using the basic Lie symmetry method, we find the most general Lie point symmetries group of the inviscid Burgers’ equation. Looking at the adjoint representation of the obtained symmetry group on its Lie algebra, we find the preliminary classification of its group-invariant solutions. The latter provides new exact solutions for the inviscid Burgers’ equation.     

Perturbation of Yamabe equation on Iwasawa N groups in presence of symmetry

Let G be a simple Lie group of real rank one and N be in the Iwasawa decomposition of G. Under the assumption of some symmetries, we obtain an existent result for the nonlinear equation △NU ＋ （1 ＋ ∈K（x, z））u2＊-1 = 0 on N, which generalizes the result of Malchiodi and Uguzzoni to the Kohn＇s subelliptic context on N in presence of symmetry.     

Lie bialgebra structures on derivation Lie algebra over quantum tori

We investigate Lie bialgebra structures on the derivation Lie algebra over the quantum torus. It is proved that, for the derivation Lie algebra W over a rank 2 quantum torus, all Lie bialgebra structures on W are the coboundary triangular Lie bialgebras. As a by-product, it is also proved that the first cohomology group H ¹(W, W ? W) is trivial.     

The Taylor map on complex path groups

Let W(G) denote the path group of an arbitrary complex connected Lie group. The existence of a heat kernel measure νt on W(G) has been shown in [M. Cecil, B.K. Driver, Heat kernel measure on loop and path groups, preprint, http://www.math.uconn.edu/~cecil/papers/p2.pdf; Infin. Dimens. Anal. Quantum Probab. Relat. Top., submitted for publication]. The present work establishes an isometric map, the Taylor map, from the space of L²(νt)-holomorphic functions on W(G) to a subspace of the dual of the universal enveloping algebra of Lie(H(G)), where H(G) is the Lie subgroup of finite energy paths. This map is shown to be surjective in the case where G is a simply connected graded Lie group.     

On Lie’s problem and differential invariants of ODEs <Emphasis Type="Italic">y</Emphasis>″ = <Emphasis Type="Italic">F</Emphasis>(<Emphasis Type="Italic">x</Emphasis>, <Emphasis Type="Italic">y</Emphasis>)

A point classification of ordinary differential equations of the form y″ = F(x, y) is considered. The algebra of differential invariants of the action of the point symmetry pseudogroup on the right-hand sides of equations of the form y″ = F(x, y) is calculated, and Lie’s problem on the point equivalence of such equations is solved.     

Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A₁, A₁⊕A₂, 3A₁, or A3,3⊕A₁. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A₁ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250].     

Analysis of a class of potential Korteweg‐de Vries‐like equations

We analyze a class of third‐order evolution equations, i.e. u_t = f(x, u_x, u_xx) u_xxx+g(x, u_x, u_xx) via the method of preliminary group classification. This method is a systematic means of analyzing the equation for symmetries. We find explicit forms of f and g, which allow for a larger dimensional Lie algebra of point symmetries. Copyright © 2009 John Wiley & Sons, Ltd.     

q-Deformation of W（2, 2） Lie Algebra Associated with Quantum Groups

The q-deformation of W (2, 2) Lie algebra is well defined based on a realization of this Lie algebra by using the famous bosonic and fermionic oscillators in physics. Furthermore, the quantum group structures on the q-deformation of W (2, 2) Lie algebra are completely determined. Finally, the 1-dimensional central extension of the q-deformed W (2, 2) Lie algebra is studied, which turns out to be coincided with the conventional W (2, 2) Lie algebra in the q → 1 limit.