Lie 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.

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.

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.

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.

Lie symmetry analysis, conservation laws and exact solutions for variable-coefficients (2+1)-dimensional dissipative long-wave system

The Lie symmetry analysis, optimal system, exact solutions and conservation laws of the (2+1)-dimensional variable-coefficients dissipative long-wave (vcDLW) system are introduced in this paper. Lie symmetry analysis method is used to obtain the vector field firstly. And the Olver method is used to obtain the optimal system of one-dimensional subalgebras of the equations. Based on the optimal system, the equations are similarly reduced. Some representative equations are selected from the equations obtained after similarly reduction, and then the expansion method is applied to these equations, so that some new exact solutions of the (2+1)-dimensional vcDLW equations can be obtained. Finally, we study the conservation laws of the system.

Lie symmetry analysis and exact solutions of the (3+1)-dimensional generalized Shallow Water-like equation

In this article, (3+1)-dimensional generalized Shallow Water-like (gSWl) equation is discussed. The infinitesimal generators of the equation are derived by using the Lie symmetry analysis method. The optimal system is obtained based on the adjoint table of the generators of the equation. Exact solutions of the equation are constructed by applying symmetry reduction, Exp−ϕ(ξ) expansion method, Exp-function expansion method, Riccati equation method, and G′/G expansion method. For analyzing the dynamical behavior of the solutions, we derive the physical structures of dark soliton, kink wave, and periodic solutions via numerical simulations.

Lie symmetry analysis, particular solutions and conservation laws for the dissipative (2 + 1)- dimensional AKNS equation

<abstract><p>The dissipative (2 + 1)-dimensional AKNS equation is considered in this paper. First, the Lie symmetry analysis method is applied to the dissipative (2 + 1)-dimensional AKNS and six point symmetries are obtained. Symmetry reductions are performed by utilizing these obtained point symmetries and four differential equations are derived, including a fourth-order ordinary differential equation and three partial differential equations. Thereafter, the direct integration approach and the $ (G'/G^{2})- $expansion method are employed to solve the ordinary differential respectively. As a result, a periodic solution in terms of the Weierstrass elliptic function is obtained via the the direct integration approach, while six kinds of including the hyperbolic function types and the hyperbolic function types are derived via the $ (G'/G^{2})- $expansion method. The corresponding graphical representation of the obtained solutions are presented by choosing suitable parametric values. Finally, the multiplier technique and the classical Noether's theorem are employed to derive conserved vectors for the dissipative (2 + 1)-dimensional AKNS respectively. Consequently, eight local conservation laws for the dissipative (2 + 1)-dimensional AKNS equation are presented by utilizing the multiplier technique and five local conservation laws are derived by invoking Noether's theorem.</p></abstract>

LIE SYMMETRY, EXACT SOLUTIONS AND CONSERVATION LAWS OF SOME FRACTIONAL PARTIAL DIFFERENTIAL EQUATIONS

In this paper, Lie symmetry analysis method is applied to spacetime fractional reaction-diffusion equations and diffusion-convection Boussinesq equations. The Lie symmetries for the governing equations are obtained and used to get group generators for reducing the space-time fractional partial differential equations(FPDEs) with Riemann-Liouville fractional derivative to space-time fractional ordinary differential equations(FODEs) with ErdélyiKober fractional derivative. Then the Laplace transformation and the power series methods are applied to derive explicit solutions for the reduced equations. Moreover, the conservation theorems and the generalization of the Noether operators are developed to acquire the conservation laws for the equations. Some figures for the obtained explicit solutions are also presented.

The lie symmetry analysis of third order Korteweg-de Vries equation

This study sought to analyse the Lie symmetry of third order Korteweg-de Vries equation. Solving nonlinear partial differential equations is of great importance in the world of dynamics. Korteweg-de Vries equations are partial differential equations arising from the theory of long waves, modelling of shallow water waves, fluid mechanics, plasma fluids and many other nonlinear physical systems, and their effects are relevant in real life. In this study, Lie symmetry analysis is demonstrated in finding the symmetry solutions of the third-order KdV equation of the form. The study systematically showed the formula to find the specific solution attained by developing prolongations, infinitesimal transformations and generators, adjoint symmetries, variation symmetries, invariant transformation and integrating factors to obtain all the lie groups presented by the equation. In conclusion, infinitesimal generators, group transformations and symmetry solutions of third-order KdV equation are acquired using a method of Lie symmetry analysis. This was achieved by generating infinitesimal generators which act on the KdV equation to form infinitesimal transformations. It can be seen from the solutions of this paper that the Lie symmetry analysis method is an effective and best mathematical technique for studying linear and nonlinear PDEs and ODEs.

Exact Solutions for the KMM System in (2+1)-Dimensions and Its Fractional Form with Beta-Derivative

Fractional calculus is useful in studying physical phenomena with memory effects. In this paper, the fractional KMM (FKMM) system with beta-derivative in (2+1)-dimensions was studied for the first time. It can model short-wave propagation in saturated ferromagnetic materials, which has many applications in the high-tech world, especially in microwave devices. Using the properties of beta-derivatives and a proper transformation, the FKMM system was initially changed into the KMM system, which is a (2+1)-dimensional generalization of the sine-Gordon equation. Lie symmetry analysis and the optimal system for the KMM system were investigated. Using the optimal system, we obtained eight (1+1)-dimensional reduction equations. Based on the reduction equations, new soliton solutions, oblique analytical solutions, rational function solutions and power series solutions for the KMM system and FKMM system were derived. Using the properties of beta-derivatives and another transformation, the FKMM system was changed into a system of ordinary differential equations. Based on the obtained system of ordinary differential equations, Jacobi elliptic function solutions and solitary wave solutions for the FKMM system were derived. For the KMM system, the results about Lie symmetries, optimal system, reduction equations, and oblique traveling wave solutions are new, since Lie symmetry analysis method has not been applied to such a system before. For the FKMM system, all of the exact solutions are new. The main novelty of the paper lies in the fact that beta-derivatives have been used to change fractional differential equations into classical differential equations. The technique can also be extended to other fractional differential equations.