Symmetry Analysis Method Research Articles

3‐Dimensional Scanning of Entire Unit Cells in Single Nanoparticles.

AbstractProperties of nanomaterials such as optical, electrical, and chemical properties are strongly correlated with lattice symmetry, making characterization of lattice symmetry essential. We introduce a symmetry analysis method using 3D atomic coordinates obtained by Brownian one‐particle 3D reconstruction. The method allows direct and quantitative analysis of symmetrical properties and delivers local structural characteristics of individual platinum (Pt) nanoparticles in unit‐cell level. Local structural deformations of the Pt nanoparticles such as lattice distortion and internal symmetry breakage are demonstrated, revealing that the crystal structure of sub‐3 nm Pt nanoparticles generally maintains FCC crystallinity and exhibits localized deviation from their bulk counterpart.

Optical solitons and modulation instability for Cubic–quartic Fokas–Lenells equation

The current study is dedicated to the optical soliton solutions and modulation instability of the Cubic–quartic Fokas–Lenells equation with nonlinear perturbation and polarization-preserving fibers. In this direction, we take the help of Lie Symmetry analysis method. First of all, we obtained the invariant condition which plays important role in the mechanism of Lie symmetry method. After that, we developed the appropriate vector fields. Consequently, with the application of these vector fields similarity solutions are obtained in form of trigonometric functions. Further, in this work, optical solitons are obtained for Cubic–quartic Fokas–Lenells equation which are in form of hyperbolic and exponential functions. In addition at the end of this work modulation instability of the Cubic–quartic Fokas–Lenells equation will be discussed.

Lie symmetry analysis and exact solutions of space-time fractional cubic Schrödinger equation

In this paper, the Lie symmetry analysis method is applied to the space-time fractional cubic Schrödinger equation. The group generators are obtained for the system of space-time fractional differential equations. They are utilized to reduce the system of fractional partial differential equations with Riemann–Liouville fractional derivative to the system of fractional ordinary differential equations with Erdélyi–Kober fractional derivative. Then, the power series method is applied to derive explicit solutions for the reduced systems. Furthermore, the traveling wave solutions of the integer-order cubic Schrödinger equation is obtained.

Closed-form invariant solutions from the Lie symmetry analysis and dynamics of solitonic profiles for (2 + 1)-dimensional modified Heisenberg ferromagnetic system

This paper studies the ([Formula: see text])-dimensional modified Heisenberg ferromagnetic (MHF) system, which describes the different nonlinear dynamical structures for long water-waves of small amplitude with weakly nonlinear restoring forces and high-frequency dispersion. This MHF system arises in the propagation of the magnetization vector of the isotropic ferromagnet and biological pattern formation. By employing the Lie symmetry analysis method, infinitesimal generators, Lie point symmetries, potential vector fields, commutation relations of infinitesimal vectors, and attractive symmetry reductions are derived. Based on the two phases of Lie similarity reductions, ([Formula: see text])-dimensional MHF system is reduced to various nonlinear ordinary differential equations (ODEs). Afterwards, with the help of symbolic computation, we solve the acquired ODEs and obtain a variety of exact closed-form solutions involving arbitrary independent functions and other constant parameters. The physical features of the obtained multi-wave soliton solutions are demonstrated to analyze the impact of the involved arbitrary independent functions on the dynamics of the solitary wave solutions via three-dimensional graphics. These exact solutions are accomplished in the shapes of single solitons, doubly solitons, multi-wave solitons, elastic behavior of multisoliton, oscillating multi-solitons, curved-shaped periodic solitons, and kink-type solitons, and so on. The newly constructed results show the trustworthiness, reliability, and efficiency of the Lie symmetry technique for obtaining the invariant closed-form solutions to nonlinear governing model. Moreover, conservation laws and self-adjoint systems have been obtained by implementing Noether’s technique. By using Lie symmetry analysis, the achieved outcomes might be helpful to understand the physical formation of this model and confirm the effectiveness and authenticity of the mentioned method.

Hyperbolic (3+1)‐Dimensional Nonlinear Schrödinger Equation: Lie Symmetry Analysis and Modulation Instability

The hyperbolic nonlinear Schrödinger equation in the (3 + 1)‐dimension depicts the evolution of the elevation of the water wave surface for slowly modulated wave trains in deep water. Many researchers have studied the applicability and practicality of this model, but the analytical approach has been virtually absent from the literature. We adapted the lie symmetry analysis method to obtain a new complex solution in this work. The obtained complex solution contains bright and dark solitons. Furthermore, modulation instability is applied to this model to explain the interplay between nonlinear and dispersive effects. As a result, the modulation instability condition and the explosive rate are also discussed.

Lie symmetry analysis and invariant solutions of 3D Euler equations for axisymmetric, incompressible, and inviscid flow in the cylindrical coordinates

Through the Lie symmetry analysis method, the axisymmetric, incompressible, and inviscid fluid is studied. The governing equations that describe the flow are the Euler equations. Under intensive observation, these equations do not have a certain solution localized in all directions (r,t,z) due to the presence of the term frac{1}{r}, which leads to the singularity cases. The researchers avoid this problem by truncating this term or solving the equations in the Cartesian plane. However, the Euler equations have an infinite number of Lie infinitesimals; we utilize the commutative product between these Lie vectors. The specialization process procures a nonlinear system of ODEs. Manual calculations have been done to solve this system. The investigated Lie vectors have been used to generate new solutions for the Euler equations. Some solutions are selected and plotted as two-dimensional plots.

Symmetry analysis of the generalized space and time fractional Korteweg–de Vries equation

In this paper, we studied the generalized space and time fractional Korteweg–de Vries (KdV) equation in the sense of the Riemann–Liouville fractional derivative. Initially, the symmetry of this considered equation through the symmetry analysis method was obtained. Next, a one-parameter Lie group of point transformation was yielded. Then, this considered fractional model can be translated into an ordinary differential equation of fractional order via the Erdélyi–Kober fractional differential operator and the Erdélyi–Kober fractional integral operator. Finally, with the help of the nonlinear self-adjointness, conservation laws of the generalized space and time fractional KdV equation can be found. This approach can provide us with a new scheme for studying space and time differential equations of fractional derivative.

Study of exact analytical solutions and various wave profiles of a new extended (2+1)-dimensional Boussinesq equation using symmetry analysis

This paper systematically investigates the exact solutions to an extended (2+1)-dimensional Boussinesq equation, which arises in several physical applications, including the propagation of shallow-water waves, with the help of the Lie symmetry analysis method. We acquired the vector fields, commutation relations, optimal systems, two stages of reductions, and exact solutions to the given equation by taking advantage of the Lie group method. The method plays a crucial role to reduce the number of independent variables by one in each stage and finally forms an ODE which is solved by taking relevant suppositions and choosing the arbitrary constants that appear therein. Furthermore, Lie symmetry analysis (LSA) is implemented for perceiving the symmetries of the Boussinesq equation and then culminating the solitary wave solutions. The behavior of the obtained results for multiple cases of symmetries is obtained in the present framework and demonstrated through three-and two-dimensional dynamical wave profiles. These solutions show single soliton, multiple solitons, elastic behavior of combo soliton profiles, and stationary waves, as can be seen from the graphics. The outcomes of the present investigation manifest that the considered scheme is systematic and significant to solve nonlinear evolution equations.

Study on the Coupled Vibration Characteristics of a Two-Stage Bladed Disk Rotor System

This paper conducts a coupled vibration analysis of a two-stage bladed disk rotor system. According to the finite element method, the bladed disk rotor system is established. The substructure modal synthesis super-element method (SMSM) with a fixed interface and free interface is presented to obtain the vibration behaviors of the rotor system. Then, the free vibration results are compared with the ones calculated by the cyclic symmetry analysis method to validate the analysis in this paper. The results show that the modes of the two-stage bladed disk not only include the modes of the first- and second-stage bladed disk, but also the coupled modes of the two-stage bladed disk.

Invariant Analysis, Analytical Solutions, and Conservation Laws for Two-Dimensional Time Fractional Fokker-Planck Equation

The main purpose of this paper is to apply the Lie symmetry analysis method for the two-dimensional time fractional Fokker-Planck (FP) equation in the sense of Riemann–Liouville fractional derivative. The Lie point symmetries are derived to obtain the similarity reductions and explicit solutions of the governing equation. By using the new conservation theorem, the new conserved vectors for the two-dimensional time fractional Fokker-Planck equation have been constructed with a detailed derivation. Finally, we obtain its explicit analytic solutions with the aid of the power series expansion method.

Adiabatic Brownian motor with a stepwise potential perturbed by a dichotomous harmonic sygnal

We obtained an analytical expression for the average motion velocity of an adiabatic Brownian motor (ratchet), which operates due to small dichotomous spatially harmonic fluctuations of a stepwise potential. The symmetry properties of the average velocity as a functional of the stationary and fluctuating components of the nanoparticle potential energy are revealed, and the ranges of values of the system parameters that ensure the rightward and leftward motion of the motor are determined. We showed that the average motor velocity is a non-monotonic function of the stepwise potential height. For a singular (infinitely high and narrow) potential barrier, the average velocity depends non-monotonically on the «power» of this barrier (the barrier width multiplied by the exponent of the ratio of the barrier height to the thermal energy). The article continues the further development of theoretical methods of symmetry analysis by applying the general approaches proposed by the authors to specific motor systems.

Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation

In soliton theory, the dynamics of solitary wave solutions may play a crucial role in the fields of mathematical physics, plasma physics, biology, fluid dynamics, nonlinear optics, condensed matter physics, and many others. The main concern of this present article is to obtain symmetry reductions and some new explicit exact solutions of the (2 + 1)-dimensional Sharma–Tasso–Olver (STO) equation by using the Lie symmetry analysis method. The infinitesimals for the STO equation were achieved under the invariance criteria of Lie groups. Then, the two stages of symmetry reductions of the governing equation are obtained with the help of an optimal system. Meanwhile, this Lie symmetry method will reduce the STO equation into new partial differential equations (PDEs) which contain a lesser number of independent variables. Based on numerical simulation, the dynamical characteristics of the solitary wave solutions illustrate multiple-front wave profiles, solitary wave solutions, kink wave solitons, oscillating periodic solitons, and annihilation of parabolic wave structures via 3D plots.

Morphometric analysis of nuclear symmetry in urothelial carcinoma for predicting tumor recurrence.

Urothelial carcinoma is the ninth most common cancer in the world. Cytological analysis of the urine is used for screening, as well as for cases suspected for neoplasia of the urinary tract. However, the sensitivity of urine cytology examination is low. The golden standard for diagnosing bladder cancer relies upon cystoscopy followed by a biopsy, which is microscopically assessed by the pathologist. Treatment decisions are based on the histological grade and stage of the tumor. Posttreatment tumor recurrence is 50%. The purpose of this study is to predict recurrence of urothelial carcinoma using a novel morphometric method of nuclear symmetry analysis. This method may help tailor the appropriate treatment and may reduce the need of invasive surgical procedures in patients. Computerized morphometry was applied to develop multiple symmetry indices of the nuclei of the tumor cells as follows: each nucleus was physically divided along its digital axis in two segments that were separately analyzed for their shape, size, optical density, and texture. Subsequently, ratios were obtained by mathematically dividing between the morphometric values of the two nuclear segments where the denominator contained the largest value of the two. These ratios were named symmetry indices and were included as variables to predict the recurrence time of the tumors. The change in the symmetry indices (loss of symmetry) of the nuclear roundness, fractal dimension and margination were the only independent predictors of recurrence time. Computerized morphometry of nuclear symmetry indices may help to predict tumor recurrence in urothelial carcinomas.

Lie symmetry analysis and similarity solutions for the Camassa–Choi equations

The method of Lie symmetry analysis of differential equations is applied to determine exact solutions for the Camassa-Choi equation and its generalization. We prove that the Camassa-Choi equation is invariant under an infinite-dimensional Lie algebra, with an essential five-dimensional Lie algebra. The application of the Lie point symmetries leads to the construction of exact similarity solutions.

Lie-Bäcklund symmetries, analytical solutions and conservation laws to the more general (2 + 1)-dimensional Boussinesq equation

The propagation of shallow water waves with small amplitudes as they propagate in a water channel of constant depth at a uniform speed is described by general Boussinesq equation. It also models the simulation of water waves in shallow seas and harbors for ocean engineering. In this work the symmetry analysis method is used to study the Lie-Bäcklund symmetry generators along with the corresponding conservation laws (cLs) for the governing equation by using a new conservation theorem. Moreover, by means of two effective analytical schemes namely the extended ShGEEM and the Kudryashov’s methods, we construct some important soliton solutions for the equation. The physical features of the acquired solutions are plotted to depict the clear outlook of the solutions.

Birkhoff’s Theorem and Lie Symmetry Analysis

Three dimensional space is said to be spherically symmetric if it admits SO(3) as the group of isometries. Under this symmetry condition, the Einstein’s Field equations for vacuum, yields the Schwarzschild Metric as the unique solution, which essentially is the statement of the well known Birkhoff’s Theorem. Geometrically speaking this theorem claims that the pseudo-Riemanian space-times provide more isometries than expected from the original metric holonomy/ansatz. In this paper we use the method of Lie Symmetry Analysis to analyze the Einstein’s Vacuum Field Equations so as to obtain the Symmetry Generators of the corresponding Differential Equation. Additionally, applying the Noether Point Symmetry method we have obtained the conserved quantities corresponding to the generators of the Schwarzschild Lagrangian and paving way to reformulate the Birkhoff’s Theorem from a different approach.

Lie symmetry analysis of fractional ordinary differential equation with neutral delay

In this paper, Lie symmetry analysis method is employed to solve the fractional ordinary differential equation with neutral delay. The Lie symmetries for the fractional ordinary differential equation with neutral delay are obtained, and the group classification of the equation is established. The obtained Lie symmetries are used to construct the exact solutions of the fractional ordinary differential equation with neutral delay.

Lie Symmetry Reductions and Wave Solutions of Coupled Equal Width Wave Equation

In the present paper, we have solved the coupled equal width wave equation by using the Lie symmetry analysis method. By applying the Lie symmetry transformation to the system of non-linear partial differential equations, we obtained the infinitesimal generators to the system and converted the system of non-linear partial differential equations into the system of non-linear ordinary differential equations. Further, the obtained system of ordinary differential equations has been solved by using the tanh method and power series method. The travelling wave solution to the coupled equal width wave equation has also been obtained. We have also found some implicit solution to the coupled equal width wave equation. The convergences of power series solutions have been shown.

Optimal portfolio for a defined-contribution pension plan under a constant elasticity of variance model with exponential utility

Based on the Lie symmetry method, we derive the explicit optimal invest strategy for an investor who seeks to maximize the expected exponential (CARA) utility of the terminal wealth in a defined-contribution pension plan under a constant elasticity of variance model. We examine the point symmetries of the Hamilton-Jacobi-Bellman (HJB) equation associated with the portfolio optimization problem. The symmetries compatible with the terminal condition enable us to transform the (2 + 1)-dimensional HJB equation into a (1 + 1)-dimensional nonlinear equation which is linearized by its infinite-parameter Lie group of point transformations. Finally, the ansatz technique based on variables separation is applied to solve the linear equation and the optimal strategy is obtained. The algorithmic procedure of the Lie symmetry analysis method adopted here is quite general compared with conjectures used in the literature.

Lie Symmetry Analysis, Exact Solutions, Conservation Laws and Bäcklund Transformations of the Gibbons-Tsarev Equation

In this paper, we mainly put the Lie symmetry analysis method on the Gibbons-Tsarev equation (GTe) to obtain some new results, including some Lie symmetries, one-parameter transformation groups, explicit invariant solutions in the form of power series. Subsequently, the self-adjointness of the GTe is singled out. It follows that the conservation laws associated with symmetries of GTe are constructed with the aid of Ibragimov’ method. Finally, we present the Bäcklund transformations so that more abundant solutions can be worked out.