Symmetry Reduction Research Articles

Symmetry analysis, closed-form invariant solutions and dynamical wave structures of the Benney-Luke equation using optimal system of Lie subalgebras

Nonlinear evolution equations (NLE-Es) arise in a variety of domains, including fluid dynamics, biological sciences, the study of solid states, optical fibres, coastal engineering, ocean physics, and nonlinear complex physical systems. This work examines the Lie symmetry study of the Benney-Luke (B-L) equation relying on two nonzero real parameters. The Lie infinitesimal generators, the one-dimensional optimal system, and the geometric vector fields are all obtained using the Lie symmetry technique. To start, we find close-form invariant solutions by employing symmetry reduction of Lie subalgebras. In some reduction cases, we transform the B-L equation into a variety of non-linear ordinary differential equations (NL-ODEs), which have the benefit of providing a significant number of closed-form solitary wave solutions. The traveling wave solutions contain special functional parameter solutions, trigonometric function solutions, rational function solutions, and hyperbolic trigonometric function solutions. The dynamical profile of closed-form wave solutions exhibits periodic solitons, dark peakon solitons, bright solitons, and bright solitons (bell shape) which we reveal in our study for the first time. By using Noether’s method, we also determine the conservation laws. Finally, three-dimensional diagrams are used to reveal the dynamical investigation of some known solutions.

Conditionally integrable PDEs, non-classical symmetries and applications

Some multi-dimensional nonlinear partial differential equations reduce to a linear equation in fewer dimensions after imposing one additional constraint. A large class of useful conditionally integrable reaction–diffusion equations follows from a single non-classical symmetry reduction, yielding an infinite-dimensional but incomplete solution space. This solution device extends further to other nonlinear equations such as higher-order reaction–diffusion, fractional-order diffusion and diffusion–convection. Applications are shown for population dynamics with or without weak Allee effects, speed-limited hyperbolic diffusion, material phase field dynamics, soil–water–plant dynamics and calcium transport on the curved surface of an oocyte. Beyond non-classical symmetry analysis, examples of other conditionally integrable equations are given; nonlinear diffusion in 1 + 2 dimensions and the Madelung–Burgers quantum fluid in 1 + 3 dimensions.

Invariant Bianchi type I cosmological models and conservation laws in f(R,T)=f1(R)+f2(R)f3(T) gravity

f(R,T) theory of gravity can explain the dark energy and the observed late time acceleration of the universe even without cosmological constant. It passes the solar system test and appears to be an alternative of dark matter. We obtain the accelerating solutions of Einstein’s field equations, corresponding to Bianchi I spacetime with perfect fluid, in f(R,T) gravity theory without shear. The non-static non-degenerating indefinite smooth solutions of field equations are derived by considering f(R,T)=f1(R)+f2(R)f3(T) with non-minimal matter–geometry coupling. The classical Lie symmetry method is used for symmetry reduction of field equations and for obtaining the exact solutions in terms of arbitrary functions. The graphical representations of solutions are shown by considering the possibilities of arbitrary functions. The conservation laws of Einstein’s equations in f(R,T) gravity are also obtained.

Multiple soliton solutions and symmetry analysis of a nonlocal coupled KP system

A nonlocal coupled Kadomtsev–Petviashivili (ncKP) system with shifted parity () and delayed time reversal () symmetries is generated from the local coupled Kadomtsev–Petviashivili (cKP) system. By introducing new dependent variables which have determined parities under the action of , the ncKP is transformed to a local system. Through this way, multiple even number of soliton solutions of the ncKPI system are generated from N-soliton solutions of the cKP system, which become breathers by choosing appropriate parameters. The standard Lie symmetry method is also applied on the ncKPII system to get its symmetry reduction solutions.

Painlevé analysis, infinite dimensional symmetry group and symmetry reductions for the (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani equation

The (2+1)-dimensional Korteweg–de Vries–Sawada–Kotera–Ramani (KdVSKR) equation is studied by the singularity structure analysis. It is proven that it admits the Painlevé property. The Lie algebras which depend on three arbitrary functions of time t are obtained by the Lie point symmetry method. It is shown that the KdVSKR equation possesses an infinite-dimensional Kac–Moody–Virasoro symmetry algebra. By selecting first-order polynomials in t, a finite-dimensional subalgebra of physical transformations is studied. The commutation relations of the subalgebra, which have been established by selecting the Laurent polynomials in t, are calculated. This symmetry constitutes a centerless Virasoro algebra which has been widely used in the field of physics. Meanwhile, the similarity reduction solutions of the model are studied by means of the Lie point symmetry theory.

Conditions for symmetry reduction of polysymplectic and polycosymplectic structures

For Hamiltonian field theories on polysymplectic manifolds with a symmetry group action and a momentum map, we explore the redundancy in a set of necessary conditions that has appeared in the literature, for a generalized version of the Marsden–Weinstein symmetry reduction theorem. Next, we prove a necessary and sufficient condition for polycosymplectic reduction. We relate polycosymplectic reduction in a one-to-one way to the reduction of an associated larger polysymplectic manifold. Throughout the paper, we provide examples and discuss special cases.

Application of symmetry analysis to viscoelastic fluid model

In the present paper, we apply Lie symmetry to the Suliciu relaxation system governed by viscoelastic shallow fluid. Utilizing the local symmetries of the governing PDEs, we derive the one-dimensional optimal system of subalgebras and construct several new exact solutions through symmetry reduction. Further, we employ the direct multipliers method to the given model to extract new conservation laws. Furthermore, we establish potential systems and inverse potential systems (IPS) to classify the nonlocal symmetry of the governing system. In addition, we develop some non-trivial exact solutions of the given model using the nonlocal symmetries of IPS. Finally, we investigate the propagation and interaction of C1-wave and characteristic shock.

Anisotropic hybridization probed by polarization dependent x-ray absorption spectroscopy in VI3 van der Waals Mott ferromagnet

Polarization dependent x-ray absorption spectroscopy was used to study the magnetic ground state and the orbital occupation in bulk-phase VI3 van der Waals crystals below and above the ferromagnetic and structural transitions. X-ray natural linear dichroism and x-ray magnetic circular dichroism spectra acquired at the V edges are compared against multiplet cluster calculations within the frame of the ligand field theory to quantify the intra-atomic electronic interactions at play and evaluate the effects of symmetry reduction occurring in a trigonally distorted VI6 unit. We observed a non zero linear dichroism proving the presence of an anisotropic charge density distribution around the V3+ ion due to the unbalanced hybridization between the vanadium and the ligand states. Such hybridization acts as an effective trigonal crystal field, slightly lifting the degeneracy of the ground state. However, the energy splitting associated to the distortion underestimates the experimental band gap, suggesting that the insulating ground state is stabilized by Mott correlation effects rather than via a Jahn–Teller mechanism. Our results clarify the role of the distortion in VI3 and establish a benchmark for the study of the spectroscopic properties of other van der Waals halides, including emerging 2D materials with mono and few-layers thickness, whose fundamental properties might be altered by reduced dimensions and interface proximity.

Approximate symmetry of time-fractional partial differential equations with a small parameter

In the sense of Riemann–Liouville fractional derivative, an approximate symmetry method of (1+1)-dimensional time-fractional partial differential equations (PDEs) with a small parameter is proposed. By first inserting the power series expansion of dependent variable in the small parameter into the fractional PDEs and then separating the equation with respect to the small parameter to get a coupled system of fractional PDEs, we define the pth-order approximate symmetries by the Lie symmetries of the first p+1 coupled fractional PDEs. After performing symmetry reductions for the coupled system of fractional PDEs, we find that zero-order reduced equation coincides with the fractional ordinary differential equation corresponding to the fractional PDE without containing the small parameter while the solutions of higher-order reduced equation can be obtained by solving linear variable coefficient fractional ordinary differential equations. We first exemplify the whole procedure of computing approximate symmetry by studying a time-fractional KdV equation with a small parameter. Specifically, we obtain the higher-order approximate symmetries and perform approximate symmetry reductions by means of an optimal system of one-dimensional approximate Lie subalgebras. Moreover, we construct an explicit power series solution and show the dynamic behaviors of the truncated power series solutions by the evolutional figures. Then in order to make the proposed approximate symmetry method be further convincing, we study an anomalous diffusion equation with a small parameter and construct an analytical solution in terms of the Mittag-Leffler function.

New symmetry reduction method for (1+1)-dimensional differential-difference equations

We propose a new symmetry reduction method for (1+1)-dimensional differential-difference equations (DDEs), namely, the λ-symmetry reduction method of solving ordinary differential equations is generalized to DDEs. Order-reduction processes are a consequence of the invariance of the given DDE under vector fields of the new class. These vector fields satisfy a new prolongation formula. A simple example of order-reduction is provided to illustrate the application.

Ultrafast carrier dynamics and symmetry reduction in bismuth by nonperturbative optical excitation in the terahertz range

Ultrafast carrier dynamics and symmetry reduction in bismuth by nonperturbative optical excitation in the terahertz range

Towards control of the chirality sign at ultrathin metal films: Bi at 2Ni/Co

Proximity effects can be used to introduce spin–orbit interactions in magnetic metallic layers in contact to a heavy metal (HM). This well known phenomenon has been exploited to induce chiral spin textures at Co ultrathin films, where the left- or right- handedness can be tuned by the HM layer position, based on the broken inversion symmetry of the film and the existence of an additive interface effect. Here we show that structural and chemical features introducing further symmetry reductions can be added to this scenario ultimately enabling control over the definition of a unique winding sense. We focus on 2Ni/Co heterostructures and Bi, a scarcely explored HM metal of large size, to combine a chemically inhomogeneous ferromagnetic stack along the normal to the surface with in-plane asymmetries. Our results are contrasted to 2Co layers combined with Ir.

General form of axially symmetric stationary metric: exact solutions and conservation laws in vacuum fields

The invariant non-static solutions of Einstein’s vacuum field equations, corresponding to the most general form of axially symmetric stationary line element that represents a non conformally flat semi-Riemannian spacetime in cylindrical coordinates, are investigated. Lie symmetry method is used for symmetry reduction as well as for obtaining exact solutions in terms of arbitrary functions. The conservation laws are obtained for vacuum equations in axially symmetric gravitational fields. The solutions of Lewis metric and Chandrasekhar metric, are derived from the obtained solutions. By considering the possibilities of arbitrary functions, the graphical behaviour of the solutions is also shown.

High-pressure-induced electronic and structural transition of superatoms.

High pressure has been recognized as an important tool in molecule and materials research, and thus, it is expected to be used to understand the evolution of electronic states and geometric structures in superatoms. In this work, by studying three characteristic axial compressions on a typical endohedral metallofullerene superatom U@C28 with Td symmetry, we find that the triplet ground electronic state is preserved when the compression moves along the direction that reduces the symmetry to D2d, but the electronic state of the structure compressed along the direction of symmetry reduction to C2v or Cs is transformed into a singlet. The transition is attributed to the distinction in the response of electron spin to different axial compressions, which results in a change in the electron occupation mode of the system. Furthermore, we also confirm the gradual evolution from stereo to near-plane superatoms and the connection between their electron structures. This is reflected in the fact that the electron density distributions of the superatomic molecular orbitals (SAMOs) with extension along the restricted degrees of freedom (Dz2, Fz3 SAMOs) gradually contract, and the delocalization destruction of special orbitals is associated with this freedom. In addition, Raman and ultraviolet-visible spectra show a hyperchromic effect and redshift of characteristic peaks during axial compression, which are expected to be used for fingerprinting the superatomic planarization. Therefore, our work provides new insights based on high pressure for future research toward the discovery of physical properties and applications of superatoms.

Lie symmetry analysis and some new exact solutions of the Fokker–Planck equation

The classical symmetry method is often employed to find precise solutions to differential equations. This method has yielded several new symmetry reductions and exact solutions for numerous theoretically and physically relevant partial differential equations. These results, as well as the symmetries of a variety of specific cases of the Fokker–Planck equation, were presented in this study using the classical Lie symmetry approach. New exact solutions to the Fokker–Planck equations are provided for each of the six cases.

A small-box approach to the local crystal structure of Y3NbO7

The local crystal structure of Y3NbO7 was probed using neutron total scattering. Five different structural models were fit to the experimental pair distribution function of this material up to 15 Å. Fits of defect fluorite (Fm3‾m), pyrochlore (Fd3‾m), and orthorhombic C weberite models (C2221) were attempted based on prior structural investigations of Y3NbO7 and related fluorite-based materials. Orthorhombic P weberite (P2221) and monoclinic weberite models (P1121) were derived via symmetry reduction of the orthorhombic C weberite structure. All models except monoclinic weberite failed to reproduce the experimental pair distribution function. Misfits were clearly visible for peaks arising from atom pairs within the first coordination shell (defect fluorite and pyrochlore) and beyond (orthorhombic C and P weberite). By contrast, monoclinic weberite (γ ≈ 91.8°) provided an excellent fit, with no major misfits observed. This unit cell features distorted YO8 dodecahedra, YO7 monocapped octahedra, and NbO6 octahedra as building blocks. Comparison between the topologies of the cationic and anionic substructures in monoclinic weberite and its parent orthorhombic C structure revealed displacements of the metal atoms along the [100] direction and a less compact spatial distribution of oxygen atoms along the a axis. The monoclinic weberite model presented herein may serve as a starting point to revisit structural modeling of fluorite-based materials with challenging local structures.

Symmetry analysis and exact Jacobi elliptic solutions for the nonlinear couple Drinfeld Sokolov Wilson dynamical system arising in shallow water waves

We adopted the Lie symmetry technique to obtain some new exact waveform solutions for the coupled Drinfeld Sokolov Wilson (DSW) system. This system emerges from phenomena such as water waves, theoretical physics, fluid dynamics, biology, and chemical sciences. We use the Jacobi elliptic function (JEF) method to deal with some of the reduced systems during the symmetry reductions, which reveals rational, exponential, and hyperbolic functions based waveform solutions. The properties of these solutions indicate that these are periodic waves, solitary waves, double periodic waves, and shock wave solutions. Several of the answers that have been found are entirely new and might be important for researchers to recognize. By assigning specific values to parameters, some chosen solutions are graphically validated. This method can be applied more broadly to handle several different types of nonlinear evolution systems.

Novel octa-graphene-like structures based on GaP and GaAs.

The discovery of graphene gave way to the search for new two-dimensional structures. In this regard, octa-graphene is a carbon allotrope consisting of 4- and 8-membered rings in a single planar sheet, drawing the research community's attention to study their inorganic analogs. Considering the promising properties of octa-graphene-like structures and the role of GaAs and GaP in semiconductor physics, this study aims to propose, for the first time, two novel inorganics buckled nanosheets based on the octa-graphene structure, the octa-GaAs and octa-GaP. This work investigated the structural, electronic, and vibrational properties of these novel octa-graphene-based materials. The octa-GaP and octa-GaAs have an indirect band gap transition with a valence band maximum between M and Г points and a conduction band minimum at Г point with energy of 3.05 eV and 2.56 eV, respectively. The QTAIMC analysis indicates that both structures have incipient covalent in their bonds. The vibrational analysis demonstrates the occurrence of ΓRaman = 6Ag + 6Bg and ΓRaman = 12A' + 12B″ for octa-GaP and octa-GaAs, respectively. The symmetry reduction of octa-GaAs leads to activating inactive modes observed in the octa-GaP structure. The frontier crystalline orbitals are composed by Ga(px) and P(py and pz) orbitals for octa-GaP and Ga(px and py) and As(s, py, and pz) for octa-GaAs in the valence bands while in the conduction bands by Ga(py, pz, and s) for both compounds and P(px and pz) and As(py). The phonon bands demonstrate the absence of the negative frequency modes and the structural stability of these new nanosheets. This report aims to reveal the fundamental properties of both newfound materials for stimulating experimental research groups in the search for synthesis routes to obtain this structure. This work used the DFT/B3LYP approach implemented in the CRYSTAL17 computational package. Ga, As, and P atomic centers were described by triple-zeta valence with polarization (TZVP) basis set. The vibrational analysis was carried out via coupled-perturbed Hartree-Fock/Kohn Sham (CPHF/KS) method, and the chemical bonds were evaluated via the quantum theory of atoms in molecules and crystals (QTAIMC).

Symmetry analysis and closed-form invariant solutions of the nonlinear wave equations in elasticity using optimal system of Lie subalgebra

Nonlinear wave equations are generated by the elastic wave propagation through inelastic material. We studied such a unidirectional nonlinear elastic wave by considering potential having geometrical nonlinearity. The nonlinear problem in elasticity is studied by symmetry method. Optimal system of non-similar one dimensional subalgebras of Lie algebra is constructed. Symmetry reductions are performed using these subalgebras and corresponding group invariant solutions are found. The problem of nonlinear elastic wave with a damping term is also studied. Multiplier approach is used for finding conservation laws of these equations.

Construction of Multivariate Polynomial Approximation Kernels via Semidefinite Programming

In this paper we construct a hierarchy of multivariate polynomial approximation kernels for uniformly continuous functions on the hypercube via semidefinite programming. We give details on the implementation of the semidefinite programs defining the kernels. Finally, we show how symmetry reduction may be performed to increase numerical tractability.