Showing papers on "Lie group published in 2021"

Lie symmetry analysis for obtaining the abundant exact solutions, optimal system and dynamics of solitons for a higher-dimensional Fokas equation

Abstract: In this article, the Lie group of transformation method via one-dimensional optimal system is proposed to obtain some more exact solutions of the (4+1)-dimensional Fokas equation. Lie infinitesimal generators, possible vector fields, and their commutative and adjoint relations are presented by employing the Lie symmetry method. An optimal system of the one-dimensional subalgebras is also constructed using Lie vectors. Meanwhile, based on the optimal system, Lie symmetry reductions of the Fokas equation is obtained. A repeated process of Lie symmetry reductions, using the single, double, triple, quadruple, and quintuple combinations between the considered vectors, transforms the Fokas equation into nonlinear ordinary differential equations which produce abundant group-invariant solutions. The same problem was studied by Sadat et al. (Chaos Solitons Fractals 140:110134, 2020) using the same Lie symmetry technique via commutative product approach but with the less number of vector fields and therefore could obtain only three exact solutions as compared to the number of analytic solutions in this paper. In order to provide rich localized structures, some solutions are supplemented via numerical simulation, which produces some breather-type solitons, oscillating multi-solitons on the parabolic-shaped surface, fractal dromions, lump-type solitons, and annihilation of different parabolic multi-solitons profiles. The dynamical behaviors of excitation-localized structures are demonstrated graphically via 3D plots for suitable values of the arbitrary free parameters and independent arbitrary functions.

Quasi-periodic, chaotic and travelling wave structures of modified Gardner equation

Abstract: In this paper, the nonlinear modified Gardner (mG) equation is under consideration which represents the super nonlinear proliferation of the ion-acoustic waves and quantum electron-positronion magneto plasmas. The considered model is investigated with the help of Lie group analysis. Lie point symmetries are computed under the invariance criteria of Lie groups and symmetry group for each generator is reported. Furthermore, the one-dimensional optimal system of subalgebras is developed by adjoint technique and then we compute the similarity reductions corresponding to each vector field present in the optimal system, with the help of similarity reduction method we have to convert the PDE into the ODE. Some exact explicit solutions of obtained ordinary differential equations were constructed by the power series technique. With the aid of the Galilean transformation, the model is transformed into a planer dynamical system and the bifurcation behaviour is recorded. All practicable types of phase portraits with regard to the parameters of the problem considered are plotted. Meantime, sensitivity is observed by utilizing sensitivity analysis. In addition, the influence of physical parameters is studied by the application of an extrinsic periodic power. With additional perturbed term, quasi-periodic and quasi-periodic-chaotic behaviours is reported.

Painlevé analysis, Lie group analysis and soliton-cnoidal, resonant, hyperbolic function and rational solutions for the modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics

Abstract: Under investigation in this paper is a modified Korteweg-de Vries-Calogero-Bogoyavlenskii-Schiff equation in fluid mechanics/plasma physics. Painleve integrability is investigated. Soliton-cnoidal and resonant solutions are derived via the consistent tanh expansion method. The equation is reduced to certain symmetry reduction equations via the Lie point symmetry generators obtained though the Lie group method. Hyperbolic function and rational solutions are derived through those symmetry reduction equations.

Symmetry-resolved entanglement entropy in Wess-Zumino-Witten models

Abstract: We consider the problem of the decomposition of the Renyi entanglement entropies in theories with a non-abelian symmetry by doing a thorough analysis of Wess-Zumino-Witten (WZW) models. We first consider SU(2)k as a case study and then generalise to an arbitrary non-abelian Lie group. We find that at leading order in the subsystem size L the entanglement is equally distributed among the different sectors labelled by the irreducible representation of the associated algebra. We also identify the leading term that breaks this equipartition: it does not depend on L but only on the dimension of the representation. Moreover, a log log L contribution to the Renyi entropies exhibits a universal prefactor equal to half the dimension of the Lie group.

Quasisymmetry Groups and Many-Body Scar Dynamics.

Abstract: In quantum systems, a subspace spanned by degenerate eigenvectors of the Hamiltonian may have higher symmetries than those of the Hamiltonian itself. When this enhanced-symmetry group can be generated from local operators, we call it a quasisymmetry group. When the group is a Lie group, an external field coupled to certain generators of the quasisymmetry group lifts the degeneracy, and results in exactly periodic dynamics within the degenerate subspace, namely, the many-body-scar dynamics (given that Hamiltonian is nonintegrable). We provide two related schemes for constructing one-dimensional spin models having on-demand quasisymmetry groups, with exact periodic evolution of a prechosen product or matrix-product state under external fields.

LieTransformer: Equivariant Self-Attention for Lie Groups

Abstract: Group equivariant neural networks are used as building blocks of group invariant neural networks, which have been shown to improve generalisation performance and data efficiency through principled parameter sharing. Such works have mostly focused on group equivariant convolutions, building on the result that group equivariant linear maps are necessarily convolutions. In this work, we extend the scope of the literature to self-attention, that is emerging as a prominent building block of deep learning models. We propose the LieTransformer, an architecture composed of LieSelfAttention layers that are equivariant to arbitrary Lie groups and their discrete subgroups. We demonstrate the generality of our approach by showing experimental results that are competitive to baseline methods on a wide range of tasks: shape counting on point clouds, molecular property regression and modelling particle trajectories under Hamiltonian dynamics.

Automorphic forms and fermion masses

Abstract: We extend the framework of modular invariant supersymmetric theories to encompass invariance under more general discrete groups Γ, that allow the presence of several moduli and make connection with the theory of automorphic forms. Moduli span a coset space G/K, where G is a Lie group and K is a compact subgroup of G, modded out by Γ. For a general choice of G, K, Γ and a generic matter content, we explicitly construct a minimal Kahler potential and a general superpotential, for both rigid and local $$ \mathcal{N} $$ = 1 supersymmetric theories. We also specialize our construction to the case G = Sp(2g, ℝ), K = U(g) and Γ = Sp(2g, ℤ), whose automorphic forms are Siegel modular forms. We show how our general theory can be consistently restricted to multi-dimensional regions of the moduli space enjoying residual symmetries. After choosing g = 2, we present several examples of models for lepton and quark masses where Yukawa couplings are Siegel modular forms of level 2.

Lie group analysis and analytic solutions for a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation in fluid mechanics and plasma physics

Abstract: Under investigation in this paper is a (2+1)-dimensional generalized Bogoyavlensky–Konopelchenko equation in fluid mechanics and plasma physics. We obtain the Lie point symmetry generators, Lie symmetry group and symmetry reductions via the Lie group method. Hyperbolic-function, power-series, trigonometric-function, soliton and rational solutions are derived via the power-series expansion, polynomial expansion and $$\left( \frac{G^{'}}{G}\right) $$ expansion method.

A Lightweight and Robust Lie Group-Convolutional Neural Networks Joint Representation for Remote Sensing Scene Classification

Abstract: The existing convolutional neural network (CNN) models have shown excellent performance in remote sensing scene classification. However, the structure of such models is becoming more and more complex, and the learning of low-level features is difficult to interpret. To address this problem, in this study, we introduce lie group machine learning into the CNN model, try to combine both approaches to extract more distinguishing ability and effective features, and propose a novel network model, namely, the lie group regional influence network (LGRIN). First, manifold space samples of the lie group are obtained by mapping, and then, the features of the lie group are extracted after the operations of image decomposition and integral image calculation. Second, the multidilation pooling is integrated into the CNN architecture. At the same time, the image regional influence network module is designed to guide the attention of the classification model by using the regional-level supervision of the decomposition. Finally, the fusion features are classified, and the predicted results are obtained. Our model takes full advantage of regional influence, lie group kernel function, and lie group feature learning. Moreover, our model produces satisfactory performance on three public and challenging data sets: Aerial Image Dataset (AID), UC Merced, and NWPU-RESISC45. The experimental results verify that, compared with the state-of-the-art methods, this method is more explanatory and achieves higher accuracy.

Some more closed-form invariant solutions and dynamical behavior of multiple solitons for the (2+1)-dimensional rdDym equation using the Lie symmetry approach

Abstract: The present study is devoted to obtaining some exact generalized solutions and new solitary wave solutions for a (2+1)-dimensional nonlinear r th dispersionless Dym (rdDym) equation by utilizing the Lie symmetry analysis. The Lie infinitesimals, geometric vector fields, the commutation relation of the symmetry Lie algebra are derived by employing the Lie group symmetry technique. Thenceforth, explicit exact closed-form solutions are achieved through two stages of symmetry reductions. These closed-form solutions include arbitrary independent functional parameters and other constant parameters, therefore exhibiting rich localized solitons and comprising the existing solutions in the literature. Furthermore, the different dynamical wave structures of established soliton solutions in the forms of oscillating periodic multisolitons, the interaction between four lump-solitons with parabolic waves, W-shaped solitons, and the interaction between two-doubly solitons with parabolic waves, periodic multi-solitons, and parabolic-shaped solitons profiles are exhibited graphically through three-dimensional plots.

Lie symmetry analysis, group-invariant solutions and dynamics of solitons to the (2+1)-dimensional Bogoyavlenskii–Schieff equation

Abstract: In the present work, abundant group-invariant solutions of ( $$2+1$$ )-dimensional Bogoyavlenskii–Schieff equation have been investigated using Lie symmetry analysis. The Lie infinitesimal generators, all the geometric vector fields, their commutative and adjoint relations are provided by utilising the Lie symmetry method. The Lie symmetry method depends on the invariance criteria of Lie groups, which results in the reduction of independent variables by one. A repeated process of Lie symmetry reductions, using the double, triple and septuple combinations between the considered vectors, converts the Bogoyavlenskii–Schieff (BS) equation into nonlinear ordinary differential equations (ODEs) which furnish numerous explicit exact solutions with the help of computerised symbolic computation. The obtained group-invariant solutions are entirely new and distinct from the earlier established findings. As far as possible, a comparison of our reported results with the previous findings is given. The dynamical behaviour of solutions is discussed both analytically as well as graphically via their evolutionary wave profiles by considering suitable choices of arbitrary constants and functions. To ensure rich physical structures, the exact closed-form solutions are supplemented via numerical simulation, which produce some bright solitons, doubly solitons, parabolic waves, U-shaped solitons and asymptotic nature.

Invariance Analysis, Exact Solution and Conservation Laws of (2 + 1) Dim Fractional Kadomtsev-Petviashvili (KP) System

Abstract: In this work, a Lie group reduction for a (2 + 1) dimensional fractional Kadomtsev-Petviashvili (KP) system is determined by using the Lie symmetry method with Riemann Liouville derivative. After reducing the system into a two-dimensional nonlinear fractional partial differential system (NLFPDEs), the power series (PS) method is applied to obtain the exact solution. Further the obtained power series solution is analyzed for convergence. Then, using the new conservation theorem with a generalized Noether’s operator, the conservation laws of the KP system are obtained.

A Variational Finite Element Discretization of Compressible Flow

Abstract: We present a finite element variational integrator for compressible flows. The numerical scheme is derived by discretizing, in a structure-preserving way, the Lie group formulation of fluid dynamics on diffeomorphism groups and the associated variational principles. Given a triangulation on the fluid domain, the discrete group of diffeomorphisms is defined as a certain subgroup of the group of linear isomorphisms of a finite element space of functions. In this setting, discrete vector fields correspond to a certain subspace of the Lie algebra of this group. This subspace is shown to be isomorphic to a Raviart–Thomas finite element space. The resulting finite element discretization corresponds to a weak form of the compressible fluid equation that does not seem to have been used in the finite element literature. It extends previous work done on incompressible flows and at the lowest order on compressible flows. We illustrate the conservation properties of the scheme with some numerical simulations.

A study of Bogoyavlenskii’s (2+1)-dimensional breaking soliton equation: Lie symmetry, dynamical behaviors and closed-form solutions

Abstract: This paper employs the Lie symmetry analysis to investigate novel closed-form solutions to a (2+1)-dimensional Bogoyavlenskii’s breaking soliton equation. This Lie symmetry technique, used in combination with Maple’s symbolic computation system, demonstrates that the Lie infinitesimals are dependent on five arbitrary parameters and two independent arbitrary functions f 1 ( t ) and f 2 ( t ) . The invariance criteria of Lie group analysis are used to construct all infinitesimal vectors, commutative relations of their examined vectors, a one-dimensional optimal system and then several symmetry reductions. Subsequently, Bogoyavlenskii’s breaking soliton (BBS) equation is reduced into several nonlinear ODEs by employing desirable Lie symmetry reductions through optimal system. Explicit exact solutions in terms of arbitrary independent functions and other constants are obtained as a result of solving the nonlinear ODEs. These established results are entirely new and dissimilar from the previous findings in the literature. The physical behaviors of the gained solutions illustrate the dynamical wave structures of multiple solitons, curved-shaped wave–wave interaction profiles, oscillating periodic solitary waves, doubly-solitons, kink-type waves, W-shaped solitons, and novel solitary waves solutions through 3D plots by selecting the suitable values for arbitrary functional parameters and free parameters based on numerical simulation. Eventually, the derived results verify the efficiency, trustworthiness, and credibility of the considered method.

Generalized Kähler almost abelian Lie groups

Abstract: We study left-invariant generalized Kahler structures on almost abelian Lie groups, ie, on solvable Lie groups with a codimension-one abelian normal subgroup In particular, we classify six-dimensional almost abelian Lie groups which admit a left-invariant complex structure and establish which of those have a left-invariant Hermitian structure whose fundamental 2-form is $$\partial {{\overline{\partial }}}$$ -closed We obtain a classification of six-dimensional generalized Kahler almost abelian Lie groups and determine the six-dimensional compact almost abelian solvmanifolds admitting an invariant generalized Kahler structure Moreover, we prove some results in relation to the existence of holomorphic Poisson structures and to the pluriclosed flow

Lie symmetry analysis, conservation laws and analytical solutions of a time-fractional generalized KdV-type equation

Abstract: Under investigation in this work is the time-fractional generalized KdV-type equation, which occurs in different contexts in mathematical physics. Lie group analysis method is presented to explicit...

Group theoretic approach to many-body scar states in fermionic lattice models

Abstract: It has been shown \cite{pakrouski2020GroupInvariantScars} that three families of highly symmetric states are many-body scars for any spin-1/2 fermionic Hamiltonian of the form $H_0+OT$, where $T$ is a generator of an appropriate Lie group. One of these families consists of the well-known $\eta$-pairing states. In addition to having the usual properties of scars, these families of states are insensitive to electromagnetic noise and have advantages for storing and processing quantum information. In this paper we show that a number of well-known coupling terms, such as the Hubbard and the Heisenberg interactions, and the Hamiltonians containing them, are of the required form and support these states as scars without fine-tuning. The explicit $H_0+OT$ decomposition for a number of most commonly used models, including topological ones, is provided. To facilitate possible experimental implementations, we discuss the conditions for the low-energy subspace of these models to be comprised solely of scars. Further, we write down all the generators $T$ that can be used as building blocks for designing new models with scars, most interestingly including the spin-flip hopping and superconducting pairing terms. We expand this framework to the non-Hermitian open systems and demonstrate that for them the scar subspace continues to undergo coherent time evolution and exhibit the ``revivals." A full numerical study of an extended 2D $tJU$ model explicitly illustrates the novel properties of the invariant scars and supports our findings.

Exact Solution of the Time Fractional Variant Boussinesq-Burgers Equations

Abstract: In the present article, we consider a nonlinear time fractional system of variant Boussinesq-Burgers equations. Using Lie group analysis, we derive the infinitesimal groups of transformations containing some arbitrary constants. Next, we obtain the system of optimal algebras for the symmetry group of transformations. Afterward, we consider one of the optimal algebras and construct similarity variables, which reduces the given system of fractional partial differential equations (FPDEs) to fractional ordinary differential equations (FODEs). Further, under the invariance condition we construct the exact solution and the physical significance of the solution is investigated graphically. Finally, we study the conservation law of the system of equations.

$$L^p$$-Bounds for Pseudo-differential Operators on Graded Lie Groups

Abstract: In this work we obtain sharp $$L^p$$ -estimates for pseudo-differential operators on arbitrary graded Lie groups. The results are presented within the setting of the global symbolic calculus on graded Lie groups by using the Fourier analysis associated to every graded Lie group which extends the usual one due to Hormander on $${\mathbb {R}}^n$$ . The main result extends the classical Fefferman’s sharp theorem on the $$L^p$$ -boundedness of pseudo-differential operators for Hormander classes on $${\mathbb {R}}^n$$ to general graded Lie groups, also adding the borderline $$\rho =\delta $$ case.

Lie group based nonlinear state errors for MEMS-IMU/GNSS/magnetometer integrated navigation

Abstract: In the integrated navigation system using extended Kalman filter (EKF), the state error conventionally uses linear approximation to tackle the commonly nonlinear problem. However, this error definition can diverge the filter in some adverse situations due to significant distortion of the linear approximation. By contrast, the nonlinear state error defined in the Lie group satisfies the autonomous equation, which thus has distinctively better convergence property. This work proposes a novel strapdown inertial navigation system (SINS) nonlinear state error defined in the Lie group and derives the SINS equations of the Lie group EKF (LG-EKF) for the MIMU/GNSS/magnetometer integrated navigation system. The corresponding measurement equations are also derived. A land vehicle field test has been conducted to evaluate the performance of EKF, ST-EKF (state transformation extended Kalman filter) and LG-EKF, which verifies LG-EKF's superior estimation accuracy of the heading angle as well as the other two horizontal angles (pitch and roll). The LG-EKF proposed in this paper is unlimited in the choice of sensors, which means it can be applied with both high-end and low-end inertial sensors.

Expanding cone and applications to homogeneous dynamics

Abstract: Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on finite volume homogeneous space $G/\Gamma$ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/\Gamma$ has a spectral gap, we prove effective multiple equidistribution and pointwise equidistribution with an error rate. In the paper we formulate the notion of the expanding cone and prove the dynamical results above in the more general setting where $H$ is a semisimple Lie group without compact factors. In the appendix, joint with Rene Ruhr, we prove a multiple ergodic theorem with an error rate.

Topological invariants of parabolic G-Higgs bundles

Abstract: For a semisimple real Lie group G, we study topological properties of moduli spaces of polystable parabolic G-Higgs bundles over a Riemann surface with a divisor of finitely many distinct points. For a split real form of a complex simple Lie group, we compute the dimension of apparent parabolic Teichmuller components. In the case of isometry groups of classical Hermitian symmetric spaces of tube type, we provide new topological invariants for maximal parabolic G-Higgs bundles arising from a correspondence to orbifold Higgs bundles. Using orbifold cohomology we count the least number of connected components of moduli spaces of such objects. We further exhibit an alternative explanation of fundamental results on counting components in the absence of a parabolic structure.

Structure and regularity for subsets of groups with finite VC-dimension

Abstract: Suppose $G$ is a finite group and $A\subseteq G$ is such that $\{gA:g\in G\}$ has VC-dimension strictly less than $k$. We find algebraically well-structured sets in $G$ which, up to a chosen $\epsilon>0$, describe the structure of $A$ and behave regularly with respect to translates of $A$. For the subclass of groups with uniformly fixed finite exponent $r$, these algebraic objects are normal subgroups with index bounded in terms of $k$, $r$, and $\epsilon$. For arbitrary groups, we use Bohr neighborhoods of bounded rank and width inside normal subgroups of bounded index. Our proofs are largely model theoretic, and heavily rely on a structural analysis of compactifications of pseudofinite groups as inverse limits of Lie groups. The introduction of Bohr neighborhoods into the nonabelian setting uses model theoretic methods related to the work of Breuillard, Green, and Tao and Hrushovski on approximate groups, as well as a result of Alekseev, Glebskii, and Gordon on approximate homomorphisms.