Lie Symmetry Analysis Method Research Articles

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

In this article, the symmetry, optimal system, exact solutions and conservation laws of (2+1)-dimensional dissipative long-wave system with variable-coefficients (vcDLW) are investigated. These are important for mathematicians and physicists to solve problems about shallow water waves. The infinitesimal generators of independent variables are solved using Lie symmetry analysis method firstly. Based on the linear combination of vector fields, the representative elements of one-dimensional subalgebras optimal system are introduced by Olver's method. Some new exact solutions of the (2+1)-dimensional vcDLW are obtained by using Riccati equation method and tanh expansion method, such as kink solutions and soliton solutions. It is shown that this system has different solutions for different reductions. These solutions are important for describing a number of physical phenomena. The difference in this article is that the coefficients of some terms are functions. The unique feature of this article is that we can obtain many different types of equations when the coefficient functions change. This makes the (2+1)-dimensional vcDLW describe more general physical phenomena. In addition, the dynamical behaviour of the exact solutions of the (2+1)-dimensional vcDLW is analysed. And the evolution of several types of solutions is studied depending on the change in time. This process can be described better with the help of figures. Finally, The conservation laws of the (2+1)-dimensional vcDLW are obtained using the new conservation theorem with two modificatory rules.

Perturbed Fokas–Lenells equation: Lie symmetry analysis, complexitons and baseband modulation instability

In this work, Lie symmetry analysis method is utilized to find the complex soliton solutions of the perturbed Fokas–Lenells equation. In this direction, first of all, we obtained the infinitesimals of the Fokas–Lenells equation with the help of the Lie symmetry method. After that, we reduced the Fokas–Lenells equation into the highly nonlinear system of an ordinary differential equation. Consequently, with the application of suitable back transformation, complex solitons are formulated for the Fokas–Lenells equation in trigonometric, hyperbolic and exponential functions. Finally, in this work, conditions of stability and instability are discussed with the aid of baseband modulation instability.

Lie symmetry analysis, exact solutions and power series solutions of the logarithmic Monge–Ampère flow evolution equation

In this paper, the logarithmic Monge–Ampère flow evolution equation is studied by the classical Lie symmetry analysis method. First, the optimal system was obtained by the adjoint representation table. Second, by performing symmetric transformation on the optimal system, the corresponding ordinary differential equations are obtained, and the Jacobian elliptic function solution, the periodic solution and the power series solution are constructed. Finally, the dynamical behaviors of the solutions are described by choosing arbitrary parameters.

Lie symmetry analysis of time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays

In this paper, Lie symmetry analysis method is applied to time fractional Burgers equation, Korteweg-de Vries equation and generalized reaction-diffusion equation with delays, respectively. The Lie symmetries for fractional partial differential equations with delays (DFPDEs) are obtained, and the group classifications of the equations are established. The obtained group generators are used to reduce the DFPDEs to fractional ordinary differential equations with delays (DFODEs). Some exact solutions constructed for the DFODEs generate group-invariant solutions of the discussed DFPDEs.

Hyperbolic (2+1)-dimensional Schrödinger equation: Similarity analysis, Optimal system and complexitons for the one-parameter group of rotations

The current study is dedicated to find the complex soliton solutions of the hyperbolic (2+1)-dimensional nonlinear Schrödinger equation. In this direction we takes the help of Lie Symmetry analysis method. First of all we obtained the invariant condition which play important role in the mechanism of Lie symmetry method. After that we obtained the symmetries of the hyperbolic Schrödinger equation. These symmetries are further used to develop the appropriate vector fields. Consequently, with the application of these vector field optimal system of subalgebras are obtained. Further, under the each subalgebras similarity solutions are obtained in form of trigonometric functions. Most important the similarity variables which are obtained with the help of Lie symmetry method are in form of hyperbolic function. This clearly indicate that the phenomenon modeled by the hyperbolic Schrödinger equation invariant under these similarity solutions and similarity variable. At the end of the study complex soliton are obtained for Schrödinger equation which are in form of trigonometric and hyperbolic functions.

The Lie symmetry analysis and dynamics of exact solutions for variable coefficients (2+1)-dimensional coupled burgers equation

In this article, the (2+1)-dimensional variable coefficients coupled Burgers equation (vcCBE) is investigated for the first time with the help of Lie symmetry analysis method. This equation is an important nonlinear physical model. The optimal system of the (2+1)-dimensional vcCBE is analyzed by Olver’s method. Then, the (2+1)-dimensional vcCBE is reduced on the basis of the optimal system to multiple sets of (1+1)-dimensional equations. Various types of soliton solutions are obtained by solving the reduced equations. The -expansion method, tanh-coth method, and Riccati equation method are used respectively. Finally, the obtained solutions are analyzed dynamically and their kinetic behavior is investigated.

Analytical and Numerical Solutions for a Kind of High-Dimensional Fractional Order Equation

In this paper, a (4+1)-dimensional nonlinear integrable Fokas equation is studied. It is rarely studied because the order of the highest-order derivative term of this equation is higher than the common generalized (4+1)-dimensional Fokas equation. Firstly, the (4+1)-dimensional time-fractional Fokas equation with the Riemann–Liouville fractional derivative is derived by the semi-inverse method and variational method. Further, the symmetry of the time-fractional equation is obtained by the fractional Lie symmetry analysis method. Based on the symmetry, the conservation laws of the time fractional equation are constructed by the new conservation theorem. Then, the G′G-expansion method is used here to solve the equation and obtain the exact traveling wave solutions. Finally, the spectral method in the spatial direction and the Gru¨nwald–Letnikov method in the time direction are considered to obtain the numerical solutions of the time-fractional equation. The numerical solutions are compared with the exact solutions, and the error results confirm the effectiveness of the proposed numerical method.

Analytic solutions for the MHD flow and heat transfer in a thin liquid film over an unsteady stretching surface with Lie symmetry and homotopy analysis method

In thin films on unsteady stretching surfaces, flow and heat transfer under different physical conditions are studied to refine countless industrial products. In this work we consider such a flow with an external magnetic field and apply Lie symmetry method on it to analyze the flow and heat transfer. We construct Lie point symmetry generators for this magnetohydrodynamic flow and heat transfer in a thin liquid film over an unsteady stretching surface. We obtain a six dimensional Lie point symmetry algebra and by employing a few of these symmetry generators we deduce invariants and similarity transformations for the flow model. We employ these transformations on the flow equations that are partial differential equations to reduce their independent variables from three to one. This procedure reveals corresponding systems of ordinary differential equations. The similarity transformations obtained here are different from those that are derived earlier for the considered flow problem. Hence by employing these transformations on the flow model leads to new classes of ordinary differential equations. It implies existence of additional classes of Lie similarity transformations to study the MHD flow and heat transfer in a thin film on an unsteady stretching sheet. We obtain analytic solutions for these reduced systems of equations with the homotopy analysis method to present effects of the Prandtl number, magnetic and unsteadiness parameters on the velocity and temperature profiles. These are new solutions for the considered flow model as Lie symmetries have not been employed earlier on it. All results are presented with the help of graphs and tables.

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.

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.

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.

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.

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.