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Journal articles on the topic 'Lie symmetry method'

Author: Grafiati

Published: 6 September 2023

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1

Ji, Lina, and Rui Wang. "Conditional Lie-Bäcklund Symmetries and Differential Constraints of Radially Symmetric Nonlinear Convection-Diffusion Equations with Source." Entropy 22, no. 8 (August 8, 2020): 873. http://dx.doi.org/10.3390/e22080873.

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A conditional Lie-Bäcklund symmetry method and differential constraint method are developed to study the radially symmetric nonlinear convection-diffusion equations with source. The equations and the admitted conditional Lie-Bäcklund symmetries (differential constraints) are identified. As a consequence, symmetry reductions to two-dimensional dynamical systems of the resulting equations are derived due to the compatibility of the original equation and the additional differential constraint corresponding to the invariant surface equation of the admitted conditional Lie-Bäcklund symmetry.

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2

YU, JUN, and HANWEI HU. "FINITE SYMMETRY GROUP AND COHERENT SOLITON SOLUTIONS FOR THE BROER–KAUP–KUPERSHMIDT SYSTEM." International Journal of Bifurcation and Chaos 23, no. 09 (September 2013): 1350156. http://dx.doi.org/10.1142/s0218127413501563.

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A modified CK direct method is generalized to find finite symmetry groups of nonlinear mathematical physics systems. For the (2 + 1)-dimensional Broer–Kaup–Kupershmidt (BKK) system, both the Lie point symmetry and the non-Lie symmetry groups are obtained by this method. While using the traditional Lie approach, one can only find the Lie symmetry groups. Furthermore, abundant localized structures of the BKK equation are also obtained from the non-Lie symmetry group.

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3

Wang, Zhenli, Liangji Sun, Rui Hua, Lihua Zhang, and Haifeng Wang. "Lie Symmetry Analysis, Particular Solutions and Conservation Laws of Benjiamin Ono Equation." Symmetry 14, no. 7 (June 25, 2022): 1315. http://dx.doi.org/10.3390/sym14071315.

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In this paper, by applying the Lie group method and the direct symmetry method, Lie algebras of the Benjiamin Ono equation are obtained, and we find that results of the two methods are same. Based on the Lie algebra, Lie symmetry groups, relationships between new solutions and old solutions, two kinds of ODEs as symmetry reductions are obtained. Making use of the power series method, the exact power series solution of the Benjiamin Ono equation has been derived. We also give the conservation laws of Benjiamin Ono equation by means of Ibragimovs new conservation Theorem.

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4

Chen, Yong, and Xiaorui Hu. "Lie Symmetry Group of the Nonisospectral Kadomtsev-Petviashvili Equation." Zeitschrift für Naturforschung A 64, no. 1-2 (February 1, 2009): 8–14. http://dx.doi.org/10.1515/zna-2009-1-202.

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The classical symmetry method and the modified Clarkson and Kruskal (C-K) method are used to obtain the Lie symmetry group of a nonisospectral Kadomtsev-Petviashvili (KP) equation. It is shown that the Lie symmetry group obtained via the traditional Lie approach is only a special case of the symmetry groups obtained by the modified C-K method. The discrete group analysis is given to show the relations between the discrete group and parameters in the ansatz. Furthermore, the expressions of the exact finite transformation of the Lie groups via the modified C-K method are much simpler than those obtained via the standard approach.

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5

Bilige, Sudao, and Yanqing Han. "Symmetry reduction and numerical solution of a nonlinear boundary value problem in fluid mechanics." International Journal of Numerical Methods for Heat & Fluid Flow 28, no. 3 (March 5, 2018): 518–31. http://dx.doi.org/10.1108/hff-08-2016-0304.

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Purpose The purpose of this paper is to study the applications of Lie symmetry method on the boundary value problem (BVP) for nonlinear partial differential equations (PDEs) in fluid mechanics. Design/methodology/approach The authors solved a BVP for nonlinear PDEs in fluid mechanics based on the effective combination of the symmetry, homotopy perturbation and Runge–Kutta methods. Findings First, the multi-parameter symmetry of the given BVP for nonlinear PDEs is determined based on differential characteristic set algorithm. Second, BVP for nonlinear PDEs is reduced to an initial value problem of the original differential equation by using the symmetry method. Finally, the approximate and numerical solutions of the initial value problem of the original differential equations are obtained using the homotopy perturbation and Runge–Kutta methods, respectively. By comparing the numerical solutions with the approximate solutions, the study verified that the approximate solutions converge to the numerical solutions. Originality/value The application of the Lie symmetry method in the BVP for nonlinear PDEs in fluid mechanics is an excellent and new topic for further research. In this paper, the authors solved BVP for nonlinear PDEs by using the Lie symmetry method. The study considered that the boundary conditions are the arbitrary functions Bi(x)(i = 1,2,3,4), which are determined according to the invariance of the boundary conditions under a multi-parameter Lie group of transformations. It is different from others’ research. In addition, this investigation will also effectively popularize the range of application and advance the efficiency of the Lie symmetry method.

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6

Zhang, Bo, and Hengchun Hu. "Similarity Reduction and Exact Solutions of a Boussinesq-like Equation." Zeitschrift für Naturforschung A 73, no. 4 (March 28, 2018): 357–62. http://dx.doi.org/10.1515/zna-2017-0442.

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AbstractThe similarity reduction and similarity solutions of a Boussinesq-like equation are obtained by means of Clarkson and Kruskal (CK) direct method. By using Lie symmetry method, we also obtain the similarity reduction and group invariant solutions of the model. Further, we compare the results obtained by the CK direct method and Lie symmetry method, and we demonstrate the connection of the two methods.

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7

Tian, Yi, and Kang-Le Wang. "Polynomial characteristic method an easy approach to lie symmetry." Thermal Science 24, no. 4 (2020): 2629–35. http://dx.doi.org/10.2298/tsci2004629t.

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Along the approach to Lie symmetry, it is always needed to solve an over-determined system, which is difficult and complex if not impossible. Here we suggest a new polynomial characteristic method combined with Lie algorithm to complete symmetry classification for a class of perturbed equations. A differential polynomial characteristic set algorithm is proposed to decompose the determining equations into a series of equations easy to be solved.

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8

Zhang, Lihua, Gangwei Wang, Qianqian Zhao, and Lingshu Wang. "Lie Symmetries and Conservation Laws of Fokas–Lenells Equation and Two Coupled Fokas–Lenells Equations by the Symmetry/Adjoint Symmetry Pair Method." Symmetry 14, no. 2 (January 26, 2022): 238. http://dx.doi.org/10.3390/sym14020238.

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The Fokas–Lenells equation and its multi-component coupled forms have attracted the attention of many mathematical physicists. The Fokas–Lenells equation and two coupled Fokas–Lenells equations are investigated from the perspective of Lie symmetries and conservation laws. The three systems have been turned into real multi-component coupled systems by appropriate transformations. By procedures of symmetry analysis, Lie symmetries of the three real systems are obtained. Explicit conservation laws are constructed using the symmetry/adjoint symmetry pair method, which depends on Lie symmetries and adjoint symmetries. The relationships between the multiplier and the adjoint symmetry are investigated.

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9

Yang, Huizhang, Wei Liu, and Yunmei Zhao. "Lie Symmetry Analysis, Traveling Wave Solutions, and Conservation Laws to the (3 + 1)-Dimensional Generalized B-Type Kadomtsev-Petviashvili Equation." Complexity 2020 (October 24, 2020): 1–8. http://dx.doi.org/10.1155/2020/3465860.

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In this paper, the (3 + 1)-dimensional generalized B-type Kadomtsev-Petviashvili(BKP) equation is studied applying Lie symmetry analysis. We apply the Lie symmetry method to the (3 + 1)-dimensional generalized BKP equation and derive its symmetry reductions. Based on these symmetry reductions, some exact traveling wave solutions are obtained by using the tanh method and Kudryashov method. Finally, the conservation laws to the (3 + 1)-dimensional generalized BKP equation are presented by invoking the multiplier method.

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10

Jadaun, Vishakha, and Sachin Kumar. "Symmetry analysis and invariant solutions of (3 + 1)-dimensional Kadomtsev–Petviashvili equation." International Journal of Geometric Methods in Modern Physics 15, no. 08 (June 22, 2018): 1850125. http://dx.doi.org/10.1142/s0219887818501256.

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Based on Lie symmetry analysis, we study nonlinear waves in fluid mechanics with strong spatial dispersion. The similarity reductions and exact solutions are obtained based on the optimal system and power series method. We obtain the infinitesimal generators, commutator table of Lie algebra, symmetry group and similarity reductions for the [Formula: see text]-dimensional Kadomtsev–Petviashvili equation. For different Lie algebra, Lie symmetry method reduces Kadomtsev–Petviashvili equation into various ordinary differential equations (ODEs). Some of the solutions of [Formula: see text]-dimensional Kadomtsev–Petviashvili equation are of the forms — traveling waves, Weierstrass’s elliptic and Zeta functions and exponential functions.

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11

Tian, Yi. "Symmetry reduction a promising method for heat conduction equations." Thermal Science 23, no. 4 (2019): 2219–27. http://dx.doi.org/10.2298/tsci1904219t.

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Though there are many approximate methods, e. g., the variational iteration method and the homotopy perturbation, for non-linear heat conduction equations, exact solutions are needed in optimizing the heat problems. Here we show that the Lie symmetry and the similarity reduction provide a powerful mathematical tool to searching for the needed exact solutions. Lie algorithm is used to obtain the symmetry of the heat conduction equations and wave equations, then the studied equations are reduced by the similarity reduction method.

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12

Nadjafikhah, Mehdi, and Seyed-Reza Hejazi. "SYMMETRY ANALYSIS OF TELEGRAPH EQUATION." Asian-European Journal of Mathematics 04, no. 01 (March 2011): 117–26. http://dx.doi.org/10.1142/s1793557111000101.

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Lie symmetry group method is applied to study the telegraph equation. The symmetry group and one-parameter group associated to the symmetries with the structure of the Lie algebra symmetries are determined. The reduced version of equation and its one-dimensional optimal system are given.

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13

Wang, Gangwei, Bo Shen, Mengyue He, Fei Guan, and Lihua Zhang. "Symmetry Analysis and PT-Symmetric Extension of the Fifth-Order Korteweg-de Vries-Like Equation." Fractal and Fractional 6, no. 9 (August 26, 2022): 468. http://dx.doi.org/10.3390/fractalfract6090468.

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In the present paper, PT-symmetric extension of the fifth-order Korteweg-de Vries-like equation are investigated. Several special equations with PT symmetry are obtained by choosing different values, for which their symmetries are obtained simultaneously. In particular, for the particular equation, its conservation laws are obtained, including conservation of momentum and conservation of energy. Reciprocal Ba¨cklund transformations of conservation laws of momentum and energy are presented for the first time. The important thing is that for the special case of ϵ=3, the corresponding time fractional case are studied by Lie group method. And what is interesting is that the symmetry of the time fractional equation is obtained, and based on the symmetry, this equation is reduced to a fractional ordinary differential equation. Finally, for the general case, the symmetry of this equation is obtained, and based on the symmetry, the reduced equation is presented. Through the results obtained in this paper, it can be found that the Lie group method is a very effective method, which can be used to deal with many models in natural phenomena.

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14

Zhang, Wenbin, Jiangbo Zhou, and Sunil Kumar. "Symmetry Reduction, Exact Solutions, and Conservation Laws of the ZK-BBM Equation." ISRN Mathematical Physics 2012 (August 15, 2012): 1–9. http://dx.doi.org/10.5402/2012/837241.

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Employing the classical Lie method, we obtain the symmetries of the ZK-BBM equation. Applying the given Lie symmetry, we obtain the similarity reduction, group invariant solution, and new exact solutions. We also obtain the conservation laws of ZK-BBM equation of the corresponding Lie symmetry.

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15

Lashkarian, Elham, Elaheh Saberi, and S. Reza Hejazi. "Symmetry reductions and exact solutions for a class of nonlinear PDEs." Asian-European Journal of Mathematics 09, no. 03 (August 2, 2016): 1650061. http://dx.doi.org/10.1142/s1793557116500613.

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This paper uses Lie symmetry group method to study a special kind of PDE. By using the Lie symmetry analysis, all of the geometric vector fields of the equation are obtained; the symmetry reductions are also presented. Some new nonlinear wave solutions, involving differentiable arbitrary functions are obtained.

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16

Nadjafikhah, Mehdi, and Mehdi Jafari. "Some General New Einstein Walker Manifolds." Advances in Mathematical Physics 2013 (2013): 1–8. http://dx.doi.org/10.1155/2013/591852.

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Lie symmetry group method is applied to find the Lie point symmetry group of a system of partial differential equations that determines general form of four-dimensional Einstein Walker manifold. Also we will construct the optimal system of one-dimensional Lie subalgebras and investigate some of its group invariant solutions.

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17

Munir, Mobeen, Muhammad Athar, Sakhi Sarwar, and Wasfi Shatanawi. "Lie symmetries of Generalized Equal Width wave equations." AIMS Mathematics 6, no. 11 (2021): 12148–65. http://dx.doi.org/10.3934/math.2021705.

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<abstract><p>Lie symmetry analysis of differential equations proves to be a powerful tool to solve or atleast to reduce the order and non-linearity of the equation. The present article focuses on the solution of Generalized Equal Width wave (GEW) equation using Lie group theory. Over the years, different solution methods have been tried for GEW but Lie symmetry analysis has not been done yet. At first, we obtain the infinitesimal generators, commutation table and adjoint table of Generalized Equal Width wave (GEW) equation. After this, we find the one dimensional optimal system. Then we reduce GEW equation into non-linear ordinary differential equation (ODE) by using the Lie symmetry method. This transformed equation can take us to the solution of GEW equation by different methods. After this, we get the travelling wave solution of GEW equation by using the Sine-cosine method. We also give graphs of some solutions of this equation.</p></abstract>

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18

Chepngetich, Winny. "The lie symmetry analysis of third order Korteweg-de Vries equation." Journal of Physical and Applied Sciences (JPAS) 1, no. 1 (November 1, 2022): 38–43. http://dx.doi.org/10.51317/jpas.v1i1.299.

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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.

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Vinita and S. Saha Ray. "Symmetry analysis with similarity reduction, new exact solitary wave solutions and conservation laws of (3 + 1)-dimensional extended quantum Zakharov–Kuznetsov equation in quantum physics." Modern Physics Letters B 35, no. 09 (January 30, 2021): 2150163. http://dx.doi.org/10.1142/s0217984921501633.

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A recently defined (3+1)-dimensional extended quantum Zakharov–Kuznetsov (QZK) equation is examined here by using the Lie symmetry approach. The Lie symmetry analysis has been used to obtain the varieties in invariant solutions of the extended Zakharov–Kuznetsov equation. Due to existence of arbitrary functions and constants, these solutions provide a rich physical structure. In this paper, the Lie point symmetries, geometric vector field, commutative table, symmetry groups of Lie algebra have been derived by using the Lie symmetry approach. The simplest equation method has been presented for obtaining the exact solution of some reduced transform equations. Finally, by invoking the new conservation theorem developed by Nail H. Ibragimov, the conservation laws of QZK equation have been derived.

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20

Nadjafikhah, Mehdi, and Mostafa Hesamiarshad. "Analysis of the Symmetries and Conservation Laws of the Nonlinear Jaulent-Miodek Equation." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/476025.

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Lie symmetry method is performed for the nonlinear Jaulent-Miodek equation. We will find the symmetry group and optimal systems of Lie subalgebras. The Lie invariants associated with the symmetry generators as well as the corresponding similarity reduced equations are also pointed out. And conservation laws of the J-M equation are presented with two steps: firstly, finding multipliers for computation of conservation laws and, secondly, symbolic computation of conservation laws will be applied.

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21

Wang, Jianping, Huijing Ba, Yaru Liu, Longqi He, and Lina Ji. "Second-Order Conditional Lie-Bäcklund Symmetry and Differential Constraint of Radially Symmetric Diffusion System." Advances in Mathematical Physics 2021 (January 15, 2021): 1–17. http://dx.doi.org/10.1155/2021/8891750.

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The classifications and reductions of radially symmetric diffusion system are studied due to the conditional Lie-Bäcklund symmetry method. We obtain the invariant condition, which is the so-called determining system and under which the radially symmetric diffusion system admits second-order conditional Lie-Bäcklund symmetries. The governing systems and the admitted second-order conditional Lie-Bäcklund symmetries are identified by solving the nonlinear determining system. Exact solutions of the resulting systems are constructed due to the compatibility of the original system and the admitted differential constraint corresponding to the invariant surface condition. For most of the cases, they are reduced to solving four-dimensional dynamical systems.

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Nadjafikhah, Mehdi, and Vahid Shirvani-Sh. "Lie Symmetry Analysis of Kudryashov-Sinelshchikov Equation." Mathematical Problems in Engineering 2011 (2011): 1–9. http://dx.doi.org/10.1155/2011/457697.

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The Lie symmetry method is performed for the fifth-order nonlinear evolution Kudryashov-Sinelshchikov equation. We will find ones and two-dimensional optimal systems of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions is investigated.

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23

Perera, Disanayakage Hashan Sanjaya, and Dilruk Gallage. "Solution Methods for Nonlinear Ordinary Differential Equations Using Lie Symmetry Groups." Advanced Journal of Graduate Research 13, no. 1 (February 28, 2023): 37–61. http://dx.doi.org/10.21467/ajgr.13.1.37-61.

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For formulating mathematical models, engineering problems and physical problems, Nonlinear ordinary differential equations(NODEs) are used widely. Nevertheless, explicit solutions can be obtained for very few NODEs, due to lack of techniques to obtain explicit solutions. Therefore methods to obtain approximate solution for NODEs are used widely. Although symmetry groups of ordinary differential equations (ODEs) can be used to obtain exact solutions however, these techniques are not widely used. The purpose of this paper is to present applications of Lie symmetry groups to obtain exact solutions of NODEs . In this paper we connect different methods,theorems and definitions of Lie symmetry groups from different references and we solve first order and second order NODEs. In this analysis three different methods are used to obtain exact solutions of NODEs. Using applications of these symmetry methods, drawbacks and advantages of these different symmetry methods are discussed and some examples have been illustrated graphically. Focus is first placed on discussing about the notion of symmetry groups of the ODEs. Focus is then changed to apply them to find general solutions for NODEs under three different methods. First we find suitable change of variables that convert given first order NODE into variable separable form using these symmetry groups. As another method to find general solutions for first order NODEs, we find particular type of solution curves called invariant solution curves under Lie symmetry groups and we use these invariant solution curves to obtain the general solutions. We find general solutions for the second order NODEs by reducing their order to first order using Lie symmetry groups.

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Mehdi Nadjafikhah and Omid Chekini. "Invariant solutions of Barlett and Whitaker’s equations." Malaya Journal of Matematik 2, no. 02 (April 1, 2014): 103–7. http://dx.doi.org/10.26637/mjm202/002.

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Lie symmetry group method is applied to study the Barlett and Whitaker’s equations. The symmetry group and its optimal system are given,and group invariant solutions associated to the symmetries are obtained. Finally the structure of the Lie algebra symmetries is determined.

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25

Ray, S. Saha. "Lie symmetries, exact solutions and conservation laws of the Oskolkov–Benjamin–Bona–Mahony–Burgers equation." Modern Physics Letters B 34, no. 01 (December 18, 2019): 2050012. http://dx.doi.org/10.1142/s0217984920500128.

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In this paper, the Oskolkov–Benjamin–Bona–Mahony–Burgers (OBBMB) equation has been investigated by Lie symmetry analysis. Lie group analysis method is implemented to derive the vector fields and symmetry reductions. The OBBMB equation has been reduced into nonlinear ordinary differential equation (ODE) by exploiting symmetry reduction method. Using the similarity reduction equation, the exact solutions are obtained by extended [Formula: see text]-method. Finally, the new conservation theorem proposed by Ibragimov has been utilized to construct the conservation laws of the aforesaid equation.

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Chatibi, Youness, El Hassan El Kinani, and Abdelaziz Ouhadan. "Lie symmetry analysis and conservation laws for the time fractional Black–Scholes equation." International Journal of Geometric Methods in Modern Physics 17, no. 01 (December 30, 2019): 2050010. http://dx.doi.org/10.1142/s0219887820500103.

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In this paper, the Lie symmetry algebra admitted by the time fractional Black–Scholes equation is obtained by using the Lie group method. The constructed symmetry generators are investigated to construct a family of exact solutions and conservation laws for the studied equation. At the same time, the family of solutions is extended by using the invariant subspace method.

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Chauhan, Astha, and Rajan Arora. "Time fractional Kupershmidt equation: symmetry analysis and explicit series solution with convergence analysis." Communications in Mathematics 27, no. 2 (December 1, 2019): 171–85. http://dx.doi.org/10.2478/cm-2019-0013.

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AbstractIn this work, the fractional Lie symmetry method is applied for symmetry analysis of time fractional Kupershmidt equation. Using the Lie symmetry method, the symmetry generators for time fractional Kupershmidt equation are obtained with Riemann-Liouville fractional derivative. With the help of symmetry generators, the fractional partial differential equation is reduced into the fractional ordinary differential equation using Erdélyi-Kober fractional differential operator. The conservation laws are determined for the time fractional Kupershmidt equation with the help of new conservation theorem and fractional Noether operators. The explicit analytic solutions of fractional Kupershmidt equation are obtained using the power series method. Also, the convergence of the power series solutions is discussed by using the implicit function theorem.

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CARIÑENA, JOSÉ F., JOANA M. NUNES DA COSTA, and PATRÍCIA SANTOS. "REDUCTION OF LIE ALGEBROID STRUCTURES." International Journal of Geometric Methods in Modern Physics 02, no. 05 (October 2005): 965–91. http://dx.doi.org/10.1142/s0219887805000909.

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Based on the ideas of Marsden–Ratiu, a reduction method for Lie algebroids is developed in such a way that the canonical projection onto the reduced Lie algebroid is a homomorphism of Lie algebroids. A relation between Poisson reduction and Lie algebroid reduction is established. Reduction of Lie algebroids with symmetry is also studied using this method.

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Ramaswamy, Rajagopalan, E. S. El-Shazly, M. S. Abdel Latif, Amr Elsonbaty, and A. H. Abdel Kader. "Exact Optical Solitons for Generalized Kudryashov’s Equation by Lie Symmetry Method." Journal of Mathematics 2023 (June 19, 2023): 1–12. http://dx.doi.org/10.1155/2023/2685547.

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In this article, we use Lie point symmetry analysis to extract some new optical soliton solutions for the generalized Kudryashov’s equation (GKE) with an arbitrary power nonlinearity. Using a traveling wave transformation, the GKE is transformed into a nonlinear second order ordinary differential equation (ODE). Using Lie point symmetry analysis, the nonlinear second-order ODE is reduced to a first-order ODE. This first-order ODE is solved in two cases to retrieve some new bright, dark, and kink soliton solutions of the GKE. These soliton solutions for the GKE are obtained here for the first time.

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Bodner, Mark, Goce Chadzitaskos, Jiří Patera, and Agnieszka Tereszkiewitz. "THE SHMUSHKEVICH METHOD FOR HIGHER SYMMETRY GROUPS OF INTERACTING PARTICLES." Acta Polytechnica 53, no. 5 (October 31, 2013): 395–98. http://dx.doi.org/10.14311/ap.2013.53.0395.

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About 60 years ago, I. Shmushkevich presented a simple ingenious method for computing the relative probabilities of channels involving the same interacting multiplets of particles, without the need to compute the Clebsch-Gordan coefficients. The basic idea of Shmushkevich is “isotopic non-polarization” of the states before the interaction and after it. Hence his underlying Lie group was <em>SU</em>(2). We extend this idea to any simple Lie group. This paper determines the relative probabilities of various channels of scattering and decay processes following from the invariance of the interactions with respect to a compact simple a Lie group. Aiming at the probabilities rather than at the Clebsch-Gordan coefficients makes the task easier, and simultaneous consideration of all possible channels for given multiplets involved in the process, makes the task possible. The probability of states with multiplicities greater than 1 is averaged over. Examples with symmetry groups <em>O</em>(5), <em>F</em>(4), and <em>E</em>(8) are shown.

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Mhlanga, Isaiah Elvis, and Chaudry Masood Khalique. "Exact Solutions of the Symmetric Regularized Long Wave Equation and the Klein-Gordon-Zakharov Equations." Abstract and Applied Analysis 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/679016.

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We study two nonlinear partial differential equations, namely, the symmetric regularized long wave equation and the Klein-Gordon-Zakharov equations. The Lie symmetry approach along with the simplest equation and exp-function methods are used to obtain solutions of the symmetric regularized long wave equation, while the travelling wave hypothesis approach along with the simplest equation method is utilized to obtain new exact solutions of the Klein-Gordon-Zakharov equations.

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BAI, CHENGMING. "LEFT-SYMMETRIC BIALGEBRAS AND AN ANALOGUE OF THE CLASSICAL YANG–BAXTER EQUATION." Communications in Contemporary Mathematics 10, no. 02 (April 2008): 221–60. http://dx.doi.org/10.1142/s0219199708002752.

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We introduce a notion of left-symmetric bialgebra which is an analogue of the notion of Lie bialgebra. We prove that a left-symmetric bialgebra is equivalent to a symplectic Lie algebra with a decomposition into a direct sum of the underlying vector spaces of two Lagrangian subalgebras. The latter is called a parakähler Lie algebra or a phase space of a Lie algebra in mathematical physics. We introduce and study coboundary left-symmetric bialgebras and our study leads to what we call "S-equation", which is an analogue of the classical Yang–Baxter equation. In a certain sense, the S-equation associated to a left-symmetric algebra reveals the left-symmetry of the products. We show that a symmetric solution of the S-equation gives a parakähler Lie algebra. We also show that such a solution corresponds to the symmetric part of a certain operator called "[Formula: see text]-operator", whereas a skew-symmetric solution of the classical Yang–Baxter equation corresponds to the skew-symmetric part of an [Formula: see text]-operator. Thus a method to construct symmetric solutions of the S-equation (hence parakähler Lie algebras) from [Formula: see text]-operators is provided. Moreover, by comparing left-symmetric bialgebras and Lie bialgebras, we observe that there is a clear analogue between them and, in particular, parakähler Lie groups correspond to Poisson–Lie groups in this sense.

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Liu, Hanze. "Symmetry Analysis and Exact Solutions to the Space-Dependent Coefficient PDEs in Finance." Abstract and Applied Analysis 2013 (2013): 1–10. http://dx.doi.org/10.1155/2013/156965.

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The variable-coefficients partial differential equations (vc-PDEs) in finance are investigated by Lie symmetry analysis and the generalized power series method. All of the geometric vector fields of the equations are obtained; the symmetry reductions and exact solutions to the equations are presented, including the exponentiated solutions and the similarity solutions. Furthermore, the exact analytic solutions are provided by the transformation technique and generalized power series method, which has shown that the combination of Lie symmetry analysis and the generalized power series method is a feasible approach to dealing with exact solutions to the variable-coefficients PDEs.

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Bibi, Khudija, and Khalil Ahmad. "New Exact Solutions of Date Jimbo Kashiwara Miwa Equation Using Lie Symmetry Groups." Mathematical Problems in Engineering 2021 (April 10, 2021): 1–8. http://dx.doi.org/10.1155/2021/5533983.

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In this article, new exact solutions of 2 + 1 -dimensional Date Jimbo Kashiwara Miwa (DJKM) equation are constructed by applying the Lie symmetry method. By considering similarity variables obtained through Lie symmetry generators, considered 2 + 1 -dimensional DJKM equation is transformed into a linear partial differential equation with reduction of one independent variable. Afterwards by using Lie symmetry generators of this linear PDE, different invariant solutions involving exponential and logarithmic functions are explored which lead to the new exact solutions of the DJKM equation. Graphical representations of the obtained solutions are also presented to show the significance of the current work.

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35

Mehdi Nadjafikhah and Omid Chekini. "Classical and partial symmetries of the Benney equation." Malaya Journal of Matematik 3, no. 01 (January 1, 2015): 86–92. http://dx.doi.org/10.26637/mjm301/008.

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Lie symmetry group method is applied to study Benney equation. The symmetry group and its optimal system are given,and group invariant solutions associated to the symmetries are obtained. Also the structure of the Lie algebra symmetries is determined. Mainly, we have compared one of the resolved analitical solutions of the Benney equation with one of it’s numerical solutions which is obtained via homotopy perturbation method in [4].

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36

KOCABIYIK, Mehmet, and Mevlude YAKIT ONGUN. "ANALYSIS OF GRANULE CELL GENERATION SYSTEM BY LIE SYMMETRY METHOD." Cumhuriyet Science Journal 41, no. 1 (March 22, 2020): 56–68. http://dx.doi.org/10.17776/csj.461819.

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37

Ray, S. Saha. "Painlevé analysis, group invariant analysis, similarity reduction, exact solutions, and conservation laws of Mikhailov–Novikov–Wang equation." International Journal of Geometric Methods in Modern Physics 18, no. 06 (March 26, 2021): 2150094. http://dx.doi.org/10.1142/s0219887821500948.

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In this paper, for the study of integrability, symmetry analysis, group invariant solutions and conservation laws, the Mikhailov–Novikov–Wang equation is considered. Firstly, Painlevé analysis is being employed to study the integrability properties for the considered equation so as to check the possibility that this equation passes the Painlevé test. Secondly, Lie group analysis is studied for finding the symmetries by using Lie classical group analysis method and to obtain its symmetry group, infinitesimal generator, Lie algebra commutation table, and similarity reductions. The vector fields and the symmetry reduction of this equation are calculated with the aid of Lie symmetry analysis. From the similarity reduction equation, some explicit exact solutions are derived. Finally, using the new conservation theorem proposed by Ibragimov [N. H. Ibragimov, A new conservation theorem, J. Math. Anal. Appl. 333 (2007) 311–328], the conservation laws of the aforesaid equation have been constructed.

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38

Iwasa, Masatomo. "Derivation of Asymptotic Dynamical Systems with Partial Lie Symmetry Groups." Journal of Applied Mathematics 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/601657.

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Lie group analysis has been applied to singular perturbation problems in both ordinary differential and difference equations and has allowed us to find the reduced dynamics describing the asymptotic behavior of the dynamical system. The present study provides an extended method that is also applicable to partial differential equations. The main characteristic of the extended method is the restriction of the manifold by some constraint equations on which we search for a Lie symmetry group. This extension makes it possible to find a partial Lie symmetry group, which leads to a reduced dynamics describing the asymptotic behavior.

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Huang, Yehui, Wei Li, Guo Wang, and Xuelin Yong. "Lie symmetry analysis of the deformed KdV equation." Modern Physics Letters B 31, no. 30 (October 26, 2017): 1750275. http://dx.doi.org/10.1142/s021798491750275x.

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The deformed KdV equation is a generalization of the classical equation that can describe the motion of the interaction between different solitary waves. In this paper, the Lie symmetry analysis is performed on the deformed KdV equation. The similarity reductions and exact solutions are obtained based on the optimal system method. The exact analytic solutions are considered by using the power series method. The conservation laws for the deformed KdV equation are presented. Finally, the analytic solutions are given and their dynamics are studied.

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40

Nadjafikhah, Mehdi, and Fatemeh Ahangari. "SYMMETRY ANALYSIS AND SIMILARITY REDUCTION OF THE KORTEWEG–DE VRIES–ZAKHAROV–KUZNETSOV EQUATION." Asian-European Journal of Mathematics 05, no. 01 (March 2012): 1250006. http://dx.doi.org/10.1142/s1793557112500064.

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In this paper, the problem of determining the largest possible set of symmetries for an important nonlinear dynamical system in mathematical physics, the Korteweg–de Vries–Zakharov–Kuznetsov (KdV–ZK) equation, is studied. By applying the basic Lie symmetry method for the KdV–ZK equation, the classical Lie point symmetry operators are obtained. Also, the structure of the Lie algebra of symmetries is discussed and the optimal system of subalgebras of the equation is constructed. The Lie invariants as well as similarity reduced equations corresponding to infinitesimal symmetries are obtained. The non-classical symmetries of the KdV–ZK equation are also investigated.

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Kumar, Sachin, Ilyas Khan, Setu Rani, and Behzad Ghanbari. "Lie Symmetry Analysis and Dynamics of Exact Solutions of the (2+1)-Dimensional Nonlinear Sharma–Tasso–Olver Equation." Mathematical Problems in Engineering 2021 (May 10, 2021): 1–12. http://dx.doi.org/10.1155/2021/9961764.

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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.

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Thiam, Lamine, and Xi-zhong Liu. "Residual Symmetry Reduction and Consistent Riccati Expansion to a Nonlinear Evolution Equation." Complexity 2019 (November 13, 2019): 1–9. http://dx.doi.org/10.1155/2019/6503564.

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The residual symmetry of a (1 + 1)-dimensional nonlinear evolution equation (NLEE) ut+uxxx−6u2ux+6λux=0 is obtained through Painlevé expansion. By introducing a new dependent variable, the residual symmetry is localized into Lie point symmetry in an enlarged system, and the related symmetry reduction solutions are obtained using the standard Lie symmetry method. Furthermore, the (1 + 1)-dimensional NLEE equation is proved to be integrable in the sense of having a consistent Riccati expansion (CRE), and some new Bäcklund transformations (BTs) are given. In addition, some explicitly expressed solutions including interaction solutions between soliton and cnoidal waves are derived from these BTs.

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Chaolu, Temuer, and Sudao Bilige. "Applications of Differential Form Wu’s Method to Determine Symmetries of (Partial) Differential Equations." Symmetry 10, no. 9 (September 3, 2018): 378. http://dx.doi.org/10.3390/sym10090378.

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In this paper, we present an application of Wu’s method (differential characteristic set (dchar-set) algorithm) for computing the symmetry of (partial) differential equations (PDEs) that provides a direct and systematic procedure to obtain the classical and nonclassical symmetry of the differential equations. The fundamental theory and subalgorithms used in the proposed algorithm consist of a different version of the Lie criterion for the classical symmetry of PDEs and the zero decomposition algorithm of a differential polynomial (d-pol) system (DPS). The version of the Lie criterion yields determining equations (DTEs) of symmetries of differential equations, even those including a nonsolvable equation. The decomposition algorithm is used to solve the DTEs by decomposing the zero set of the DPS associated with the DTEs into a union of a series of zero sets of dchar-sets of the system, which leads to simplification of the computations.

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Hejazi, Seyed Reza. "Lie group analysis, Hamiltonian equations and conservation laws of Born–Infeld equation." Asian-European Journal of Mathematics 07, no. 03 (September 2014): 1450040. http://dx.doi.org/10.1142/s1793557114500405.

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Lie symmetry group method is applied to study the Born–Infeld equation. The symmetry group is given, and similarity solutions associated to the symmetries are obtained. Finally the Hamiltonian equations including Hamiltonian symmetry group and conservation laws are determined.

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45

Wang, Gangwei, Yixing Liu, Shuxin Han, Hua Wang, and Xing Su. "Generalized Symmetries and mCK Method Analysis of the (2+1)-Dimensional Coupled Burgers Equations." Symmetry 11, no. 12 (December 3, 2019): 1473. http://dx.doi.org/10.3390/sym11121473.

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In this paper, generalized symmetries and mCK method are employed to analyze the (2+1)-dimensional coupled Burgers equations. Firstly, based on the generalized symmetries method, the corresponding symmetries of the (2+1)-dimensional coupled Burgers equations are derived. And then, using the mCK method, symmetry transformation group theorem is presented. From symmetry transformation group theorem, a great many of new solutions can be derived. Lastly, Lie algebra for given symmetry group are considered.

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46

Liu, Xi-Zhong. "Conservation Laws Related to the Kac-Moody-Virasoro Structure of the Potential Nizhnik-Novikov-Veselov Equation." Zeitschrift für Naturforschung A 66, no. 5 (May 1, 2011): 297–303. http://dx.doi.org/10.1515/zna-2011-0505.

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We derive the symmetry group by the standard Lie symmetry method and prove it constitutes to the Kac-Moody-Virasoro algebra. Then we construct the conservation laws corresponding to the Kac- Moody-Virasoro symmetry algebra up to second-order group invariants

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47

Wang, Rui, and Lina Ji. "Conditional Lie–Bäcklund Symmetries and Functionally Generalized Separation of Variables to Quasi-Linear Diffusion Equations with Source." Symmetry 12, no. 5 (May 21, 2020): 844. http://dx.doi.org/10.3390/sym12050844.

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The conditional Lie–Bäcklund symmetry method is applied to investigate the functionally generalized separation of variables for quasi-linear diffusion equations with a source. The equations and the admitted conditional Lie–Bäcklund symmetries related to invariant subspaces are identified. The exact solutions possessing the form of the functionally generalized separation of variables are constructed for the resulting equations due to the corresponding symmetry reductions.

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48

Naderifard, Azadeh, Elham Dastranj, and S. Reza Hejazi. "Exact solutions for time-fractional Fokker–Planck–Kolmogorov equation of Geometric Brownian motion via Lie point symmetries." International Journal of Financial Engineering 05, no. 02 (June 2018): 1850009. http://dx.doi.org/10.1142/s2424786318500093.

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In this paper, the transition joint probability density function of the solution of geometric Brownian motion (GBM) equation is obtained via Lie group theory of differential equations (DEs). Lie symmetry analysis is applied to find new solutions for time-fractional Fokker–Planck–Kolmogorov equation of GBM. This analysis classifies the forms of the solutions for the equation by the similarity variables arising from the symmetry operators. Finally, an analytic method called invariant subspace method is applied in order to find another exact solution.

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49

Sinkala, W., and M. Chaisi. "Using Lie Symmetry Analysis to Solve a Problem That Models Mass Transfer from a Horizontal Flat Plate." Mathematical Problems in Engineering 2012 (2012): 1–9. http://dx.doi.org/10.1155/2012/232698.

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We use Lie symmetry analysis to solve a boundary value problem that arises in chemical engineering, namely, mass transfer during the contact of a solid slab with an overhead flowing fluid. This problem was earlier tackled using Adomian decomposition method (Fatoorehchi and Abolghasemi 2011), leading to the Adomian series form of solution. It turns out that the application of Lie group analysis yields an elegant form of the solution. After introducing the governing mathematical model and some preliminaries of Lie symmetry analysis, we compute the Lie point symmetries admitted by the governing equation and use these to construct the desired solution as an invariant solution.

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Jyoti, Divya, and Sachin Kumar. "General form of axially symmetric stationary metric: exact solutions and conservation laws in vacuum fields." Classical and Quantum Gravity 40, no. 14 (June 23, 2023): 145011. http://dx.doi.org/10.1088/1361-6382/acdb3e.

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Abstract The invariant non-static solutions of Einstein’s vacuum field equations, corresponding to the most general form of axially symmetric stationary line element that represents a non conformally flat semi-Riemannian spacetime in cylindrical coordinates, are investigated. Lie symmetry method is used for symmetry reduction as well as for obtaining exact solutions in terms of arbitrary functions. The conservation laws are obtained for vacuum equations in axially symmetric gravitational fields. The solutions of Lewis metric and Chandrasekhar metric, are derived from the obtained solutions. By considering the possibilities of arbitrary functions, the graphical behaviour of the solutions is also shown.

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