Lie Algebra Of Symmetries Research Articles

Partially invariant solution with an arbitrary surface of blow-up for the gas dynamics equations admitting pressure translation

We applied a method of symmetry reduction to the gas dynamics equations with a special form of the equation of state. This equation of state is a pressure represented as the sum of a density and an entropy functions. The symmetry Lie algebra of the system is 12-dimensional. One, two and three-dimensional subalgebras were considered. In this article, four-dimensional subalgebras are considered for the first time. Specifically, invariants are calculated for 50 four-dimensional subalgebras. Using invariants of one of the subalgebras, a symmetry reduction of the original system is calculated. The reduced system is a partially invariant submodel because one gas-dynamic function cannot be expressed in terms of the invariants. The submodel leads to two families of exact solutions, one of which describes the isochoric motion of the media, and the other solution specifies an arbitrary blow-up surface. For the first family of solutions, the particle trajectories are parabolas or rays; for the second family of solutions, the particles move along cubic parabolas or straight lines. From each point of the blow-up surface, particles fly out at different speeds and end up on a straight line at any other fixed moment in time. A description of the motion of particles for each family of solutions is given.

Invariant analysis of the multidimensional Martinez Alonso–Shabat equation

Abstract This present study is concerned with the group-invariant solutions of the (3 + 1)-dimensional Martinez Alonso–Shabat equation by using the Lie symmetry method. The Lie transformation technique is used to deduce the infinitesimals, Lie symmetry operators, commutation relations, and symmetry reductions. The optimal system for the obtained Lie symmetry algebra is obtained by using the concept of the adjoint map. As for now, the considered model equation is converted into nonlinear ordinary differential equations (ODEs) in two cases in the symmetry reductions. The exact closed-form solutions are obtained by applying constraint conditions on the symmetry generators. Due to the presence of arbitrary functional parameters, these group-invariant solutions are displayed based on suitable numerical simulations. The conservation laws are obtained by using the multiplier method. The conclusion is accounted for toward the end.

Dissimilar subalgebras of symmetry algebra of plasticity equations

In this paper we construct the optimal sets of dissimilar subalgebras up to dimension three for the Lie algebra of point symmetries of the system of three-dimensional stationary equations of perfect plasticity with the Huber–von Mises yield condition. The obtained results can be used to solve the problem of determining all invariant solutions of this system. It was necessary to design algorithms to facilitate some steps of the classification of subalgebras. The computational algebraic system SageMath was chosen to implement these algorithms. The most used functions and procedures are listed. The developed algorithms can be adapted to classify subalgebras of higher dimensions.

Group analysis and invariant solutions of the (3+1)-dimensional defocusing Gardner-KP equation

In this paper, the (3+1)-dimensional defocusing Gardner-KP equation is investigated again with the help of Lie symmetry method since the results are either incorrect, or incomplete, or misleading in the literature. For the Lie algebra of infinitesimal symmetries spanned by eight vector fields, the one-dimensional optimal system of subalgebra is established by leveraging the fundamental invariants and general adjoint transformation representation. Meanwhile, by virtue of some infinitesimal generators in the optimal system, several symmetry reductions and exact solutions are presented.

Spin-charged point particle in a non-Abelian external field with the generalized uncertainty relation

The dynamics of a particle carrying a non-Abelian charge is studied in the presence of a minimal length. By choosing an appropriate non-Abelian gauge field, the system identifies with the Hamiltonian of the Jaynes-Cummings model whose solutions can be determined algebraically. The model has an underlying graded Lie algebra symmetry reminiscent of supersymmetric quantum mechanics. We calculate the energy levels and associated eigenstates using conservation of the number of excitations of the system. Then, we present the effect of the minimal length on the dynamics of the system and we are particularly interested in two particular cases, that of Rabi oscillations and that of the collapse-revival of the wave function. The results show that the higher the deformation parameter, the faster the oscillatory behavior of the atomic inversion.

Extension of Exactly-Solvable Hamiltonians Using Symmetries of Lie Algebras.

Exactly solvable Hamiltonians that can be diagonalized by using relatively simple unitary transformations are of great use in quantum computing. They can be employed for the decomposition of interacting Hamiltonians either in Trotter-Suzuki approximations of the evolution operator for the quantum phase estimation algorithm or in the quantum measurement problem for the variational quantum eigensolver. One of the typical forms of exactly solvable Hamiltonians is a linear combination of operators forming a modestly sized Lie algebra. Very frequently, such linear combinations represent noninteracting Hamiltonians and thus are of limited interest for describing interacting cases. Here, we propose an extension in which the coefficients in these combinations are substituted by polynomials of the Lie algebra symmetries. This substitution results in a more general class of solvable Hamiltonians, and for qubit algebras, it is related to the recently proposed noncontextual Pauli Hamiltonians. In fermionic problems, this substitution leads to Hamiltonians with eigenstates that are single Slater determinants but with different sets of single-particle states for different eigenstates. The new class of solvable Hamiltonians can be measured efficiently using quantum circuits with gates that depend on the result of a midcircuit measurement of the symmetries.

Symmetry algebra classification of scalar n$$ n $$th‐order ordinary differential equations

We obtain a complete classification of scalar th‐order ordinary differential equations for all subalgebras of vector fields in the real plane. While softwares like Maple can compute invariants of a given order, our results are for a general . The cases are well‐known in the literature. Further, it is known that there are three types of th‐order equations depending upon the point symmetry algebra they possess, namely, first‐order equations which admit an infinite dimensional Lie algebra of point symmetries, second‐order equations possessing the maximum 8‐point symmetries, and higher‐order, , admitting the maximum dimensional point symmetry algebra. We show that scalar th‐order equations for do not admit maximally an dimensional real Lie algebra of point symmetries. Moreover, we prove that for , equations can admit two types of dimensional real Lie algebra of point symmetries: one type resulting in nonlinear equations which are not linearizable via a point transformation and the second type yielding linearizable (via point transformation) equations. Furthermore, we present the types of maximal real dimensional and higher than ‐dimensional point symmetry algebras admissible for equations of order and their canonical forms. The types of lower‐dimensional point symmetry algebras which can be admitted are shown, and the equations are constructible as well. We state the relevant results in tabular form and in theorems.

Optimal System, Symmetry Reductions and Exact Solutions of the (2 + 1)-Dimensional Seventh-Order Caudrey–Dodd–Gibbon–KP Equation

In this paper, the (2+1)-dimensional seventh-order Caudrey–Dodd–Gibbon–KP equation is investigated through the Lie group method. The Lie algebra of infinitesimal symmetries, commutative and adjoint tables, and one-dimensional optimal systems is presented. Then, the seventh-order Caudrey–Dodd–Gibbon–KP equation is reduced to nine types of (1+1)-dimensional equations with the help of symmetry subalgebras. Finally, the unified algebra method is used to obtain the soliton solutions, trigonometric function solutions, and Jacobi elliptic function solutions of the seventh-order Caudrey–Dodd–Gibbon–KP equation.

GROUP OF SYMMETRIES FOR THE DYNAMICS SYSTEM OF EQUATIONS OF RAREFIED TWO-PHASE MEDIUM

The symmetries of the system of equations of a two-phase medium are found, where the first phase is gas and the second one is solid particles. The second phase is considered rarefied, it is expressed in the absence of pressure in the equations of motion of the second phase. The medium is assumed to be nonisothermal. Using the methods of group analysis, the Lie algebras of symmetries of the model under study are found in the one-dimensional and three-dimensional cases. The paper describes in detail the process of searching for symmetries in the case of equations of state for a perfect gas. Some partially invariant solutions are found for the one-domensional system.

New Approximate Symmetry Theorems and Comparisons With Exact Symmetries

Three new approximate symmetry theories are proposed. The approximate symmetries are contrasted with each other and with the exact symmetries. The theories are applied to nonlinear ordinary differential equations for which exact solutions are available. It is shown that from the symmetries, approximate solutions as well as exact solutions in some restricted cases can be retrievable. Depending on the specific approximate theory and the equations considered, the approximate symmetries may expand the Lie Algebra of the exact symmetries, may be a perturbed form of the exact symmetries or may be a subalgebra of the exact symmetries. Exact and approximate solutions are retrieved using the symmetries.

Novel solitonic structure, Hamiltonian dynamics and lie symmetry algebra of biofilm

In this study, the Lie point symmetries and optimal system have been established. We discuss the biofilm model’s soliton solutions. To examine a nonlinear dynamical biofilm system, which is simply a bistable Allen–Cahn equation with quartic potential, to determine solitary wave profiles using the considered equation, a new auxiliary equation technique is used. A suitable variable transformation is used to convert the governing equation into a nonlinear ordinary differential equation. The unified approach is utilized to evaluate periodic solutions, solitary and soliton solutions as well as several newly discovered exact solitary wave solutions, it can be accomplished via Mathematica (https://www.wolfram.com/mathematica/online/). The new auxiliary equation approach is an efficient method for creating unique wave profiles based on a variety of soliton families. Also, the results are graphically visualized, by using the appropriate parametric settings. The outcomes are shown graphically in two dimensions, three dimensions, and contour form. The Hamiltonian conditions are satisfied by the planer dynamical framework of equations to ensure as the system that was generated is a conservative Hamiltonian dynamical system which includes all traveling wave structures. A sensitivity evaluation is used to explore the governing model’s thoroughly dynamical properties. It is demonstrated that the model becomes more dependent on the beginning conditions than the variables. The strategy could be used to look for exact solutions to various nonlinear evolution equations. We believe that this study will have important applications in different areas of science.

Symmetry analysis of the canonical connection on Lie groups: six-dimensional case with abelian nilradical and one-dimensional center

<abstract><p>In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by $ A_{6, 20} $–$ A_{6, 27} $. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified.</p></abstract>

Linearization of Second-Order Non-Linear Ordinary Differential Equations: A Geometric Approach

Using the coefficients of a system semilinear cubic in the first derivative second order differential equations one defines a connection in the space of the independent and dependent variables, which is specified modulo two free parameters. In this way, to any such equation one associates an affine space which is not necessarily Riemannian, that is, a metric is not required. If such a metric exists, then under the Cartan parametrization the geodesic equations of the metric coincide with the system of the considered semilinear equations. In the present work, we consider semilinear cubic in the first derivative second order differential equations whose Lie symmetry algebra is the sl(3,R). The covariant condition for these equations is the vanishing of the curvature tensor. We demonstrate the method in the solution of the Painlevé-Ince equation and in a system of two equations. Because the approach is geometric, the number of equations in the system is not important besides the complication in the calculations. It is shown that it is possible to linearize an equation in this form using a different covariant condition, for example, assuming the space to be of constant non-vanishing curvature. Finally, it is shown that one computes the associated metric to a semilinear cubic in the first derivatives differential equation using the inverse transformation derived from the transformation of the connection.

Generalized teleparallel de Sitter geometries

Theories of gravity based on teleparallel geometries are characterized by the torsion, which is a function of the coframe, derivatives of the coframe, and a zero curvature and metric compatible spin-connection. The appropriate notion of a symmetry in a teleparallel geometry is that of an affine symmetry. Due to the importance of the de Sitter geometry and Einstein spaces within General Relativity, we shall describe teleparallel de Sitter geometries and discuss their possible generalizations. In particular, we shall analyse a class of Einstein teleparallel geometries which have a 4-dimensional Lie algebra of affine symmetries, and display two one-parameter families of explicit exact solutions.

Diffeomorphisms as quadratic charges in 4d BF theory and related TQFTs

We present a Sugawara-type construction for boundary charges in 4d BF theory and in a general family of related TQFTs. Starting from the underlying current Lie algebra of boundary symmetries, this gives rise to well-defined quadratic charges forming an algebra of vector fields. In the case of 3d BF theory (i.e. 3d gravity), it was shown in [1] that this construction leads to a two-dimensional family of diffeomorphism charges which satisfy a certain modular duality. Here we show that adapting this construction to 4d BF theory first requires to split the underlying gauge algebra. Surprisingly, the space of well-defined quadratic generators can then be shown to be once again two-dimensional. In the case of tangential vector fields, this canonically endows 4d BF theory with a diff(S2) × diff(S2) or diff(S2) ⋉ vect(S2)ab algebra of boundary symmetries depending on the gauge algebra. The prospect is to then understand how this can be reduced to a gravitational symmetry algebra by imposing Plebański simplicity constraints.

The smallest interacting universe

We study a mechanism by which the most basic structures of quantum physics can emerge from the most meager of starting points, a Hilbert space, lacking any preassigned structure such as a tensor decomposition, and a loss function. In a simple toy model of the universe, we hypothesize a fundamental loss functional for the combined Hamiltonian and quantum state, and then minimize this loss functional by gradient descent. We find that this minimization gives rise to a co-emergence of locality, i.e. a tensor product structure simultaneously respected by both the Hamiltonian and the state, suggesting that locality can emerge by a process analogous to spontaneous symmetry breaking. We discuss the relevance of this program to the arrow of time problem.In our toy model, we interpret the emergence of a tensor factorization as the appearance of individual degrees of freedom within a previously undifferentiated (raw) Hilbert space. Earlier work [5, 6] looked at the emergence of locality in Hamiltonians only, and in that context found strong numerical confirmation of the hypothesis that raw Hilbert spaces of dim = n are unstable and prefer to settle on tensor factorization when n is not prime, expressing, for example, n = pq, and in [6] even primes were seen to “factor” after first shedding a small summand, e.g. 7 = 1 + 2 · 3. This was found in the context of a rather general potential functional F on the space of metrics {gij} on mathfrak{su} (n), the Lie algebra of symmetries. This emergence of qunits through operator-level spontaneous symmetry breaking (SSB) may help us understand why the world seems to consist of myriad interacting degrees of freedom. But understanding why the universe has an initial Hamiltonian H0 with a many-body structure is of limited conceptual value unless the initial state, ∣ψ0〉, is also structured by this tensor decomposition. Here we adapt F to become a functional on {g, | ψ0〉} = (metrics) × (initial states), and find SSB now produces a conspiracy between g and ∣ψ0〉, where they simultaneously attain low entropy by jointly settling on the same qubit decomposition. Extreme scaling of the computational problem has confined us to studying ℂ4 breaking to ℂ2 ⊗ ℂ2 and ℂ8 breaking to ℂ2 ⊗ ℂ4 or ℂ2 ⊗ ℂ2 ⊗ ℂ2.

Classification of the symmetry Lie algebras for six-dimensional co-dimension two Abelian nilradical Lie algebras

<abstract><p>In this paper, we consider the symmetry algebra of the geodesic equations of the canonical connection on a Lie group. We mainly consider the solvable indecomposable six-dimensional Lie algebras with co-dimension two abelian nilradical that have an abelian complement. In dimension six, there are nineteen such algebras, namely, $ A_{6, 1} $–$ A_{6, 19} $ in Turkowski's list. For each algebra, we give the geodesic equations, a basis for the symmetry Lie algebra in terms of vector fields, and finally we identify the symmetry Lie algebra from standard lists.</p></abstract>

Reduce-Order Modeling and Higher Order Numerical Solutions for Unsteady Flow and Heat Transfer in Boundary Layer with Internal Heating

We obtain similarity transformations to reduce a system of partial differential equations representing the unsteady fluid flow and heat transfer in a boundary layer with heat generation/absorption using Lie symmetry algebra. There exist seven Lie symmetries for this system of differential equations having three independent and three dependent variables. We use these Lie symmetries for the reduced-order modeling of the flow equations by constructing invariants corresponding to linear combinations of these Lie point symmetries. This procedure reduces one independent variable of the concerned fluid flow model when applied once. Double reductions are achieved by employing invariants twice that lead to ordinary differential equations with one independent and two dependent variables. Similarity transformations are constructed using these two sets of derived invariants corresponding to linear combinations of the Lie point symmetries. These similarity transformations have not been obtained earlier for this flow model. Moreover, the corresponding reduced systems of ordinary differential equations are different from those which exist in the literature for fluid flow and heat transfer that we have been dealing with. We obtain multiple similarity transformations which lead us to new classes of systems of ordinary differential equations. Accurate numerical solutions of these systems are obtained using the combination of an adaptive fourth-order Runge–Kutta method and shooting procedure. Effects of variation of unsteadiness parameter, Prandtl number and heat generation/absorption on fluid velocity, skin friction, surface temperature and heat flux are studied and presented with the help of tables and figures.

Null-forms of conic systems in R3 are determined by their symmetries

We address the problem of characterisation of null-forms of conic 3-dimensional systems, that is, control-affine systems whose field of admissible velocities forms a conic (without parameters) in the tangent space. Those systems have been previously identified as the simplest control systems under a conic nonholonomic constraint or as systems of zero curvature. In this work, we propose a direct characterisation of null-forms of conic systems among all control-affine systems by studying the Lie algebra of infinitesimal symmetries. Namely, we show that the Lie algebra of infinitesimal symmetries characterises uniquely null-forms of conic systems.

Symmetries and Casimirs of radial compressible fluid flow and gas dynamics inn > 1 dimensions

Symmetries and Casimirs are studied for the Hamiltonian equations of radial compressible fluid flow inn>1dimensions. An explicit determination of all Lie point symmetries is carried out, from which a complete classification of all maximal Lie symmetry algebras is obtained. The classification includes all Lie point symmetries that exist only for special equations of state. For a general equation of state, the hierarchy of advected conserved integrals found in the recent work is proved to consist of Hamiltonian Casimirs. A second hierarchy that holds only for an entropic equation of state is explicitly shown to comprise non-Casimirs, which yield a corresponding hierarchy of generalized symmetries through the Hamiltonian structure of the equations of radial fluid flow. The first-order symmetries are shown to generate a non-abelian Lie algebra. Two new kinematic conserved integrals found in the recent work are likewise shown to yield additional first-order generalized symmetries holding for a barotropic equation of state and an entropic equation of state. These symmetries produce an explicit transformation group acting on solutions of the fluid equations. Since these equations are well known to be equivalent to the equations of gas dynamics, all of the results obtained forn-dimensional radial fluid flow carry over to radial gas dynamics.