Lie Symmetry Group Research Articles

On some non-static plane symmetric perfect fluid solutions in [formula omitted] gravity

We address the problem of finding non-static plane symmetric perfect fluid solutions in the f(R,T) theory of gravity, where R and T respectively are the Ricci scalar and trace of the energy momentum tensor. In order to obtain solutions, a classification has been proposed for such spacetimes. Solutions are then calculated in each of these classes, which include special Bianchi-Type I and static plane symmetric geometries. The static geometries are no surprise, but importantly, non-static spacetime metrics have also been found. It is well known that the class of isometries are the source of deriving the conservation laws. Therefore, we work out the Killing symmetries to determine the Lie symmetry groups admitted by the obtained spacetimes. As a result of our analysis, we find that the obtained spacetimes admit 10, 7, 6, 5, 4 and 3 Killing vector fields.

Non‐classical Lie symmetries for nonlinear time‐fractional Heisenberg equations

The current work is based on performing the non‐classical and classical Lie group analysis for a system of nonlinear time‐fractional Heisenberg equation. Here, we apply analysis of the Lie group symmetry as one of the powerful tools dealing with the large class of fractional order differential equations in the Riemann–Liouville (RL) sense. Indeed, classical and non‐classical Lie symmetries group is used to similarity reduction of nonlinear time‐fractional Heisenberg equation. Especially, based on the obtained Lie symmetry generators, we construct conservation laws for the classical and non‐classical vector fields related to the Heisenberg time‐fraction equation.

Solving the initial value problem of ordinary differential equations by Lie group based neural network method.

To combine a feedforward neural network (FNN) and Lie group (symmetry) theory of differential equations (DEs), an alternative artificial NN approach is proposed to solve the initial value problems (IVPs) of ordinary DEs (ODEs). Introducing the Lie group expressions of the solution, the trial solution of ODEs is split into two parts. The first part is a solution of other ODEs with initial values of original IVP. This is easily solved using the Lie group and known symbolic or numerical methods without any network parameters (weights and biases). The second part consists of an FNN with adjustable parameters. This is trained using the error back propagation method by minimizing an error (loss) function and updating the parameters. The method significantly reduces the number of the trainable parameters and can more quickly and accurately learn the real solution, compared to the existing similar methods. The numerical method is applied to several cases, including physical oscillation problems. The results have been graphically represented, and some conclusions have been made.

Lie group analysis for a higher-order Boussinesq-Burgers system

Boussinesq-Burgers (BB)-type equations have been proposed to model the shallow water waves. Under investigation in this Letter is a higher-order BB system. We obtain the Lie point symmetry generators, Lie symmetry groups and symmetry reductions for that system via the Lie group method. Hyperbolic-function, trigonometric-function and rational solutions for that system are derived.

Symmetry group analysis and conservation laws of the potential modified KdV equation using the scaling method

In this paper, using the Lie symmetry method, we obtain the Lie symmetry group of the potential modified Korteweg–de Vries (potential mKdV) equation, and we construct the optimal system of one-dimensional subalgebras for the Lie symmetry algebra of this equation. Furthermore, similarity reductions related to the infinitesimal generators of the Lie symmetry group are obtained. Also, we construct new conservation laws for this equation by the scaling method. This method is algorithmic and based on the linear algebra tools and calculus of variations. The density of the conservation laws is constructed using the Euler operator and scaling symmetry of the equation and also the associated flux is calculated by the homotopy operator. This density-flux pair renders a conservation law for the potential mKdV equation. We specify the density-flux pairs of conservation laws in ranks 4, 8, 12, 16 and 20 for the potential mKdV equation.

Emergence of Lie group symmetric classical spacetimes in the canonical tensor model

AbstractWe analyze a wave function of a tensor model in the canonical formalism, when the argument of the wave function takes Lie group invariant or nearby values. Numerical computations show that there are two phases, which we call the quantum and the classical phases. In the classical phase fluctuations are suppressed, and configurations emerge which are discretizations of classical geometric spaces invariant under Lie group symmetries. This is explicitly demonstrated for emergence of Sn (n = 1, 2, 3) for SO(n + 1) symmetries by checking the topological and geometric (Laplacian) properties of the emerging configurations. The transition between the two phases has the form of splitting/merging of distributions of variables, resembling a matrix model counterpart, namely the transition between one-cut and two-cut solutions. However, this resemblance is obscured by a difference in the mechanism of distribution in our setup from that in the matrix model. We also discuss this transition as a replica symmetry breaking. We perform various preliminary studies of the properties of the phases and the transition for such values of the argument.

Majors About the New Symmetry of a Nonlinear Acoustics Model

The Lie symmetry group of the Zabolotskaya-Khokhlov equation has been seriously studied, earlier than this. Eventually, it has been endowed with a general model by N.J.C. Ndogmo, in 2008. This research is devoted to introducing the algebra, the group, and the reductions of a new symmetry which is an exception of that model.

Symmetries of optimal system, various closed-form solutions, and propagation of different wave profiles for the Boussinesq–Burgers system in ocean waves

Water waves, a common natural phenomenon, have been influential in various fields, such as energy development, offshore engineering, mechanical engineering, and hydraulic engineering. To describe the shallow water waves near an ocean coast or in a lake, we use the (1 + 1)-dimensions Boussinesq–Burgers system. By means of Lie symmetry analysis, symmetry groups and infinitesimal generators are obtained for the (1 + 1)-dimension Boussinesq–Burgers system. For the sake of finding the invariant solutions of the Boussinesq–Burgers system, the optimal one-dimensional subalgebra system is computed. Furthermore, using similarity reduction and the generalized Kudryashov method, we attain the abundant wave solutions of the Boussinesq–Burgers system presented in this research paper. Additionally, the exact solutions, which illustrate the effectiveness of the proposed method, also reveal the physical interpretation of the nonlinear models. To demonstrate the significance of interaction phenomena, dynamical behaviors of some attained solutions are depicted geometrically and theoretically through suitable parameter values. Consequently, kink, singular, periodic, solitary wave solutions, and their elastic nature have been shown to validate these solutions with physical phenomena. With the aid of the obtained results, the researchers could gain an understanding of the different modes of shallow water waves nearby an ocean beach. The computational work ascertained that the imposed methods are sturdy, precise, modest, and widely applicable.

Lie Group Symmetry Analysis and Invariant Difference Schemes of the Two-Dimensional Shallow Water Equations in Lagrangian Coordinates

Lie Group Symmetry Analysis and Invariant Difference Schemes of the Two-Dimensional Shallow Water Equations in Lagrangian Coordinates

Optimal System and Exact Solutions of Monge-Ampere Equation

A solution that remains unchanged when transformed under Lie group of point symmetries of the differential equation is an invariant solution of the differential equation. Optimal system of Lie group of point symmetry generators provide all possible invariant solutions of differential equation. Here, using optimal system of Lie point symmetry generators group invariant solutions are obtained. Using these solutions, exact solutions of non-homogeneous Monge-Ampere equation have been presented here.

Lie group symmetries' complete classification for a generalized Chazy equation and its equivalence group

In this work, a complete classification of the Lie group symmetries for a generalization of Chazy equation was carried out and the equivalence group for the generalized Chazy equation is calculated and used to present the principal algebra of the equation.

On quantum jumps and attractors of the Maxwell–Schrödinger equations

Our goal is the discussion of the problem of mathematical interpretation of basic postulates (or “principles”) of Quantum Mechanics: transitions to quantum stationary orbits, the wave-particle duality, and the probabilistic interpretation, in the context of semiclassical self-consistent Maxwell–Schrödinger equations. We discuss possible dynamical interpretation of these postulates relying on a new general mathematical conjecture on global attractors of G-invariant nonlinear Hamiltonian partial differential equations with a Lie symmetry group G. This conjecture is inspired by the results on global attractors of nonlinear Hamiltonian PDEs obtained by the author together with his collaborators since 1990 for a list of model equations with three basic symmetry groups: the trivial group, the group of translations, and the unitary group \(\mathbf {U}(1)\). We sketch these results.

Optimal system, reductions and Lie algebra classification for Kudryashov–Sinelshchikov equations of second order

In this work, the optimal system associated to the second order Kudryashov–Sinelshchikov equation is calculated. Such optimal system is obtained from the Lie symmetry group corresponding to the equation previously mentioned and using this algebra we are able to reduce this equation to known first order equations. In this way we find a new solutions, a wave travelling solution, which is represented graphically. Moreover, we classify the Lie algebra associated to the given equation.

Analysis of pressure and heat distribution in a dilating or contracting filter chamber with two outlets using multivariate spectral quasilinearization method

AbstractThe development of efficient filters is an essential part of industrial machinery design, specifically to increase the lifespan of a machine. In the filter chamber design considered in this study, the magnetic material is placed along the horizontal surface of the filter chamber. The inside of the filter chamber is layered with a porous material to restrict the outflow of unwanted particles. This study aims to investigate the flow, pressure, and heat distribution in a dilating or contracting filter chamber with two outlets driven by injection through a permeable surface. The proposed model of the fluid dynamics within the filter chamber follows the conservation equations in the form of partial differential equations. The model equations are further reduced to a steady case through Lie's symmetry group of transformation. They are then solved using a multivariate spectral‐based quasilinearization method on the Chebyshev–Gauss–Lobatto nodes. Insights and analyses of the thermophysical parameters that drive optimal outflow during the filtration process are provided through the graphs of the numerical solutions of the differential equations. We find, among other results, that expansion of the filter chamber leads to an overall decrease in internal pressure and an increase in heat distribution inside the filter chamber. The results also show that shrinking the filter chamber increases the internal momentum inside the filter, which leads to more outflow of filtrates.

On the Symmetry of Blast Waves

This paper presents a brief historical review of G. I. Taylor’s solution of the point blast wave problem which was applied to the Trinity test of the first atomic bomb. Lie group symmetry techniques (also referred to throughout this paper as geometric techniques) are used to derive Taylor’s famous two-fifths law that relates the position of a blast wave to the time after the explosion and the total energy released. The theory of exterior differential systems is combined with the method of characteristics to demonstrate that the solution of the blast wave problem is directly related to the basic relationships that exist between the symmetry (or geometry) and the physics of wave propagation through the equations of motion. The point blast wave model is cast in terms of two exterior differential systems, and both systems are shown to be integrable with local solutions for the velocity, pressure, and density along curves in space and time behind the blast wave. This work is dedicated to the memory of Professor Roy Axford, who introduced many of his students to the topic of symmetry analysis of differential equations.

Optimal system, invariant solutions and complete classification of Lie group symmetries for a generalized Kummer-Schwarz equation and its Lie algebra representation

We obtain the complete classification of the Lie symmetry group and the optimal system’s generating operators associated with a particular case of the generalized Kummer - Schwarz equation. Using those operators we characterize all invariant solutions, alternative solutions were found for the equation studied and the Lie algebra associated with the symmetry group is classified.

Two Isospectral-Nonisospectral Super-Integrable Hierarchies and Related Invariant Solutions

In this article, we adopt two kinds of loop algebras corresponding to the Lie algebra B(0,1) to introduce two line spectral problems with different numbers of even and odd superfunctions. Through generalizing the time evolution λt to a polynomial of λ, two isospectral-nonisospectral super integrable hierarchies are derived in terms of Tu scheme and zero-curvature equation. Among them, the first super integrable hierarchy is further reduced to generalized Fokker–Plank equation and special bond pricing equation, as well as an explicit super integrable system under the choice of specific parameters. More specifically, a super integrable coupled equation is derived and the corresponding integrable properties are discussed, including the Lie point symmetries and one-parameter Lie symmetry groups as well as group-invariant solutions associated with characteristic equation.

A time-dependent energy-momentum method

We devise a generalisation of the energy momentum-method for studying the stability of non-autonomous Hamiltonian systems with a Lie group of Hamiltonian symmetries. A generalisation of the relative equilibrium point notion to a non-autonomous realm is provided and studied. Relative equilibrium points of a class of non-autonomous Hamiltonian systems are described via foliated Lie systems, which opens a new field of application of such systems of differential equations. We reduce non-autonomous Hamiltonian systems via the Marsden–Weinstein theorem and we provide conditions ensuring the stability of the projection of relative equilibrium points to the reduced space. As a byproduct, a geometrical extension of notions and results from Lyapunov stability theory on linear spaces to manifolds is provided. As an application, we study a class of mechanical systems, the hereafter called almost-rigid bodies, which covers rigid bodies as a particular instance.

Variational Bayesian Iteration-Based Invariant Kalman Filter for Attitude Estimation on Matrix Lie Groups

Motivated by the rapid progress of aerospace and robotics engineering, the navigation and control systems on matrix Lie groups have been actively studied in recent years. For rigid targets, the attitude estimation problem is a benchmark one with its states defined as rotation matrices on Lie groups. Based on the invariance properties of symmetry groups, the invariant Kalman filter (IKF) has been developed by researchers for matrix Lie group systems; however, the limitation of the IKF is that its estimation performance is prone to be degraded if the given knowledge of the noise statistics is not accurate. For the symmetry Lie group attitude estimation problem, this paper proposes a new variational Bayesian iteration-based adaptive invariant Kalman filter (VBIKF). In the proposed VBIKF, the a priori error covariance is not propagated by the conventional steps but directly calibrated in an iterative manner based on the posterior sequences. The main advantage of the VBIKF is that the statistics parameter of the system process noise is no longer required and so the IKF’s hard dependency on accurate process noise statistics can be reduced significantly. The mathematical foundation for the new VBIKF is presented and its superior performance in adaptability and simplicity is further demonstrated by numerical simulations.

Invariant analysis and conservation laws of the time-fractional [formula omitted]-family peakon equations

In this paper, we further extend the theories of Lie symmetry group and conservation law to study the time-fractional b-family peakon equations. The main distinction of the equations with the usual time-fractional partial differential equations is the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order x-derivative. Thus we first give a prolongation formula of the infinitesimal generator for the case of mixed derivative, then after finding the Lie symmetries, we use them to transform the equations into fractional and integer-order ordinary differential equations respectively. Some exact solutions and power series solutions are constructed. Finally, a general conservation law formula is given based on the idea of nonlinear self-adjointness and some nontrivial conservation laws of the equations are presented.