Lie Algebra Of Symmetries Research Articles

Lie remarkable partial differential equations characterized by Lie algebras of point symmetries

Within the framework of inverse Lie problem, we give some non-trivial examples of coupled Lie remarkable equations, i.e., classes of differential equations that are in correspondence with their Lie point symmetries. In particular, we determine hierarchies of second order partial differential equations uniquely characterized by affine transformations of Rn+m, and a system of two third order partial differential equations in two independent variables uniquely determined by the Lie algebra of projective transformations of R4.

A mechanical algorithm for constructing structural constants of the Lie algebra of symmetry of differential equations based on Wu's method

A mechanical algorithm for constructing structural constants of the Lie algebra of symmetry of differential equations based on Wu's method

Leibniz–Chern–Simons Theory and Phases of Exceptional Field Theory

We discuss a generalization of Chern-Simons theory in three dimensions based on Leibniz (or Loday) algebras, which are generalizations of Lie algebras. Special cases of such theories appear in gauged supergravity, where the Leibniz algebra is defined in terms of the global (Lie) symmetry algebra of the ungauged limit and an embedding tensor. We show that the Leibniz algebra of generalized diffeomorphisms in exceptional field theory can similarly be obtained from a Lie algebra that describes the enhanced symmetry of an `ungauged phase' of the theory. Moreover, we show that a `topological phase' of ${\rm E}_{8(8)}$ exceptional field theory can be interpreted as a Chern-Simons theory for an algebra unifying the three-dimensional Poincar\'e algebra and the Leibniz algebra of ${\rm E}_{8(8)}$ generalized diffeomorphisms.

An Evolutionary Approach to the Automatic Classification of Automorphisms of Lower-Dimensional Lie Algebras

We implement two algorithms in MATHEMATICA for classifying automorphisms of lower-dimensional non-Abelian Lie algebras. The first algorithm is a brute-force approach whereas the second is an evolutionary strategy. These algorithms are delivered as the MATHEMATICA package cwsAutoClass. In order to facilitate the application of this package to symmetry Lie algebras of differential equations, we also provide a package, cwsLieSymTools, for manipulating finite-dimensional Lie algebras of vector fields. In particular, this package allows the computations of Lie brackets, structure constants, and the visualization of commutator tables. Several examples are provided to illustrate the pertinence of our approach. These computational experiments show that the evolutionary strategy outperformed consistently the brute-force method.

Reduced Pre-Lie Algebraic Structures, the Weak and Weakly Deformed Balinsky–Novikov Type Symmetry Algebras and Related Hamiltonian Operators

The Lie algebraic scheme for constructing Hamiltonian operators is differential-algebraically recast and an effective approach is devised for classifying the underlying algebraic structures of integrable Hamiltonian systems. Lie–Poisson analysis on the adjoint space to toroidal loop Lie algebras is employed to construct new reduced pre-Lie algebraic structures in which the corresponding Hamiltonian operators exist and generate integrable dynamical systems. It is also shown that the Balinsky–Novikov type algebraic structures, obtained as a Hamiltonicity condition, are derivations on the Lie algebras naturally associated with differential toroidal loop algebras. We study nonassociative and noncommutive algebras and the related Lie-algebraic symmetry structures on the multidimensional torus, generating via the Adler–Kostant–Symes scheme multi-component and multi-dimensional Hamiltonian operators. In the case of multidimensional torus, we have constructed a new weak Balinsky–Novikov type algebra, which is instrumental for describing integrable multidimensional and multicomponent heavenly type equations. We have also studied the current algebra symmetry structures, related with a new weakly deformed Balinsky–Novikov type algebra on the axis, which is instrumental for describing integrable multicomponent dynamical systems on functional manifolds. Moreover, using the non-associative and associative left-symmetric pre-Lie algebra theory of Zelmanov, we also explicate Balinsky–Novikov algebras, including their fermionic version and related multiplicative and Lie structures.

Symmetry Reduction and Soliton-Like Solutions for the Generalized Korteweg-De Vries Equation

We analyze the gKdV equation, a generalized version of Korteweg-de Vries with an arbitrary function f(u). In general, for a function f(u) the Lie algebra of symmetries of gKdV is the 2-dimensional Lie algebra of translations of the plane xt. This implies the existence of plane wave solutions. Indeed, for some specific values of f(u) the equation gKdV admits a Lie algebra of symmetries of dimension grater than 2. We compute the similarity reductions corresponding to these exceptional symmetries. We prove that the gKdV equation has soliton-like solutions under some general assumptions, and we find a closed formula for the plane wave solutions, that are of hyperbolic secant type.

Generalized Squeeze Equation and Its Symmetry

I study the solutions, symmetry, and the map under time inversion (t ↦ 1/t) of the following generalized squeeze equation (GSQE) $$_{n,m} u \equiv \left[ {\frac{\partial }{{\partial t}} - k\left( {\sum\limits_{k = 1}^n {\frac{{\partial ^2 }}{{\partial x_k^2 }} - \frac{\gamma }{{t^2 }}\sum\limits_{r = 1}^m {\frac{{\partial ^2 }}{{\partial p_r^2 }}} } }\right)} \right]u\left( {t,x_k ,p_r } \right) = 0,$$ which is a formal generalization of the squeeze equation SQE, $$Q \equiv \left[ {\partial t - \frac{1}{4}\partial x^2 + \frac{1}{{4t^2 }}\partial p^2 } \right]Q\left( {t,x,p} \right) = 0.$$ I determine the Lie symmetry algebra gn,m of the GSQE, which yields a deeper understanding of the Lie symmetry algebra gn,0 of the n-dimensional heat equation. I introduced the parameter γ to obtain an ‘internal contraction’ of so(n,m) to iso(n,m), similar to that of the hydrogen atom.

Symmetry Lie algebra and exact solutions of some fourth-order difference equations

Symmetry Lie algebra and exact solutions of some fourth-order difference equations

Lie symmetry analysis of a variable coefficient Calogero–Degasperis equation

We consider a general class of variable coefficient Calogero–Degasperis equations. The complete Lie group classification is performed with the aid of the appropriate equivalence group. Lie symmetries are used to derive a number of reductions by constructing the corresponding optimal lists of one-dimensional subalgebras of the Lie symmetry algebras. Furthermore, a number of non-Lie reductions are given. One of the reduced equations is the variable coefficient potential KdV equation which is studied from the point of view of Lie group analysis.

Linearization of third‐order ordinary differential equations via point transformations

The linearization problem for scalar third‐order ordinary differential equations via point transformations was solved partially in the works of Al‐Dweik et al by the use of the Cartan equivalence method. In order to solve this problem completely, the Cartan equivalence method is applied to provide an invariant characterization of the linearizable third‐order ordinary differential equation , which admits a four‐dimensional point symmetry Lie algebra. The invariant characterization is given in terms of function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained invariant is also presented. The approach provides auxiliary functions, which can be effectively utilized to determine the point transformation that does the reduction to the equivalent canonical form. Furthermore, illustrations to the main theorem and applications are given.

Invariant functions of vector field realizations of Lie algebras and some applications to representation theory and dynamical systems

Realizations of Lie algebras by means vector fields associated to a linear representation and their corresponding invariant functions are inspected from the perspective of the embedding problem of Lie algebras and the branching rules for the subduction of representations. Some applications concerning the construction of dynamical systems with prescribed Lie algebras of point symmetries and the symmetry breaking with respect to embeddings of algebras into (N) are discussed.

Nonlocal Symmetries of Integrable Linearly Degenerate Equations: A Comparative Study

We continue the study of Lax integrable equations. We consider four three-dimensional equations: (1) the rdDym equation uty = uxuxy − uyuxx, (2) the Pavlov equation uyy = utx + uyuxx − uxuxy, (3) the universal hierarchy equation uyy = utuxy − uyutx, and (4) the modified Veronese web equation uty = utuxy − uyutx. For each equation, expanding the known Lax pairs in formal series in the spectral parameter, we construct two differential coverings and completely describe the nonlocal symmetry algebras associated with these coverings. For all four pairs of coverings, the obtained Lie algebras of symmetries manifest similar (but not identical) structures; they are (semi)direct sums of the Witt algebra, the algebra of vector fields on the line, and loop algebras, all of which contain a component of finite grading. We also discuss actions of recursion operators on shadows of nonlocal symmetries.

Random k-Body Ensembles for Chaos and Thermalization in Isolated Systems.

Embedded ensembles or random matrix ensembles generated by k-body interactions acting in many-particle spaces are now well established to be paradigmatic models for many-body chaos and thermalization in isolated finite quantum (fermion or boson) systems. In this article, briefly discussed are (i) various embedded ensembles with Lie algebraic symmetries for fermion and boson systems and their extensions (for Majorana fermions, with point group symmetries etc.); (ii) results generated by these ensembles for various aspects of chaos, thermalization and statistical relaxation, including the role of q-hermite polynomials in k-body ensembles; and (iii) analyses of numerical and experimental data for level fluctuations for trapped boson systems and results for statistical relaxation and decoherence in these systems with close relations to results from embedded ensembles.

Invariant characterization of third-order ordinary differential equations [formula omitted] with five-dimensional point symmetry group

The Cartan equivalence method is applied to provide an invariant characterization of the third-order ordinary differential equation u″′=f(x,u,u′,u″) which admits a five-dimensional point symmetry Lie algebra. The invariant characterization is given in terms of the function f in a compact form. A simple procedure to construct the equivalent canonical form by use of an obtained constant invariant is also presented. We also show how one obtains the point transformation that does the reduction to linear form. Moreover, some applications are provided.

A cosmological study of Einstein–Skyrme model in anisotropic Kantowski–Sachs spacetime using Lie and Noether symmetries

The present work deals with anisotropic (but zero heat flux) Skyrme fluid in a locally rotational Kantowski–Sachs (KS) spacetime in the background of Einstein gravity. The Lie point symmetry is imposed to the system of Einstein field equations and unknown parameters are either determined or interrelated by the imposition of the symmetry. Subsequently, Noether symmetry, a point-like symmetry of the Lagrangian is used and it is found that the Lie algebra of the Noether symmetry is a subalgebra of the corresponding Lie algebra of the Lie symmetry. Then a point transformation in the 2D augmented space is taken in such a manner that one of the variables becomes cyclic and hence the Lagrangian as well as the evolution equations are simplified to a great extent. Finally, solutions to the physical system are presented and are analyzed physically.

Exhaustive Classification of the Invariant Solutions for a Specific Nonlinear Model Describing Near Planar and Marginally Long-Wave Unstable Interfaces for Phase Transition

Problems of thermodynamic phase transition originate inherently in solidification, combustion and various other significant fields. If the transition region among two locally stable phases is adequately narrow, the dynamics can be modeled by an interface motion. This paper is devoted to exhaustive analysis of the invariant solutions for a modified Kuramoto-Sivashinsky equation in two spatial and one temporal dimensions is presented. This nonlinear partial differential equation asymptotically characterizes near planar interfaces, which are marginally long-wave unstable. For this purpose, by applying the classical symmetry method for this model the classical symmetry operators are attained. Moreover, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras, which yields the preliminary classification of group invariant solutions is constructed. Mainly, the Lie invariants corresponding to the infinitesimal symmetry generators as well as associated similarity reduced equations are also pointed out. Furthermore, the nonclassical symmetries of this nonlinear PDE are also comprehensively investigated.

Invariant characterization of scalar third‐order ODEs that admit the maximal point symmetry Lie algebra

The Cartan equivalence method is used to deduce an invariant characterization of the scalar third‐order ordinary differential equation , which admits the maximal 7‐dimensional point symmetry Lie algebra. The method provides auxiliary functions that can be used to efficiently obtain the point transformation that does the reduction to the simplest linear equation . Moreover, examples are given to illustrate the method.

Invariant solutions to the Khokhlov-Zabolotskaya singular manifold equation and their application

We find the Lie algebra of point symmetries for the singular manifold equation associated with the Khokhlov–Zabolotskaya equation (KhZ), classify one-dimensional subalgebras of this algebra, and find corresponding invariant solutions . Then we use the Backlund transformation between the singular manifold equation and KhZ to find new exact solutions to the last equation.

Lie symmetry based-analytical and numerical approach for modified Burgers-KdV equation

Abstract In this work, the variable-coefficient modified Burgers-KdV equation, which arises in modeling various physical phenomena, is studied for exact and numerical solution based on Lie symmetry. The infinitesimals of the group of transformations which leaves this equation invariant are furnished along with the admissible forms of the variable coefficients. The optimal systems of one-dimensional subalgebras of the Lie symmetry algebras are determined with the adjoint action of the symmetry group. These are then used to establish new power series solution and exact solutions of variable-coefficient modified Burgers-KdV equation. Further, RK4 (e.g. Fourth Order Runge Kutta) method is applied to the reduced ODE for constructing numerical solutions of the modified Burger-KdV equation.

A Note on the Lie Algebra of the Invariants in the CBS Nonlinear Equation

In this short note, a particular realization of the vector fields that form a Lie Algebra of symmetries for the Calogero-Bogoyavleskii-Schiff equation is found. The Lie Algebra is examined and the result is a semidirect product of two Lie Groups. The structure of the semidirect product is examined through the table of commutation rules. Two reductions are made with the help of two sets of generators and the final outcome for the solution is related to the elliptic Painlevé P(ξ)-function.