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Showing papers on "Lie group published in 2022"


Journal ArticleDOI
TL;DR: In this paper , the Lie point symmetry generators, Lie symmetry groups and symmetry reductions for a higher-order Boussinesq-Burgers (BB) system were derived via the Lie group method.

50 citations


Journal ArticleDOI
TL;DR: In this article , a robust numerical technique known as successive linearization approach (SLM) is used to solve the nonlinear coupled formulated equations, which shows more efficient results compared with other similar methods.

31 citations



Journal ArticleDOI
TL;DR: In this article , a (2+1)-dimensional system of Broer-Kaup-Kupershmidt (BKK) equations, which describes the nonlinear, and long gravity waves in a dispersive system, is investigated by applying the method of Lie group of invariance.

22 citations


Journal ArticleDOI
TL;DR: In this paper , the authors studied a higher-dimensional space and time fractional model, namely, the (3+1)-dimensional dissipative Burgers equation which can be used to describe the shallow water waves phenomena.
Abstract: In this paper, we studied a higher-dimensional space and time fractional model, namely, the (3+1)-dimensional dissipative Burgers equation which can be used to describe the shallow water waves phenomena. Here, the analyzed tool is the Lie symmetry scheme in the sense of the Riemann–Liouville fractional derivative. First of all, the symmetry of this considered equation was yielded. Then, based on the above obtained symmetry, the one-parameter Lie group was obtained. Subsequently, this model can be changed into the lower-dimensional equation with the Erdélyi–Kober fractional operators. Lastly, conservation laws of this studied equation via a new conservation theorem were also received. After such a series of processing, these new results play an important role in our understanding of this higher-dimensional space and time differential equations.

15 citations


Journal ArticleDOI
TL;DR: In this paper , the robust technique of the Lie group theory of differential equation was invoked to achieve analytic solutions to the equation, which is used in a systematic way to generate the Lie point symmetries of the equation under study.

10 citations


Journal ArticleDOI
TL;DR: In this paper , the generalized Calogero-Bogoyavlenskii-Schiff equation (GCBSE) was analyzed by the Lie symmetry method, and the symmetry generators of the GCBSE and commutation relation were computed.
Abstract: This work is focused to analyze the generalized Calogero–Bogoyavlenskii–Schiff equation (GCBSE) by the Lie symmetry method. GCBS equation has been utilized to explain the wave profiles in soliton theory. GCBSE was constructed by Bogoyavlenskii and Schiff in different ways (explained in the introduction section). With the aid of Lie symmetry analysis, we have computed the symmetry generators of the GCBSE and commutation relation. We observed from the commutator table, translational symmetries make an Abelian algebra. Then by using the theory of Lie, we have discovered the similarity variables, which are used to convert the supposed nonlinear partial differential equation (NLPDE) into a nonlinear ordinary differential equation (NLODE). Using the new auxiliary method (NAM), we have to discover some new wave profiles of GCBSE in the type of few trigonometric functions. These exits some parameters which we give to some suitable values to attain the different diagrams of some obtained solutions. Further, the GCBSE is presented by non-linear self-adjointness, and conserved vectors are discovered corresponding to each generator.

7 citations


Journal ArticleDOI
TL;DR: In this article , a nonlinear strapdown inertial navigation system (SINS)/global navigation satellite system (GNSS) integrated navigation estimation algorithm based on the Lie group manifold space is proposed.
Abstract: In this article, in order to improve the performance of the micro inertial measurement unit (MIMU) based on low-accuracy navigation system under the condition of initial large misalignment angle, a nonlinear strapdown inertial navigation system (SINS)/global navigation satellite system (GNSS) integrated navigation estimation algorithm based on the Lie group manifold space is proposed. The proposed nonlinear algorithm is based on unscented Kalman filter (UKF), and its core is to realize the propagation of Lie algebra state error variable sigma points between Lie group space and Lie algebra space through the retraction operation and inverse retraction operation. Meanwhile, the sensor bias state error variable sigma points are always propagated through linear operations in the Euclidean space. Finally, the covariance of the two forms of state error variable is calculated in the Euclidean space, from which the lie algebra state error variable and sensor bias state error variable are estimated through the measurement updating process. The simulation and experimental results show that the proposed algorithm has higher accuracy and faster convergence speed compared with the existing state-of-the-art integrated navigation algorithms, and it has good estimation consistency.

7 citations


Journal ArticleDOI
TL;DR: In this paper , an energy-optimal geometric switching control method for a class of rigid body systems with switching submodes is proposed for the first time, where the attitude control problem is investigated.
Abstract: In this article, the attitude control problem for a class of rigid body systems with switching submodes is investigated. To address such issue, an energy-optimal geometric switching control method is proposed for the first time. First, a switching Lie group method and a switching Lie group discrete variational integrator method based on the variational principle are developed. Then, continuous-time and discrete-time attitude switching dynamics models for rigid bodies are globally expressed on a special orthogonal matrix group (i.e., SO(3) group) to avoid the locality, singularity, and ambiguity caused by using the traditional Euler angle method, minimum representation method, or quaternion method. Second, the switching process of rigid bodies from initial attitude and angular velocity to desired attitude and angular velocity within a fixed maneuver time and the minimum energy consumption of rigid bodies with control saturation are solved. Furthermore, by solving the energy-optimal geometric control problem of attitude switching dynamics models, the global optimal geometric switching control and optimal switching time conditions under continuous-time and discrete-time are first derived. The obtained criteria ensure that the intrinsic geometric properties of attitude switching dynamics models will not be lost during the optimization process. Finally, some simulation results are presented to demonstrate the feasibility of the proposed techniques.

6 citations


Journal ArticleDOI
TL;DR: In this article , the authors apply the Radu-Mihet method (RMM) derived from an alternative fixed point theorem, and obtain the existence of a unique solution and the H-U-R stability (Hyers-Ulam-Rassias) for the homomorphisms and Jordan homomorphism on Lie matrix valued fuzzy algebras with ς members.
Abstract: In this work, by considering a class of matrix valued fuzzy controllers and using a (κ,ς)-Cauchy–Jensen additive functional equation ((κ,ς)-CJAFE), we apply the Radu–Mihet method (RMM), which is derived from an alternative fixed point theorem, and obtain the existence of a unique solution and the H–U–R stability (Hyers–Ulam–Rassias) for the homomorphisms and Jordan homomorphisms on Lie matrix valued fuzzy algebras with ς members (ς-LMVFA). With regards to each theorem, we consider the aggregation function as a matrix value fuzzy control function and investigate the results obtained.

6 citations



Journal ArticleDOI
TL;DR: In this article , a systematic investigation of Lie group analysis of non-linear time-fractional Black-Scholes equation including numerical approximations is presented, and conservation laws for the intended equation are derived by using a modified version of Noether's theorem.

Journal ArticleDOI
TL;DR: In this article, a systematic investigation of Lie group analysis of non-linear time-fractional Black-Scholes equation including numerical approximations is presented, and conservation laws for the intended equation are derived by using a modified version of Noether's theorem.

Journal ArticleDOI
TL;DR: In this article , the generalized spin mapping representation for non-adiabatic dynamics is presented, where the Stratonovich-Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group.
Abstract: We present the rigorous theoretical framework of the generalized spin mapping representation for non-adiabatic dynamics. Our work is based upon a new mapping formalism recently introduced by Runeson and Richardson [J. Chem. Phys. 152, 084110 (2020)], which uses the generators of the su(N) Lie algebra to represent N discrete electronic states, thus preserving the size of the original Hilbert space. Following this interesting idea, the Stratonovich-Weyl transform is used to map an operator in the Hilbert space to a continuous function on the SU(N) Lie group, i.e., a smooth manifold which is a phase space of continuous variables. We further use the Wigner representation to describe the nuclear degrees of freedom and derive an exact expression of the time-correlation function as well as the exact quantum Liouvillian for the non-adiabatic system. Making the linearization approximation, this exact Liouvillian is reduced to the Liouvillian of several recently proposed methods, and the performance of this linearized method is tested using non-adiabatic models. We envision that the theoretical work presented here provides a rigorous and unified framework to formally derive non-adiabatic quantum dynamics approaches with continuous variables and connects the previous methods in a clear and concise manner.

Journal ArticleDOI
24 Jan 2022-Symmetry
TL;DR: In this article, Lie symmetry analysis was employed to recover cubic-quartic optical soliton solutions to the Lakshmanan-Porsezian-Daniel model in birefringent fibers.
Abstract: This paper employs Lie symmetry analysis to recover cubic–quartic optical soliton solutions to the Lakshmanan–Porsezian–Daniel model in birefringent fibers. The results are a sequel to the previously reported work on the same model in unpolarized fibers. Dark, singular, and straddled optical solitons that emerged from the scheme are presented.



Journal ArticleDOI
TL;DR: In this article , it was shown that the space of Dolbeault harmonic (1,1)-forms on compact quotients of 4-dimensional Lie groups admits a left invariant almost Hermitian structure (J,ω).

Journal ArticleDOI
TL;DR: In this paper , an unscented Kalman filter (UKF) for matrix Lie groups is proposed where the time propagation of the state is formulated on the Lie algebra and the sigma points can then be expressed as logarithms in vector form.
Abstract: An unscented Kalman filter (UKF) for matrix Lie groups is proposed where the time propagation of the state is formulated on the Lie algebra. This is done with the kinematic differential equation of the logarithm, where the inverse of the right Jacobian is used. The sigma points can then be expressed as logarithms in vector form, and time propagation of the sigma points and the computation of the mean and the covariance can be done on the Lie algebra. The resulting formulation is to a large extent based on logarithms in vector form and is, therefore, closer to the UKF for systems in $\mathbb {R}^n$ . This gives an elegant and well-structured formulation, which provides additional insight into the problem, and which is computationally efficient. The proposed method is in particular formulated and investigated on the matrix Lie group $SE(3)$ . A discussion on right and left Jacobians is included, and a novel closed-form solution for the inverse of the right Jacobian on $SE(3)$ is derived, which gives a compact representation involving fewer matrix operations. The proposed method is validated in simulations.

Journal ArticleDOI
TL;DR: In this paper, a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups is presented, and blow-up type results and global in t -boundedness of solutions of nonlinear equations for the heat p -sub-Laplacian on stratified Lie groups are obtained.
Abstract: In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in t -boundedness of solutions of nonlinear equations for the heat p -sub-Laplacian on stratified Lie groups.

Journal ArticleDOI
TL;DR: In this paper , the authors extend the Skyrme model to the case that the global isospin group is a generic compact connected Lie group and analyze the corresponding field equations in (3+1) dimensions from a group theory point of view.

Journal ArticleDOI
TL;DR: In this paper , a (2+1)-dimensional generalized Bogoyavlensky-Konopelchenko equation was studied and the authors obtained a five-dimensional Lie algebra of the equation through Lie group analysis.
Abstract: The nonlinear phenomena in numbers are modelled in a wide range of fields such as chemical physics, ocean physics, optical fibres, plasma physics, fluid dynamics, solid-state physics, biological physics and marine engineering. This research article systematically investigates a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation. We achieve a five-dimensional Lie algebra of the equation through Lie group analysis. This, in turn, affords us the opportunity to compute an optimal system of fourteen-dimensional Lie subalgebras related to the underlying equation. As a consequence, the various subalgebras are engaged in performing symmetry reductions of the equation leading to many solvable nonlinear ordinary differential equations. Thus, we secure different types of solitary wave solutions including periodic (Weierstrass and elliptic integral), topological kink and anti-kink, complex, trigonometry and hyperbolic functions. Moreover, we utilize the bifurcation theory of dynamical systems to obtain diverse nontrivial travelling wave solutions consisting of both bounded as well as unbounded solution-types to the equation under consideration. Consequently, we generate solutions that are algebraic, periodic, constant and trigonometric in nature. The various results gained in the study are further analyzed through numerical simulation. Finally, we achieve conservation laws of the equation under study by engaging the standard multiplier method with the inclusion of the homotopy integral formula related to the obtained multipliers. In addition, more conserved currents of the equation are secured through Noether’s theorem.

Journal ArticleDOI
01 Feb 2022
TL;DR: In this paper , a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups is presented, and blow-up type results and global in t-boundedness of solutions of nonlinear equations for the heat p-sub-Laplacian on stratified Lie groups are obtained.
Abstract: In this paper we present a comparison principle for higher order nonlinear hypoelliptic heat operators on graded Lie groups. Moreover, using the comparison principle we obtain blow-up type results and global in t-boundedness of solutions of nonlinear equations for the heat p-sub-Laplacian on stratified Lie groups.

Journal ArticleDOI
TL;DR: A survey of left-invariant optimal control problems on Lie groups can be found in this article , where extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis are discussed.
Abstract: Abstract Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing. The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed. Bibliography: 91 titles.

Journal ArticleDOI
Ko Nakao1
TL;DR: In this paper , the Sobolev embedding constant on general non-compact Lie groups is estimated for sub-Riemannian inhomogeneous Soboleve spaces endowed with a left invariant measure.
Abstract: In this paper we estimate the Sobolev embedding constant on general noncompact Lie groups, for sub-Riemannian inhomogeneous Sobolev spaces endowed with a left invariant measure. The bound that we obtain, up to a constant depending only on the group and its sub-Riemannian structure, reduces to the best known bound for the classical inhomogeneous Sobolev embedding constant on R d . As an application, we prove local and global Moser–Trudinger inequalities.

Journal ArticleDOI
TL;DR: In this article , the existence of invariant SKT, balanced and generalized Kähler structures on compact quotients Γ﹨G, where G is an almost nilpotent Lie group whose nilradical has one-dimensional commutator and Γ is a lattice of G, was investigated.

Journal ArticleDOI
TL;DR: In this paper , it was shown that the pentagram map can be seen as refactorization-type mappings in the Poisson-Lie group of pseudo-difference operators.

Posted ContentDOI
TL;DR: In this paper, the analytical solution of fractional order K(m, n) type equation with variable coefficient was investigated, which is an extended type of KdV equations into a genuinely nonlinear dispersion regime.
Abstract: We investigated the analytical solution of fractional order K(m, n) type equation with variable coefficient which is an extended type of KdV equations into a genuinely nonlinear dispersion regime. By using the Lie symmetry analysis, we obtain the Lie point symmetries for this type of time-fractional partial differential equations (PDE). Also we present the corresponding reduced fractional differential equations (FDEs) corresponding to the time-fractional K(m, n) type equation.

Journal ArticleDOI
TL;DR: In this article , the discreteness of the spectrum of Schrodinger operators on a Lie group was investigated and sufficient conditions for arbitrary potentials were obtained for any potential that belongs to a local Muckenhoupt class.

Journal ArticleDOI
TL;DR: In this paper , the affine generalized Ricci solitons on three-dimensional Lorentzian Lie groups associated with canonical connections and Kobayashi-Nomizu connections were studied.
Abstract: Abstract In this paper, we study the affine generalized Ricci solitons on three-dimensional Lorentzian Lie groups associated canonical connections and Kobayashi-Nomizu connections and we classifying these left-invariant affine generalized Ricci solitons with some product structure.