Point Symmetry Research Articles

Optical solutions for a quintic derivative nonlinear Schrödinger equation using symmetry analysis

In this paper, the derivative nonlinear Schrödinger equation with time-dependent-coefficient coefficients (DNLS) has been studied. This model is widely used in the hydrodynamic wave packets studies and optical fiber optics. The DNLS will be solved using a traveling wave transformation and Lie point symmetry analysis which is considered as a powerful tool that can be used in finding exact solutions of nonlinear partial differential equations (NLPDEs). The Lie symmetry technique provides a general way to obtain invariant solutions for NLPDEs. Some new exact solutions of the DNLS are obtained in the form of hyperbolic functions, Jacobi elliptic functions and Weierstrass elliptic function. The obtained waves have the forms of bright, dark and rogue waves soliton solutions which are considered as new solutions of the studied equation. The results established in this paper are new and extend the numbers of previously published results.

Lie Symmetries of the Wave Equation on the Sphere Using Geometry

A semilinear quadratic equation of the form Aij(x)uij=Bi(x,u)ui+F(x,u) defines a metric Aij; therefore, it is possible to relate the Lie point symmetries of the equation with the symmetries of this metric. The Lie symmetry conditions break into two sets: one set containing the Lie derivative of the metric wrt the Lie symmetry generator, and the other set containing the quantities Bi(x,u),F(x,u). From the first set, it follows that the generators of Lie point symmetries are elements of the conformal algebra of the metric Aij, while the second set serves as constraint equations, which select elements from the conformal algebra of Aij. Therefore, it is possible to determine the Lie point symmetries using a geometric approach based on the computation of the conformal Killing vectors of the metric Aij. In the present article, the nonlinear Poisson equation Δgu−f(u)=0 is studied. The metric defined by this equation is 1 + 2 decomposable along the gradient Killing vector ∂t. It is a conformally flat metric, which admits 10 conformal Killing vectors. We determine the conformal Killing vectors of this metric using a general geometric method, which computes the conformal Killing vectors of a general 1+(n−1) decomposable metric in a systematic way. It is found that the nonlinear Poisson equation Δgu−f(u)=0 admits Lie point symmetries only when f(u)=ku, and in this case, only the Killing vectors are admitted. It is shown that the Noether point symmetries coincide with the Lie point symmetries. This approach/method can be used to study the Lie point symmetries of more complex equations and with more degrees of freedom.

Thermal Stability and Non-Linear Optical and Dielectric Properties of Lead-Free K0.5Bi0.5TiO3 Ceramics.

Lead-free K0.5Bi0.5TiO3 (KBT) ceramics with high density (~5.36 g/cm3, 90% of X-ray density) and compositional purity (up to 90%) were synthesized using a solid-state reaction method. Strongly condensed KBT ceramics revealed homogenous local microstructures. TG/DSC (Thermogravimetry-differential scanning calorimetry) techniques characterized the thermal and structural stability of KBT. High mass stability (>0.4%) has proven no KBT thermal decomposition or other phase precipitation up to 1000 °C except for the co-existing K2Ti6O13 impurity. A strong influence of crystallites size and sintering conditions on improved dielectric and non-linear optical properties was reported. A significant increase (more than twice) in dielectric permittivity (εR), substantial for potential applications, was found in the KBT-24h specimen with extensive milling time. Moreover, it was observed that the second harmonic generation (λSHG = 532 nm) was activated at remarkably low fundamental beam intensity. Finally, spectroscopic experiments (Fourier transform Raman and far-infrared spectroscopy (FT-IR)) were supported by DFT (Density functional theory) calculations with a 2 × 2 × 2 supercell (P42mc symmetry and C4v point group). Moreover, the energy band gap was calculated (Eg = 2.46 eV), and a strong hybridization of the O-2p and Ti-3d orbitals at Eg explained the nature of band-gap transition (Γ → Γ).

High precision 3 × 3 coupler demodulation algorithm with an additional phase shift judgment module

Ellipse fitting algorithms (EFAs) have been widely used in 3×3 coupler demodulation systems to reduce the requirement for symmetry of the 3×3 couplers. Based on the relative stability of the splitting ratio and phase difference after the establishment of the 3×3 coupler demodulation system, we solve the problem that EFA fails to work when the stimulating signal is small. Depending on the existence of a symmetry point about the origin, an additional phase shift judgment module is used to determine whether the Lissajous figure is larger than π rad. If the elliptical arc exceeds π rad, the EFA is executed. Otherwise, the previous parameters are used to correct the ellipse. Parameters are updated in real time to ensure high precision. The experimental results show that the total harmonic distortion (THD) of the ameliorated algorithm is improved by 1.28% compared to the EFA without the judgment module with a stimulus amplitude of 30 mV. The proposed scheme can effectively improve the dynamic range of the 3×3 coupler demodulation to reach 125.64 dB @ 1 kHz and 1% THD. The algorithm ensures the effective operation of the EFA under small phase shift conditions and improves the accuracy of phase demodulation.

Homogeneous Quandles with Abelian Inner Automorphism Groups and Vertex-Transitive Graphs

A quandle is an algebraic system originated in knot theory, and can be regarded as a generalization of symmetric spaces. The inner automorphism group of a quandle is defined as the group generated by the point symmetries (right multiplications). In this paper, starting from any simple graphs, we construct quandles whose inner automorphism groups are abelian. We also prove that the constructed quandle is homogeneous if and only if the graph is vertex-transitive. This shows that there is a wide family of quandles with abelian inner automorphism groups, even if we impose the homogeneity. The key examples of such quandles are realized as subquandles of oriented real Grassmannian manifolds.

The physical attributes of rubidium-based Rb2TlRhF6 double perovskite halide: A first-principles investigation

Rubidium cations (Rb+) significantly improve the overall performance of organic-inorganic hybrid perovskite based devices and have gained exceptional attention due to their good thermal and structural stability. In this paper, we have evaluated the physical attributes of rubidium based Rb2TlRhF6 via DFT code using TB-mBJ potential. Structural stability has been confirmed through negative formation energy and calculated tolerance factor. Band structure and TDOS profile confirm semiconductor nature with 3.2 eV direct bandgap at (L-L) symmetry points. A ductile nature with metallic bonding is observed through the investigation of elastic attributes. Optical analysis of Rb2TlRhF6 indicates great absorption in the UV region. Low thermal conductivity and electrical resistivity, along with a high value for power factor, resulted in high ZT value of 0.8 and 800 K. The combined qualities provide a stunning and exciting picture of this compound for use in thermoelectric advanced technologies.

Ba2MgXO6 (X = S, Se), Chalcogenide‐Based Double Perovskite Materials for Green Optoelectronics and Thermoelectric Applications

Chalcogenide‐based double perovskite materials have recently attracted attention of researchers because of their excellent properties and eco‐friendly nature. These compounds have proven to comprise better oxidation resistance, high‐temperature stability, and lower processing cost. In this report, the first‐principles study on chalcogenides double perovskites Ba2MgXO6 (X = S, Se) is described. It is shown that these perovskites can be synthesized, as they satisfy the structural, dynamical, and thermodynamical stability criteria. It is revealed that these double perovskites are inherently semiconductors in nature with low bandgap values, 0.33 eV for Ba2MgSO6 and 0.66 eV for Ba2MgSeO6 along Γ symmetry point. The absorption coefficient ≈105 cm−1 and the reflectivity below 12% in visible region are calculated. Therefore, the Ba2MgXO6 for solar cell applications is recommended. Also, on the basis of calculated larger values of figure of merit, these perovskites are proposed for thermoelectric applications at high temperatures.

A DFT approach to explore the structural, optoelectronic, mechanical, and thermoelectric attributes of lead-free RbInX3 (X = I, Br, Cl) halide perovskites for renewable energy applications

Halide perovskites are significant due to their versatility and high performance in optoelectronics and green technologies. This paper thoroughly investigates the structural, elastic, optoelectronic and transport characteristics of RbInX3 (X = I, Br, Cl) perovskites using density functional theory (DFT) embedded in Wein2K. Modified Becke Johnson as an exchange correlation potential is used to evaluate the physical characteristics. Structural stability is achieved with the tolerance factor and octahedral tilting along with the optimization curves. A metallic character is detected in RbInI3. However, the replacement of I with Br and Cl atoms results in an indirect band gap of 0.9 eV and 1.95 eV at (L-X) symmetry points, respectively. The elastic constants are utilized to determine the elastic parameters that ensure the structural stability of perovskites. Cauchy pressure and Poisson’s ratio indicate the ductile nature of all studied phases. The thermal behavior is evaluated through the computation of Debye temperature. Furthermore, an insight into the material’s interaction with electromagnetic radiation is accessed through the optical parameters. A high absorption at 1.49 eV for RbInI3, 1.65 for RbInBr3 and 2.57 for RbInCl3 show their potential for light emitting devices. The semi-classical Boltzmann theory is employed to compute thermoelectric properties, which show RbInI3, RbInBr3 and RbInCl3 possess high electrical conductivities of 3.068 ×1019 S m–1, 1.88 ×1019 S m–1 and 1.76 ×1019 S m–1 at 1200 K, respectively. The ZT value for RbInBr3 is 0.74 and RbInCl3 is 0.73, while RbInI3 exhibit the lowest ZT of 0.04 at 1200 K. The examined characteristics indicate that RbInBr3 and RbInCl3 possess promising potential for renewable energy applications.

Extended symmetry analysis of (1+2)‐dimensional fine Kolmogorov backward equation

AbstractWithin the class of (1+2)‐dimensional ultraparabolic linear equations, we distinguish a fine Kolmogorov backward equation with a quadratic diffusivity. Modulo the point equivalence, it is a unique equation within the class whose essential Lie invariance algebra is five‐dimensional and nonsolvable. Using the direct method, we compute the point symmetry pseudogroup of this equation and analyze its structure. In particular, we single out its essential subgroup and classify its discrete elements. We exhaustively classify all subalgebras of the corresponding essential Lie invariance algebra up to inner automorphisms and up to the action of the essential point‐symmetry group. This allowed us to classify Lie reductions and Lie invariant solutions of the equation under consideration. We also discuss the generation of its solutions using point and linear generalized symmetries and carry out its peculiar generalized reductions. As a result, we construct wide families of its solutions parameterized by an arbitrary finite number of arbitrary solutions of the (1+1)‐dimensional linear heat equation or one or two arbitrary solutions of (1+1)‐dimensional linear heat equations with inverse square potentials.

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.

Observation of Kekulé vortices around hydrogen adatoms in graphene

Fractional charges are one of the wonders of the fractional quantum Hall effect. Such objects are also anticipated in two-dimensional hexagonal lattices under time reversal symmetry—emerging as bound states of a rotating bond texture called a Kekulé vortex. However, the physical mechanisms inducing such topological defects remain elusive, preventing experimental realization. Here, we report the observation of Kekulé vortices in the local density of states of graphene under time reversal symmetry. The vortices result from intervalley scattering on chemisorbed hydrogen adatoms. We uncover that their 2π winding is reminiscent of the Berry phase π of the massless Dirac electrons. We can also induce a Kekulé pattern without vortices by creating point scatterers such as divacancies, which break different point symmetries. Our local-probe study thus confirms point defects as versatile building blocks for Kekulé engineering of graphene’s electronic structure.

Exact FLRW cosmological solutions via invariants of the symmetry groups

Until now, various methods have been demonstrated to solve the Friedmann-Lamaítre-Robertson-Walker (FLRW) equations in the spatially flat (k = 0) cosmological model. In this study, in order to solve the field equations of the spatially flat FLRW cosmological model in the presence of Λ, a new method based on the invariants of the symmetry groups which we called ISG-method, is presented. This method is based on the extended Prelle-Singer (PS) method and it uses the Lie point symmetry, λ-symmetry and Darboux polynomials (DPs). We employ this method to extract systematically the two independent first integrals (or invariants) such as I1(a,ȧ)=c1 and I2(t,a,ȧ)=c2 associated to the group of the Lie point transformations keeping the Friedmann-Einstein dynamical equation (DE), ä=ϕ(t,a,ȧ) , invariant. The obtained solutions from solving the DEs of the FLRW cosmological model by the ISG-method are explicitly written, so that they are suitable for cosmological applications. Finally, as an application of the solutions we look at the age of universe in the presence of cosmological constant where the dominant matter of the Universe is considered to be fluid with the state parameter w. In this regard, we calculate the age of universe when the dominant matter is dust.

Pseudo-Polymorphism in 2-Pyridylmethoxy Cone Derivatives of p-tert-butylcalix[4]arene and p-tert-butylhomooxacalix[n]arenes

This paper investigates pseudo-polymorphism in 2-pyridylmethoxy derivatives of p-tert-butylcalix[4]arene (PyC4), p-tert-butyldihomooxa-calix[4]arenes (PyHOC4), and p-tert-butylhexahomotrioxacalix[3]arenes (PyHO3C3), presenting 11 crystal structures with 15 crystallographically independent molecules. The macrocycle of PyC4 is smaller and less flexible with respect to those of PyHOC4 and PyHO3C3, and in solution, the cone conformation of these three molecules exhibits different point symmetries: C4, Cs, and C3, respectively. A correlation is observed between the macrocycle’s structural rigidity and the number of pseudo-polymorphs formed. The more rigid PyC4 displays a higher number (six) of pseudo-polymorphs compared to PyHOC4 and PyHO3C3, which exhibit a smaller number of crystalline forms (three and two, respectively). The X-ray structures obtained show that the conformation of the macrorings is primarily influenced by the presence of an acetonitrile guest molecule within the cavity, with limited impact from crystal packing and intermolecular co-crystallized solvent molecules. Notably, both calix[4]arene derivatives produce a host–guest complex with acetonitrile, while the most flexible and less aromatic PyHO3C3 does not give crystals with acetonitrile as the guest. Intertwined 1D and 2D solvent channel networks were observed in the PyHOC4-hexane and in the PyHO3C3-H2O-MeOH crystal structures, respectively, while the other pseudopolymorphs of PyHOC4 and PyHO3C3 and all PyC4 crystal forms exhibit closely packed crystal structures without open channels.

Dynamics of the breather and solitary waves in plasma using the generalized variable coefficient Gardner equation

This paper presents an analytical approach to study the generalized variable coefficient Gardner equation, aiming to investigate the behaviour of plasma phenomena, which presents challenges due to the non-linear nature of the Gardner equation. The point symmetry group associated with the proposed equation is determined by Lie symmetry analysis. The Exp-function method is employed to construct exact travelling wave solutions and to obtain anomalous solutions with non-zero amplitude. Additionally, conservation laws are derived and analysed from the equation in the frame of Noether symmetry. Through simulation, breather, periodic, and localized solitary wave solutions are also presented in graphical form. Our results are expected to elucidate the impact of the equation in the study of the propagation of Alfvén waves with non-uniform density, temperature, and wave-particle interactions in a magnetized plasma.

Conserved vectors and symmetry solutions of the Landau–Ginzburg–Higgs equation of theoretical physics

This paper is devoted to the investigation of the Landau–Ginzburg–Higgs equation (LGHe), which serves as a mathematical model to understand phenomena such as superconductivity and cyclotron waves. The LGHe finds applications in various scientific fields, including fluid dynamics, plasma physics, biological systems, and electricity-electronics. The study adopts Lie symmetry analysis as the primary framework for exploration. This analysis involves the identification of Lie point symmetries that are admitted by the differential equation. By leveraging these Lie point symmetries, symmetry reductions are performed, leading to the discovery of group invariant solutions. To obtain explicit solutions, several mathematical methods are applied, including Kudryashov’s method, the extended Jacobi elliptic function expansion method, the power series method, and the simplest equation method. These methods yield solutions characterized by exponential, hyperbolic, and elliptic functions. The obtained solutions are visually represented through 3D, 2D, and density plots, which effectively illustrate the nature of the solutions. These plots depict various patterns, such as kink-shaped, singular kink-shaped, bell-shaped, and periodic solutions. Finally, the paper employs the multiplier method and the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a crucial role in the study of physical quantities, such as the conservation of energy and momentum, and contribute to the understanding of the underlying physics of the system.

The formation of invariant optical soliton structures to electric-signal in the telegraph lines on basis of the tunnel diode and chaos visualization, conserved quantities: Lie point symmetry approach

This study critically examines the nonlinear Lonngren wave model, a mathematical representation used to describe waves in shallow water environments such as beaches, lakes, and rivers, as well as electrical signals in semiconductor materials like tunnel diodes. The symmetry generators take into account the Lie invariance criteria. A three-dimensional Lie algebra can be obtained by using the suggested methodology. It links translation symmetries in space and time, respectively, with conservation of mass and energy. The third symmetry pertains to scaling or dilation. Recognizing the constraints of the inverse scattering transform in addressing the Cauchy problem for this equation, the investigation adopts new auxiliary equation approach to ordinary differential equations. This approach yields closed-form analytical solutions for the wave model. The obtained solutions include a plane solution, mixed hyperbolic solution, periodic and mixed periodic solutions, mixed trigonometric solution, smooth soliton, trigonometric solution, mixed singular solution, complex solitary and singular solution. Furthermore, the study develops the Hamiltonian function of the dynamical system to discuss the total energy of the system in terms of its position and momentum. Chaos analysis is performed to describe periodic, quasi-periodic, and chaotic behaviours, and a sensitivity analysis highlights the model’s sensitivity to different wave numbers. Graphical representations of selected solutions are presented in both 3-D and 2-D, illustrating the behaviour under specific physical conditions. The study concludes by identifying a comprehensive set of local conservation laws for the nonlinear Lonngren wave equation, applicable to arbitrary constant coefficients, using conservation laws multipliers.

Nonlocal symmetries, soliton-cnoidal wave solution and soliton molecules to a (2+1)-dimensional modified KdV system

A (2+1)-dimensional modified KdV (2DmKdV) system is considered from several perspectives. Firstly, residue symmetry, a type of nonlocal symmetry, and the Bäcklund transformation are obtained via the truncated Painlevé expansion method. Subsequently, the residue symmetry is localized to a Lie point symmetry of a prolonged system, from which the finite transformation group is derived. Secondly, the integrability of the 2DmKdV system is examined under the sense of consistent tanh expansion solvability. Simultaneously, explicit soliton-cnoidal wave solutions are provided. Finally, abundant patterns of soliton molecules are presented by imposing the velocity resonance condition on the multiple-soliton solution.

A study on the effects of objective weighting methods on TOPSIS-based parametric optimization of non-traditional machining processes

Developing advanced engineering materials with enhanced mechanical and metallurgical properties has caused non-traditional machining (NTM) processes to emerge as cost-effective and energy-efficient options. Improper settings of the input parameters of those processes may often lead to inferior quality and accuracy of the finished products. Multi-criteria decision-making (MCDM) methods appear as potent tools for multi-objective optimization of those processes. Determining criteria weights is a crucial step in any MCDM problem; subjective approaches for evaluating criteria weights may result in inconsistent judgments. In this paper, the effects of ten different objective weighting methods, i.e., equal weight, entropy weight, Criteria Importance Through Inter-criteria Correlation (CRITIC), Method based on Removal Effects of Criteria (MEREC), Logarithmic Percentage Change-driven Objective Weighting (LOPCOW), Symmetry Point of Criterion (SPC), Criteria Impact Loss (CILOS), Integrated Determination of Objective Criteria Weights (IDOCRIW), Principal Component Analysis (PCA) and Weighted Principal Component Analysis (WPCA) are studied on Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS)-based optimization of two NTM processes. In the first example, the optimal parametric combination computed using equal weight, entropy, MEREC, LOPCOW, PCA, and WPCA is noticed to be the same. In contrast, a different set of optimal parameters is derived by applying CRITIC, SPC, CILOS, and IDOCRIW. In the second example, seven out of ten approaches obtain a unique set of parametric combinations, while CILOS, IDOCRIW, and PCA lead to a different set of intermixes. It can be concluded that the ranks of the experiment trials for both the examples are almost similar irrespective of the objective criteria weighting technique employed.

Application of conserved quantities using the formal Lagrangian of a nonlinear integro partial differential equation through optimal system of one-dimensional subalgebras in physics and engineering

Abstract This research article analytically investigates a soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences and engineering. The soliton equation of high dimensions, particularly with applications, and precisely in the fields of physical sciences along with engineering, is examined with a view to securing various pertinent results of interest. For the first time, the conserved currents of an integrodifferential equation (especially those of higher dimensions) are calculated using a detailed optimal system of one-dimensional subalgebras. Infinitesimal generators of diverse structures ascribed to Lie point symmetries of the understudy model are first calculated via Lie group analysis technique. Additionally, we construct various commutations along Lie-adjoint representation tables connected to the nine-dimensional Lie algebra achieved. Further to that, detailed and comprehensive computation of the optimal system of one-dimensional subalgebras linked to the algebra is also unveiled for the under-investigated model. This, in consequence, engenders the calculation of abundant conserved currents for the soliton equation through Ibragimov’s conserved vector theorem by utilizing its formal Lagrangian. Later, the applications of our results are highlighted.

Nonlinear long waves in shallow water for normalized Boussinesq equations

In this work, we investigate a partial differential equation (PDE) derived from a Boussinesq system of two equations that describe the wave motion in shallow water when the wavelength is small compared to wave amplitude. The Lie point symmetries of the PDE are utilized to reduce it to an ordinary differential equation (ODE) the travelling wave solutions of which are obtained. These solutions are of trigonometric, hyperbolic and exponential types. Graphical representation of some solutions exhibiting solitary and cnoidal wave phenomena are presented.