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Journal articles on the topic 'Lie Symmetry group of SDE'

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Published: 1 February 2025

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1

Muniz, Michelle, Matthias Ehrhardt, and Michael Günther. "Approximating Correlation Matrices Using Stochastic Lie Group Methods." Mathematics 9, no. 1 (January 4, 2021): 94. http://dx.doi.org/10.3390/math9010094.

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Specifying time-dependent correlation matrices is a problem that occurs in several important areas of finance and risk management. The goal of this work is to tackle this problem by applying techniques of geometric integration in financial mathematics, i.e., to combine two fields of numerical mathematics that have not been studied yet jointly. Based on isospectral flows we create valid time-dependent correlation matrices, so called correlation flows, by solving a stochastic differential equation (SDE) that evolves in the special orthogonal group. Since the geometric structure of the special orthogonal group needs to be preserved we use stochastic Lie group integrators to solve this SDE. An application example is presented to illustrate this novel methodology.

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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

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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4

Kötz, H. "A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. I. Optimal Systems of Solvable Lie Subalgebras." Zeitschrift für Naturforschung A 47, no. 11 (November 1, 1992): 1161–74. http://dx.doi.org/10.1515/zna-1992-1114.

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Abstract Lie group analysis is a powerful tool for obtaining exact similarity solutions of nonlinear (integro-) differential equations. In order to calculate the group-invariant solutions one first has to find the full Lie point symmetry group admitted by the given (integro-)differential equations and to determine all the subgroups of this Lie group. An effective, systematic means to classify the similarity solutions afterwards is an "optimal system", i.e. a list of group-invariant solutions from which every other such solution can be derived. The problem to find optimal systems of similarity solutions leads to that to "construct" the optimal systems of subalgebras for the Lie algebra of the known Lie point symmetry group. Our aim is to demonstrate a practicable technique for determining these optimal subalgebraic systems using the invariants relative to the group of the inner automorphisms of the Lie algebra in case of a finite-dimensional Lie point symmetry group. Here, we restrict our attention to optimal subsystems of solvable Lie subalgebras. This technique is applied to the nine-dimensional real Lie point symmetry group admitted by the two-dimensional non-stationary ideal magnetohydrodynamic equations

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5

SHIRKOV, DMITRIJ V. "RENORMALIZATION GROUP SYMMETRY AND SOPHUS LIE GROUP ANALYSIS." International Journal of Modern Physics C 06, no. 04 (August 1995): 503–12. http://dx.doi.org/10.1142/s0129183195000356.

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We start with a short discussion of the content of a term Renormalisation Group in modern use. By treating the underlying solution property as a reparametrisation symmetry, we relate it with the self-similarity symmetry well-known in mathematical physics and explain the notion of Functional Self-similarity. Then we formulate a program of constructing a regular approach for discovering RG-type symmetries in different problems of mathematical physics. This approach based upon S. Lie group analysis allows one to analyse a wide class of boundary problems for different type of equations. Several examples are mentioned.

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6

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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7

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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8

Almutiben, Nouf, Ryad Ghanam, G. Thompson, and Edward L. Boone. "Symmetry analysis of the canonical connection on Lie groups: six-dimensional case with abelian nilradical and one-dimensional center." AIMS Mathematics 9, no. 6 (2024): 14504–24. http://dx.doi.org/10.3934/math.2024705.

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<abstract><p>In this article, the investigation into the Lie symmetry algebra of the geodesic equations of the canonical connection on a Lie group was continued. The key ideas of Lie group, Lie algebra, linear connection, and symmetry were quickly reviewed. The focus was on those Lie groups whose Lie algebra was six-dimensional solvable and indecomposable and for which the nilradical was abelian and had a one-dimensional center. Based on the list of Lie algebras compiled by Turkowski, there were eight algebras to consider that were denoted by $ A_{6, 20} $–$ A_{6, 27} $. For each Lie algebra, a comprehensive symmetry analysis of the system of geodesic equations was carried out. For each symmetry Lie algebra, the nilradical and a complement to the nilradical inside the radical, as well as a semi-simple factor, were identified.</p></abstract>

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9

Johnpillai, Andrew G., Abdul H. Kara, and Anjan Biswas. "Exact Group Invariant Solutions and Conservation Laws of the Complex Modified Korteweg–de Vries Equation." Zeitschrift für Naturforschung A 68, no. 8-9 (September 1, 2013): 510–14. http://dx.doi.org/10.5560/zna.2013-0027.

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We study the scalar complex modified Korteweg-de Vries (cmKdV) equation by analyzing a system of partial differential equations (PDEs) from the Lie symmetry point of view. These systems of PDEs are obtained by decomposing the underlying cmKdV equation into real and imaginary components. We derive the Lie point symmetry generators of the system of PDEs and classify them to get the optimal system of one-dimensional subalgebras of the Lie symmetry algebra of the system of PDEs. These subalgebras are then used to construct a number of symmetry reductions and exact group invariant solutions to the system of PDEs. Finally, using the Lie symmetry approach, a couple of new conservation laws are constructed. Subsequently, respective conserved quantities from their respective conserved densities are computed.

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10

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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11

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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Hussain, Akhtar, Muhammad Usman, Fiazuddin Zaman, and Ahmed M. M. Zidan. "Lie group analysis and its invariants for the class of multidimensional nonlinear wave equations." Nonlinear Analysis: Modelling and Control 29, no. 6 (December 1, 2024): 1167–85. https://doi.org/10.15388/namc.2024.29.37853.

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We systematically classify Lie symmetries of a class of (2 + 1)-dimensional nonlinear wave equations. Our methodology proposes a symmetry classification for Lie generators applicable to four distinct cases inherent within this equation. For each identified category, we comprehensively analyze symmetry reduction and delineate the invariant solutions. Furthermore, we extend our Lie symmetry analysis to encompass reduced 1 + 1 partial differential equations (PDEs). Through our investigations, we establish local conservation laws corresponding to each conserved vector, employing the formal Lagrangian approach. Significantly, this classification constitutes a novel contribution to the scientific discourse, as it remains absent from extant literature to date.

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13

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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14

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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15

Reggiani, Silvio. "The index of symmetry of three-dimensional Lie groups with a left-invariant metric." Advances in Geometry 18, no. 4 (October 25, 2018): 395–404. http://dx.doi.org/10.1515/advgeom-2017-0061.

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Abstract We determine the index of symmetry of 3-dimensional unimodular Lie groups with a left-invariant metric. In particular, we prove that every 3-dimensional unimodular Lie group admits a left-invariant metric with positive index of symmetry. We also study the geometry of the quotients by the so-called foliation of symmetry, and we explain in what cases the group fibers over a 2-dimensional space of constant curvature.

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16

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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17

Owino, Joseph Owuor. "GROUP ANALYSIS OF A NONLINEAR HEAT-LIKE EQUATION." INTERNATIONAL JOURNAL OF MATHEMATICS AND COMPUTER RESEARCH 11, no. 01 (January 13, 2023): 3113–31. http://dx.doi.org/10.47191/ijmcr/v11i1.03.

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We study a nonlinear heat like equation from a lie symmetry stand point. Heat equation have been employed to study ow of current, information and propagation of heat. The Lie group approach is used on the system to obtain symmetry reductions and the reduced systems studied for exact solutions. Solitary waves have been constructed by use of a linear span of time and space translation symmetries. We also compute conservation laws using multiplier approach and by a conservation theorem due to Ibragimov.

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18

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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19

Kötz, H. "A Technique to Classify the Similarity Solutions of Nonlinear Partial (Integro-)Differential Equations. II. Full Optimal Subalgebraic Systems." Zeitschrift für Naturforschung A 48, no. 4 (April 1, 1993): 535–50. http://dx.doi.org/10.1515/zna-1993-0401.

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"Optimal systems" of similarity solutions of a given system of nonlinear partial (integro-)differential equations which admits a finite-dimensional Lie point symmetry group Gare an effective systematic means to classify these group-invariant solutions since every other such solution can be derived from the members of the optimal systems. The classification problem for the similarity solutions leads to that of "constructing" optimal subalgebraic systems for the Lie algebra Gof the known symmetry group G. The methods for determining optimal systems of s-dimensional Lie subalgebras up to the dimension r of Gvary in case of 3 ≤ s ≤ r, depending on the solvability of G. If the r-dimensional Lie algebra Gof the infinitesimal symmetries is nonsolvable, in addition to the optimal subsystems of solvable subalgebras of Gone has to determine the optimal subsystems of semisimple subalgebras of Gin order to construct the full optimal systems of s-dimensional subalgebras of Gwith 3 ≤ s ≤ r. The techniques presented for this classification process are applied to the nonsolvable Lie algebra Gof the eight-dimensional Lie point symmetry group Gadmitted by the three-dimensional Vlasov-Maxwell equations for a multi-species plasma in the non-relativistic case.

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20

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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21

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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22

Almutiben, Nouf, Edward L. Boone, Ryad Ghanam, and G. Thompson. "Classification of the symmetry Lie algebras for six-dimensional co-dimension two Abelian nilradical Lie algebras." AIMS Mathematics 9, no. 1 (2023): 1969–96. http://dx.doi.org/10.3934/math.2024098.

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<abstract><p>In this paper, we consider the symmetry algebra of the geodesic equations of the canonical connection on a Lie group. We mainly consider the solvable indecomposable six-dimensional Lie algebras with co-dimension two abelian nilradical that have an abelian complement. In dimension six, there are nineteen such algebras, namely, $ A_{6, 1} $–$ A_{6, 19} $ in Turkowski's list. For each algebra, we give the geodesic equations, a basis for the symmetry Lie algebra in terms of vector fields, and finally we identify the symmetry Lie algebra from standard lists.</p></abstract>

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23

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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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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Aksenov, Alexander V., and Anatoly A. Kozyrev. "Group Classification of the Unsteady Axisymmetric Boundary Layer Equation." Mathematics 12, no. 7 (March 26, 2024): 988. http://dx.doi.org/10.3390/math12070988.

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Unsteady equations of flat and axisymmetric boundary layers are considered. For the unsteady axisymmetric boundary layer equation, the problem of group classification is solved. It is shown that the kernel of symmetry operators can be extended by no more than four-dimensional Lie algebra. The kernel of symmetry operators of the unsteady flat boundary layer equation is found and it is shown that it can be extended by no more than a five-dimensional Lie algebra. The non-existence of the unsteady analogue of the Stepanov–Mangler transformation is proved.

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26

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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Zhao, Weidong, Muhammad Mobeen Munir, Hajra Bashir, Daud Ahmad, and Muhammad Athar. "Lie symmetry analysis for generalized short pulse equation." Open Physics 20, no. 1 (January 1, 2022): 1185–93. http://dx.doi.org/10.1515/phys-2022-0212.

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Abstract Lie symmetry analysis (LSA) is one of the most common, effective, and estimation-free methods to find the symmetries and solutions of the differential equations (DEs) by following an algorithm. This analysis leads to reduce the order of partial differential equations (PDEs). Many physical problems are converted into non-linear DEs and these DEs or system of DEs are then solved with several methods such as similarity methods, Lie Bäcklund transformation, and Lie group of transformations. LSA is suitable for providing the conservation laws corresponding to Lie point symmetries or Lie Bäcklund symmetries. Short pulse equation (SPE) is a non-linear PDE, used in optical fibers, computer graphics, and physical systems and has been generalized in many directions. We will find the symmetries and a class of solutions depending on one-parameter (ε) obtained from Lie symmetry groups. Then we will construct the optimal system for the Lie algebra and invariant solutions (called similarity solutions) from Lie subalgebras of generalized SPE.

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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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BOBYLEV, A. V. "THE BOLTZMANN EQUATION AND THE GROUP TRANSFORMATIONS." Mathematical Models and Methods in Applied Sciences 03, no. 04 (August 1993): 443–76. http://dx.doi.org/10.1142/s0218202593000230.

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This paper is devoted to the investigation of group properties of the nonlinear Boltzmann equation. The complete Lie group of invariant transformations for the spatially inhomogeneous Boltzmann equation is constructed. The generalization to the Lie-Backlund groups is given for the spatially homogeneous case. It is shown that there are only two non-trivial group transformations for the Boltzmann equation in the wide class of Lie and Lie-Backlund transformations. Some consequences of these symmetry properties are discussed. The special role of Galileo group and the analogy between the spatially homogeneous Boltzmann equation and the full equation are also investigated.

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Vinita and Santanu Saha Ray. "Optical soliton group invariant solutions by optimal system of Lie subalgebra with conservation laws of the resonance nonlinear Schrödinger equation." Modern Physics Letters B 34, no. 35 (August 25, 2020): 2050402. http://dx.doi.org/10.1142/s0217984920504023.

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In this article, the resonance nonlinear Schrödinger equation is studied, which elucidates the propagation of one-dimensional long magnetoacoustic waves in a cold plasma, dynamic of solitons and Madelung fluids in various nonlinear systems. The Lie symmetry analysis is used to achieve the invariant solution and similarity reduction of the resonance nonlinear Schrödinger equation. The infinitesimal generators, symmetry groups, commutator table and adjoint table have been obtained by the aid of invariance criterion of Lie symmetry. Also, one-dimensional system of subalgebra is constructed with the help of adjoint representation of a Lie group on its Lie algebra. By one-dimensional optimal subalgebra, the main equations are reduced to ordinary differential equations and their invariant solutions are provided. The general conservation theorem has been used to establish a set of non-local and non-trivial conservation laws.

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Rezaei-Aghdam, A., and M. Sephid. "Jacobi–Lie symmetry and Jacobi–Lie T-dual sigma models on group manifolds." Nuclear Physics B 926 (January 2018): 602–13. http://dx.doi.org/10.1016/j.nuclphysb.2017.12.003.

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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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Han, Y. L., X. X. Wang, M. L. Zhang, and L. Q. Jia. "Lie Symmetry and Approximate Hojman Conserved Quantity of Lagrange Equations for a Weakly Nonholonomic System." Journal of Mechanics 30, no. 1 (August 8, 2013): 21–27. http://dx.doi.org/10.1017/jmech.2013.47.

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ABSTRACTThe Lie symmetry and Hojman conserved quantity of Lagrange equations for a weakly nonholonomic system and its first-degree approximate holonomic system are studied. The differential equations of motion for the system are established. Under the special infinitesimal transformations of group in which the time is invariable, the definition of the Lie symmetry for the weakly nonholonomic system and its first-degree approximate holonomic system are given, and the exact and approximate Hojman conserved quantities deduced directly from the Lie symmetry are obtained. Finally, an example is given to study the exact and approximate Hojman conserved quantity for the system.

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Green, HS. "A Cyclic Symmetry Principle in Physics." Australian Journal of Physics 47, no. 1 (1994): 25. http://dx.doi.org/10.1071/ph940025.

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Many areas of modern physics are illuminated by the application of a symmetry principle, requiring the invariance of the relevant laws of physics under a group of transformations. This paper examines the implications and some of the applications of the principle of cyclic symmetry, especially in the areas of statistical mechanics and quantum mechanics, including quantized field theory. This principle requires invariance under the transformations of a finite group, which may be a Sylow 7r-group, a group of Lie type, or a symmetric group. The utility of the principle of cyclic invariance is demonstrated in finding solutions of the Yang-Baxter equation that include and generalize known solutions. It is shown that the Sylow 7r-groups have other uses, in providing a basis for a type of generalized quantum statistics, and in parametrising a new generalization of Lie groups, with associated algebras that include quantized algebras.

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ĐAPIĆ, N., M. KUNZINGER, and S. PILIPOVIĆ. "SYMMETRY GROUP ANALYSIS OF WEAK SOLUTIONS." Proceedings of the London Mathematical Society 84, no. 3 (April 29, 2002): 686–710. http://dx.doi.org/10.1112/s0024611502013436.

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Methods of Lie group analysis of differential equations are extended to weak solutions of (linear and non-linear) partial differential equations, where the term `weak solution' comprises the following settings: distributional solutions; solutions in generalized function algebras; solutions in the sense of association (corresponding to a number of weak or integral solution concepts in classical analysis). Factorization properties and infinitesimal criteria that allow the treatment of all three settings simultaneously are developed, thereby unifying and extending previous work in this area.2000 Mathematical Subject Classification: 46F30, 22E70, 35Dxx, 35A30.

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Acevedo-Agudelo, Yeisson Alexis, Danilo Andrés García-Hernández, Oscar Mario Londoño-Duque, and Gabriel Ignacio Loaiza-Ossa. "Lie algebra classification for the Chazy equation and further topics related with this algebra." Revista Politécnica 17, no. 34 (November 9, 2021): 101–9. http://dx.doi.org/10.33571/rpolitec.v17n34a7.

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It is known that the classification of the Lie algebras is a classical problem. Due to Levi’s Theorem the question can be reduced to the classification of semi-simple and solvable Lie algebras. This paper is devoted to classify the Lie algebra generated by the Lie symmetry group of the Chazy equation. We also present explicitly the one parame-ter subgroup related to the infinitesimal generators of the Chazy symmetry group. Moreover the classification of the Lie algebra associated to the optimal system is investigated. La clasificación de las álgebras de Lie es un problema clásico. Acorde al teorema de Levi la cuestión puede reducirse a la clasificación de álgebras de Lie semi-simples y solubles. Este artículo está dedicado a clasificar el álgebra de Lie generada por el grupo de simetría de Lie para la ecuación de Chazy. También presentamos explícitamente los subgrupos a un parámetro relacionados con los generadores de las simetrías del grupo de Chazy. Además, la clasificación de la álgebra de Lie asociada al sistema optimo es investigada.

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Johnpillai, I. Kenneth, Scott W. McCue, and James M. Hill. "Lie group symmetry analysis for granular media stress equations." Journal of Mathematical Analysis and Applications 301, no. 1 (January 2005): 135–57. http://dx.doi.org/10.1016/j.jmaa.2004.07.010.

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38

Wockel, Christoph. "Lie group structures on symmetry groups of principal bundles." Journal of Functional Analysis 251, no. 1 (October 2007): 254–88. http://dx.doi.org/10.1016/j.jfa.2007.05.016.

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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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Huang, Kefu, and Houguo Li. "Group Invariant Solutions of the Full Plastic Torsion of Rod with Arbitrary Shaped Cross Sections." Advances in Applied Mathematics and Mechanics 4, no. 03 (June 2012): 382–88. http://dx.doi.org/10.4208/aamm.10-m1201.

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AbstractBased on the theory of Lie group analysis, the full plastic torsion of rod with arbitrary shaped cross sections that consists in the equilibrium equation and the non-linear Saint Venant-Mises yield criterion is studied. Full symmetry group admitted by the equilibrium equation and the yield criterion is a finitely generated Lie group with ten parameters. Several subgroups of the full symmetry group are used to generate invariants and group invariant solutions. Moreover, physical explanations of each group invariant solution are discussed by all appropriate transformations. The methodology and solution techniques used belong to the analytical realm.

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A. Alaidarous, Eman Salem. "Group analysis and variational principle for nonlinear (3+1) schrodinger equation." Material Science Research India 7, no. 1 (June 25, 2010): 115–22. http://dx.doi.org/10.13005/msri/070113.

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The generators of the admitted variational Lie symmetry groups are derived and conservation laws for the conserved currents are obtained via Noether's theorem. Moreover, the consistency of a functional integral are derived for the nonlinear Schrödinger equation. In addition to this analysis functional integral are studied using Lie groups.

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Bocko, Jozef, Iveta Glodová, and Pavol Lengvarský. "Some Differential Equations of Elasticity and their Lie Point Symmetry Generators." Acta Mechanica et Automatica 8, no. 2 (August 10, 2014): 99–102. http://dx.doi.org/10.2478/ama-2014-0018.

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Abstract The formal models of physical systems are typically written in terms of differential equations. A transformation of the variables in a differential equation forms a symmetry group if it leaves the differential equation invariant. Symmetries of differential equations are very important for understanding of their properties. It can be said that the theory of Lie group symmetries of differential equations is general systematic method for finding solutions of differential equations. Despite of this fact, the Lie group theory is relatively unknown in engineering community. The paper is devoted to some important questions concerning this theory and for several equations resulting from the theory of elasticity their Lie group infinitesimal generators are given.

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Bibi, Khudija, and Khalil Ahmad. "Exact Solutions of Newell-Whitehead-Segel Equations Using Symmetry Transformations." Journal of Function Spaces 2021 (January 25, 2021): 1–8. http://dx.doi.org/10.1155/2021/6658081.

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In this article, Lie and discrete symmetry transformation groups of linear and nonlinear Newell-Whitehead-Segel (NWS) equations are obtained. By using these symmetry transformation groups, several group invariant solutions of considered NWS equations have been constructed. Furthermore, some more group invariant solutions are generated by using discrete symmetry transformation group. Graphical representations of some obtained solutions are also presented.

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Folly-Gbetoula, Mensah, Nkosingiphile Mnguni, and A. H. Kara. "A Group Theory Approach towards Some Rational Difference Equations." Journal of Mathematics 2019 (December 1, 2019): 1–9. http://dx.doi.org/10.1155/2019/1505619.

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Qin, Maochang, Fengxiang Mei, and Xuejun Xu. "Nonclassical Potential Symmetries and New Explicit Solutions of the Burgers Equation." Zeitschrift für Naturforschung A 60, no. 1-2 (February 1, 2005): 17–22. http://dx.doi.org/10.1515/zna-2005-1-203.

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Several new nonclassical potential symmetry generators to the Burgers equation are derived. Some explicit solutions, which cannot be derived from the Lie symmetry group of Burgers or its adjoined equation, are obtained by using these nonclassical potential symmetry generators.

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Ray, S. Saha. "Lie symmetry analysis and reduction for exact solution of (2+1)-dimensional Bogoyavlensky–Konopelchenko equation by geometric approach." Modern Physics Letters B 32, no. 11 (April 18, 2018): 1850127. http://dx.doi.org/10.1142/s0217984918501270.

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In this paper, the symmetry analysis and similarity reduction of the (2[Formula: see text]+[Formula: see text]1)-dimensional Bogoyavlensky–Konopelchenko (B–K) equation are investigated by means of the geometric approach of an invariance group, which is equivalent to the classical Lie symmetry method. Using the extended Harrison and Estabrook’s differential forms approach, the infinitesimal generators for (2[Formula: see text]+[Formula: see text]1)-dimensional B–K equation are obtained. Firstly, the vector field associated with the Lie group of transformation is derived. Then the symmetry reduction and the corresponding explicit exact solution of (2[Formula: see text]+[Formula: see text]1)-dimensional B–K equation is obtained.

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Moitsheki, Raseelo J. "Lie Group Analysis of a Flow with Contaminant-Modified Viscosity." Journal of Applied Mathematics 2007 (2007): 1–10. http://dx.doi.org/10.1155/2007/38278.

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A class of coupled system of diffusion equations is considered. Lie group techniques resulted in a rich array of admitted point symmetries for special cases of the source term. We also employ potential symmetry methods for chosen cases of concentration and a zero source term. Some invariant solutions are constructed using both classical Lie point and potential symmetries.

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Gazizov, R. K., A. A. Kasatkin, and S. Yu Lukashchuk. "Symmetry properties of fractional order transport equations." Proceedings of the Mavlyutov Institute of Mechanics 9, no. 1 (2012): 59–64. http://dx.doi.org/10.21662/uim2012.1.010.

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In the paper some features of applying Lie group analysis methods to fractional differential equations are considered. The problem related to point change of variables in the fractional differentiation operator is discussed and some general form of transformation that conserves the form of Riemann-Liouville fractional operator is obtained. The prolongation formula for extending an infinitesimal operator of a group to fractional derivative with respect to arbitrary function is presented. Provided simple example illustrates the necessity of considering both local and non-local symmetries for fractional differential equations in particular cases including the initial conditions. The equivalence transformation forms for some fractional differential equations are discussed and results of group classification of the wave-diffusion equation are presented. Some examples of constructing particular exact solutions of fractional transport equation are given, based on the Lie group methods and the method of invariant subspaces.

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Yu, Demin, Chan Jiang, and Jiejing Ma. "Study on Poisson Algebra and Automorphism of a Special Class of Solvable Lie Algebras." Symmetry 15, no. 5 (May 19, 2023): 1115. http://dx.doi.org/10.3390/sym15051115.

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We define a four-dimensional Lie algebra g in this paper and then prove that this Lie algebra is solvable but not nilpotent. Due to the fact that g is a Lie algebra, ∀x,y∈g,[x,y]=−[y,x], that is, the operation [,] has anti symmetry. Symmetry is a very important law, and antisymmetry is also a very important law. We studied the structure of Poisson algebras on g using the matrix method. We studied the necessary and sufficient conditions for the automorphism of this class of Lie algebras, and give the decomposition of its automorphism group by Aut(g)=G3G1G2G3G4G7G8G5, or Aut(g)=G3G1G2G3G4G7G8G5G6, or Aut(g)=G3G1G2G3G4G7G8G5G3, where Gi is a commutative subgroup of Aut(g). We give some subgroups of g’s automorphism group and systematically studied the properties of these subgroups.

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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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