Symmetry Analysis Method Research Articles

Lie symmetries, exact solutions and conservation laws of time fractional coupled (2+1)-dimensional nonlinear Schrödinger equations

ABSTRACT In this paper, Lie symmetry analysis method is applied to time fractional coupled (2 + 1)-dimensional nonlinear Schrödinger equations. We obtain all the Lie symmetries and reduce the (2 + 1)-dimensional fractional partial differential equations with Riemann-Liouville fractional derivative to (1 + 1)-dimensional counterparts with Erdélyi-Kober fractional derivative. Then we obtain the power series solutions and prove their convergence. In addition, the conservation laws for the governing equations are constructed by using the generalization of the Noether operators and the new conservation theorem.

Lie symmetries, exact solution and conservation laws of (2 + 1)-dimensional time fractional Kadomtsev–Petviashvili system

Abstract In this paper, Lie symmetry analysis method is applied to the ( 2 + 1 ) (2+1) -dimensional time fractional Kadomtsev–Petviashvili (KP) system, which is an important model in mathematical physics. We obtain all the Lie symmetries admitted by the KP system and use them to reduce the ( 2 + 1 ) (2+1) -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to some ( 1 + 1 ) (1+1) -dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative or Riemann–Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the system studied.

Lie symmetries, exact solutions and conservation laws of time fractional Boussinesq–Burgers system in ocean waves

In this paper, the Lie symmetry analysis method is applied to the time-fractional Boussinesq–Burgers system which is used to describe shallow water waves near an ocean coast or in a lake. We obtain all the Lie symmetries admitted by the system and use them to reduce the fractional partial differential equations with a Riemann–Liouville fractional derivative to some fractional ordinary differential equations with an Erdélyi–Kober fractional derivative, thereby getting some exact solutions of the reduced equations. For power series solutions, we prove their convergence and show the dynamic analysis of their truncated graphs. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Lie symmetries, exact solutions and conservation laws of (2+1)-dimensional time fractional cubic Schrödinger equation

Abstract In this paper, Lie symmetry analysis method is applied to ( 2 + 1 ) {(2+1)} -dimensional time fractional cubic Schrödinger equation. We obtain all the Lie symmetries and reduce the ( 2 + 1 ) {(2+1)} -dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to ( 1 + 1 ) (1+1) -dimensional counterparts with Erdélyi–Kober fractional derivative. Then we obtain the power series solutions of the reduced equations and prove their convergence. In addition, the conservation laws for the governing model are constructed by the new conservation theorem and the generalization of Noether operators.

Lie symmetry analysis, traveling wave solutions and conservation laws of a Zabolotskaya-Khokholov dynamical model in plasma physics

This article analyzes the analytic and solitary wave solutions of the one-dimensional Zabolotskaya-Khokholov (ZK) dynamical model which provides information about the propagation of sound beam or confined wave beam in nonlinear media and studies of beam deformation. By the Lie symmetry analysis method, we acquire the vector fields, commutation relations, optimal system, reduction, and analytic solutions to the specified equation by exerting the Lie group method. Moreover, the solitary wave solutions of the ZK model are procured by exerting the new auxiliary equation method (NAEM). The behavior of the acquired outcomes for several cases is exhibited graphically through two and three-dimensional dynamical wave profiles. Furthermore, the conservation laws of the ZK model are acquired by Ibragimov’s new conservation theorem.

Residual Symmetry and Interaction Solutions of the (2+1)-Dimensional Generalized Calogero–Bogoyavlenskii–Schiff Equation

In this paper, we investigate the (2+1)-dimensional generalized Calogero–Bogoyavlenskii–Schiff (gCBS) equation by using the residual symmetry analysis and consistent Riccati expansion (CRE) method, respectively. The residual symmetry of the gCBS equation is localized into a Lie point symmetry in a prolonged system and a new Bäcklund transformation of this equation is obtained. By applying the standard Lie symmetry method to the prolonged gCBS system, new symmetry reduction solutions of the gCBS equation are obtained. The gCBS equation is proved to be CRE integrable and new Bäcklund transformations of it are obtained, from which interaction solutions between solitons and periodic waves are generated and analyzed.

Symmetry analysis, optimal system, conservation laws and exact solutions of time-fractional diffusion-type equation

In Liu et al. [On the generalized time fractional diffusion equation: Symmetry analysis, conservation laws, optimal system and exact solutions, Int. J. Geom. Methods Mod. Phys. 17(01) (2020) 2050013], the generalized time-fractional diffusion equation [Formula: see text] is studied by the symmetry analysis method. Liu et al. obtained two group generators [Formula: see text] [Formula: see text] and only one trivial solution [Formula: see text] for the equation. In this paper, we classify the Lie symmetry group admitted by the equation, and for the case [Formula: see text], we obtain a richer set of group generators and some nontrivial solutions, including unprecedented exact solutions and power series solutions. In addition, we construct the one-dimensional optimal system of the Lie symmetry group admitted by the time-fractional diffusion-type equation by Olver’s method, and obtain the conservation laws for all the obtained Lie symmetries using the general method developed by Ibragimov. For the novel exact solutions and power series solutions, we analyze their dynamic behavior graphically.

Lie symmetry analysis of (2+1)-dimensional time fractional Kadomtsev-Petviashvili equation

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional Kadomtsev-Petviashvili (KP) equation with the mixed derivative of Riemann-Liouville time-fractional derivative and integer-order $x$-derivative. We obtained all the Lie symmetries admitted by the KP equation and used them to reduce the (2+1)-dimensional fractional partial differential equation with Riemann-Liouville fractional derivative to some (1+1)-dimensional fractional partial differential equations with Erd\'{e}lyi-Kober fractional derivative or Riemann-Liouville fractional derivative, thereby getting some exact solutions of the reduced equations. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equation studied.

Invariant analysis, invariant subspace method and conservation laws of the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system

In this paper, we investigate the (2+1)-dimensional mixed fractional Broer–Kaup–Kupershmidt system (MFBKKS) with Riemann–Liouville time fractional derivative and integer order y-derivative. This system models dispersive and nonlinear long gravity waves propagating in shallow water in two horizontal directions with time memories. Lie symmetry analysis and invariant subspace method are distinctly employed to construct the exact solutions to the MFBKKS. Initially, the Lie algebras admitted by MFBKKS are obtained with the help of Lie symmetry analysis. Then, we establish the commutative table, adjoint relations and adjoint transformation matrix. Specifically, the one-dimensional optimal system is established correspondingly, and symmetry reduction is performed. In particular, (2+1)-dimensional MFBKKS is reduced to (1+1)-dimensional mixed fractional system using Erdélyi–Kober fractional differential operator. Further, the power series solution of MFBKKS is constructed via power series method. By applying invariant subspace method, we obtain more exact solutions of MFBKKS. In addition, the conservation laws are derived by new Noether theorem. Finally, the three-dimensional diagrams of some obtained solutions are demonstrated utilizing Matlab for visualization, and some of the calculations were verified using computational packages Maple for symbolic computation.

Advanced Excitation Symmetry Analysis Method (ESAM) for improved automated ultrasonic testing of polished materials

One of the strategic plans of the international NDT community is to define standards for developing advanced and digitalized non-destructive testing for automated set-up in mass production [1,2]. Excitation Symmetry Analysis Method (ESAM) are new coded signal processing tools exploring symmetry properties [3]. ESAM method aims to extract the nonlinearity from an ultrasound output by studying the property of symmetry of the transfer function. More precisely, the extraction of nonlinearity is done using basis function associated to a group which is associated to an a priori transfer function. As soon as the linear tomography uses the invariant properties of amplitude dependence, nonlinear tomography should exploit the invariance properties (or symmetry properties) of the complex system. Consequently, the invariance of the stationary properties of a complex medium would be supposed to be associated to a signature of the degradation, that will be extracted with advanced signal processing tools. The idea remains the same: studying the symmetry and more precisely the time reversal one and the inversion. A signal processing is used to measure discrepancies in the autocorrelation function (output signal called chirp coda) for two inputs: a signal and its inversion (pulse inversion). The experiments were done on different sample with and without cracks. It has been shown that when the nonlinearities are extracted from the chirp coda amplitude are non- negligible in crack area and nonlinearities are out of phase. Then, mixing symmetries like time reversal and pulse inversion enable the detection of defect that are difficult to detect with conventional ultrasound method. Indeed, for such crack a destructive mechanical analysis called Jomini is done where the material is polished and examined with microscope. The sensitivity improvement of chirp-coded signal processing has been validated in various domains. Coded excitation techniques, used in communication systems such as radar and sonar provide improved SNR without increasing the amplitude of excitation. The typical test equipment consists of a preamplifier Juvitek TRA-02 (0.02 - 5 MHz) connected to a computer, an amplifier ENI model A150 (55 dB at 0.3-35 MHz), a shear wave transducer Technisonic (2.25 MHz), and a longitudinal wave transducer Panametrics V155 (5 MHz). The experimental device was tested with the V3 calibration block, improved, and specially scaled in order to access to a wide range of multivalued parameters: mechanical properties, ultrasonic parameters (celerity and attenuation) and local geometric data.

Lie Symmetry Analysis, Power Series Solutions and Conservation Laws of (2+1)-Dimensional Time Fractional Modified Bogoyavlenskii–Schiff Equations

In this paper, Lie symmetry analysis method is applied to the (2+1)-dimensional time fractional modified Bogoyavlenskii–Schiff equations, which is an important model in physics. The one-dimensional optimal system composed by the obtained Lie symmetries is utilized to reduce the system of (2+1)-dimensional fractional partial differential equations with Riemann–Liouville fractional derivative to the system of (1+1)-dimensional fractional partial differential equations with Erdélyi–Kober fractional derivative. Then the power series method is applied to derive explicit power series solutions for the reduced system. In addition, the new conservation theorem and the generalization of Noether operators are developed to construct the conservation laws for the equations studied.

Lie Symmetry Analysis and Conservation Laws of Fractional Benjamin–Ono Equation

In this paper, the fractional Benjamin–Ono differential equation with a Riemann–Liouville fractional derivative is considered using the Lie symmetry analysis method. Two symmetries admitted by the equation are obtained. Then, the equation is reduced to a fractional ordinary differential equation with an Erdélyi–Kober fractional derivative by one of the symmetries. Finally, conservation laws for the equations are constructed using the new conservation theorem.

On the generalized time fractional reaction–diffusion equation: Lie symmetries, exact solutions and conservation laws

In this paper, Lie symmetry analysis method is applied to the generalized time fractional reaction–diffusion equation. We obtain a conditional symmetric group and some conservation laws of the governing equation. The obtained Lie symmetries are used to reduce the studied fractional partial differential equation to some fractional ordinary differential equations with Riemann–Liouville fractional derivative or Erdélyi-Kober fractional derivative. Furthermore, we obtained asymptotic stable solutions and convergent power series solutions for the reduced equations. The dynamic behavior of these exact solutions is presented graphically.

Detection of iodixanol-induced allergic reaction signals in Chinese inpatients: a multi-center retrospective database study using prescription sequence symmetry analysis.

This study aimed to explore the signal detection method for allergic reactions induced by inpatient iodixanol injection. A database of 3,719,217 hospitalized patients from 20 large Chinese general hospitals was processed and analyzed using the prescription sequence symmetry analysis (PSSA) method. 126,680 inpatients who used iodixanol and were concurrently treated with anti-allergic drugs were analyzed. In the medical records of these patients, only 32 had documented iodixanol allergies. Statistical analysis identified 22 drugs in 4 categories-calcium preparations, adrenergic/dopaminergic agents, glucocorticoids, and antihistamines-as marker drugs. With time intervals of 3, 7, and 28days, the adjusted sequence ratios (aSRs) for all anti-allergics and the 4 categories were greater than 1. The 7-day aSRs were 2.12 (95% CI: 2.08-2.15), 1.70 (95% CI: 1.68-1.73), 3.85 (95% confidence interval [CI]: 3.75-2.30), 2.30 (95% CI: 2.26-2.35), and 1.95 (95% CI: 1.89-2.02), respectively. The proportions of adverse drug events indicated by each signal were as follows: all anti-allergics (2.92%-3%), calcium gluconate (0.19%-0.52%), adrenergic/dopaminergic agents (2.20%-3.37%), glucocorticoids (3.13%-3.76%), and antihistamines (1.05%-1.32%). This first multi-center Chinese inpatient database study detected iodixanol-induced allergy signals, revealing that reactions may be much higher than those in collected spontaneous reports. Iodixanol risk exposure was closer to actual pharmaceutical care findings. PSSA application with ≤7-day intervals appears better suited for monitoring late allergic reaction signals with these drugs.

Lie symmetry analysis, optimal system and exact solutions for variable-coefficients Boiti-Leon-Manna-Pempinelli equation

In this paper, the constant coefficients are extended to variable coefficients in the Boiti-Leon-Manna-Pempinelli(BLMP) equation, and its optimal system, exact solutions, conservation laws are studied. First, the infinitesimal generators for independent variables of the variable-coefficient Boiti-Leon-Manna-Pempinelli(vcBLMP) equation are solved by using the Lie symmetry analysis method. Subsequently, the optimal system of the vcBLMP equation is solved, and then the equations are subjected to similarity reduction, the partial differential equations(PDEs) are gained. Based on these PDEs, the representative equations are converted into ordinary differential equations(ODEs) by traveling wave transformation, and some new exact solutions of vcBLMP equation is obtained by using G′/G− expansion method and tanh-function method, such as the kink solutions, periodic solutions and soliton solutions. According to the obtained solutions, the corresponding images are plotted in order to facilitate the presentation of structural features of the solutions. At the end, the vcBLMP equation is proved to be nonlinear self-adjoint equation and the conservation laws of vcBLMP are solved.

Lie Symmetry Analysis for the Fractal Bond-Pricing Model of Mathematical Finance

The classical bond-pricing models, as important financial tools, show strong vitality in bond pricing. However, these models also expose their theoretical defects, which leads to inconsistencies with the actual observation results and usually causes the theoretical prices of bonds to be lower than the actual market prices in the financial market. In order to change this situation, considering that the price change of the underlying is regarded as a fractal transmission system, the fractal derivative is introduced into the bond-pricing equation. In order to solve the fractal bond-pricing equation, we first convert it into an equivalent equation by using a fractal two-scale transform. Only in this case can we start to study it by means of the Lie symmetry analysis method. Then the geometric vector fields, the symmetry reductions, and the exact solution to the equations are obtained. Furthermore, the dynamic behaviors of the fractal bond-pricing equation are discussed. The results show that the fractal dimension bond-pricing formula can better explain price changes in the capital market than the classical one. That is to say, the classical bond-pricing equation is only a special case of the fractal-bond pricing equation, which makes up for the defect that the theoretical bond price given by the classical bond-pricing equation is often lower than the actual market price. The results of this paper provide a basis for bond pricing in the financial market in order to seek a more appropriate and real price.

The Lie symmetry analysis, optimal system and exact solutions of the (2+1)-dimensional variable coefficients integrable coupled Burgers equations

The variable coefficients integrable coupled Burgers equations (vcICBs) are researched by the Lie symmetry analysis method. The infinitesimal generators, commutative relations and adjoint relations are given. Optimal system of vcICBs are studied by Olver’s method. Based on the optimal system, the (2+1)-dimensional vcICB equations are reduced to (1+1)-dimensional equations. Some new exact solutions of vcICB equations are derived. Some kink solutions, breathe wave solutions and soliton solutions are got by utilising -expansion method, Riccati equation method, Tanh-expansion method and Painlevé analysis method. By altering different coefficient functions, many different types of exact solutions can be derived. The effect of the coefficient functions on the solution is analysed. By studying the evolution of the solution with time t, the dynamic behaviors of the solutions are analysed.

Implicit Quiescent Optical Solitons for the Dispersive Concatenation Model with Nonlinear Chromatic Dispersion by Lie Symmetry

The primary purpose of this paper is to investigate and recover implicit quiescent optical solitons in the context of the dispersive concatenation model in nonlinear optics. Specifically, the study focuses on a model that incorporates nonlinear chromatic dispersion and includes Kerr and power-law self-phase modulation effects. The objective is to identify and characterize these soliton solutions within this complex optical system. To achieve this purpose, we employ the Lie symmetry analysis method. Lie symmetry analysis is a powerful mathematical technique commonly used in physics and engineering to identify symmetries and invariance properties of differential equations. In this context, it is used to uncover the underlying symmetries of the nonlinear optical model, which in turn aids in the recovery of the quiescent optical solitons. This method involves mathematical derivations and calculations to determine the solutions. The outcomes of the current paper include the successful recovery of implicit quiescent optical solitons for the dispersive concatenation model with nonlinear chromatic dispersion, Kerr, and power-law self-phase modulation. The study provides mathematical expressions and constraints on the model’s parameters that yield upper and lower bounds for these solutions. Essentially, this paper presents a set of mathematical descriptions for the optical solitons that can exist within the described optical system. The present paper contributes to the field of nonlinear optics by exploring the behavior of optical solitons in a model that combines multiple nonlinear effects. This extends our understanding of complex optical systems.

Anomalous Nernst Effect Induced Terahertz Emission in a Single Ferromagnetic Film.

Despite its important role in understanding ultrafast spin dynamics and revealing novel spin/orbit effects, the mechanism of the terahertz (THz) emission from a single ferromagnetic nanofilm upon a femtosecond laser pump still remains elusive. Recent experiments have shown exotic symmetry, which is not expected from the routinely adopted mechanism of ultrafast demagnetization. Here, by developing a bidirectional pump-THz emission spectroscopy and associated symmetry analysis method, we set a benchmark for the experimental distinction of the THz emission induced by various mechanisms. Our results unambiguously unveil a new mechanism─anomalous Nernst effect (ANE) induced THz emission due to the ultrafast temperature gradient created by a femtosecond laser. Quantitative analysis shows that the THz emission exhibits interesting thickness dependence where different mechanisms dominate at different thickness ranges. Our work not only clarifies the origin of the ferromagnetic-based THz emission but also offers a fertile platform for investigating the ultrafast optomagnetism and THz spintronics.

A symmetry analysis methodology for general energy conversion systems

Symmetry is a useful concept that has guided many scientific developments in fields such as structural engineering, data, and materials science. Here we apply a symmetry analysis method to explore the relationship between symmetry, output work and efficiency in macroscopic energy conversion systems. Brayton cycle is used as an example. A specific potential-displacement-energy (PDE) diagram was established for system symmetry analysis. Results prove that the symmetry of thermodynamic cycles could predict the output work and the efficiency. Stronger rotational symmetry generates more work while reflection symmetry leads to higher efficiency at constant specific heat capacity (cp). The condition for varied cp to keep intermediate maximum-work temperature constant is greatly broaden. A more symmetrical cycle with higher efficiency and larger output work is designed based on the symmetry analysis results. The results could also be applied to other thermodynamic cycles, such as Carnot cycle, which provides insights to design more efficient energy conversion cycles.