Journal articles: 'Infinitesimal generator of a Lie group'

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Li, Fu-zhi, Jia-li Yu, Yang-rong Li, and Gan-shan Yang. "Lie Group Solutions of Magnetohydrodynamics Equations and Their Well-Posedness." Abstract and Applied Analysis 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/8183079.

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Based on classical Lie Group method, we construct a class of explicit solutions of two-dimensional ideal incompressible magnetohydrodynamics (MHD) equation by its infinitesimal generator. Via these explicit solutions we study the uniqueness and stability of initial-boundary problem on MHD.

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Tryhuk, V., V. Chrastinová, and O. Dlouhý. "The Lie Group in Infinite Dimension." Abstract and Applied Analysis 2011 (2011): 1–35. http://dx.doi.org/10.1155/2011/919538.

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A Lie group acting on finite-dimensional space is generated by its infinitesimal transformations and conversely, any Lie algebra of vector fields in finite dimension generates a Lie group (the first fundamental theorem). This classical result is adjusted for the infinite-dimensional case. We prove that the (local,C∞smooth) action of a Lie group on infinite-dimensional space (a manifold modelled onℝ∞) may be regarded as a limit of finite-dimensional approximations and the corresponding Lie algebra of vector fields may be characterized by certain finiteness requirements. The result is applied to the theory of generalized (or higher-order) infinitesimal symmetries of differential equations.

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Alfaro, Ricardo, and Jim Schaeferle. "Coefficients of prolongations for symmetries of ODEs." International Journal of Mathematics and Mathematical Sciences 2004, no. 51 (2004): 2741–53. http://dx.doi.org/10.1155/s016117120430904x.

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Sophus Lie developed a systematic way to solve ODEs. He found that transformations which form a continuous group and leave a differential equation invariant can be used to simplify the equation. Lie's method uses the infinitesimal generator of these point transformations. These are symmetries of the equation mapping solutions into solutions. Lie's methods did not find widespread use in part because the calculations for the infinitesimals were quite lengthy, needing to calculate the prolongations of the infinitesimal generator. Nowadays, prolongations are obtained using Maple or Mathematica, and Lie's theory has come back to the attention of researchers. In general, the computation of the coefficients of the (n)-prolongation is done using recursion formulas. Others have given methods that do not require recursion but use Fréchet derivatives. In this paper, we present a combinatorial approach to explicitly write the coefficients of the prolongations. Besides being novel, this approach was found to be useful by the authors for didactical and combinatorial purposes, as we show in the examples.

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Schürmann, Michael, and Michael Skeide. "Infinitesimal Generators on the Quantum Group SUq(2)." Infinite Dimensional Analysis, Quantum Probability and Related Topics 01, no. 04 (October 1998): 573–98. http://dx.doi.org/10.1142/s0219025798000314.

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Quantum Lévy processes on a quantum group are, like classical Lévy processes with values in a Lie group, classified by their infinitesimal generators. We derive a formula for the infinitesimal generators on the quantum group SU q(2) and decompose them in terms of an infinite-dimensional irreducible representation and of characters. Thus we obtain a quantum Lévy–Khintchine formula.

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Cai, J. L., and F. X. Mei. "Conformal Invariance and Conserved Quantity of the Higher-Order Holonomic Systems by Lie Point Transformation." Journal of Mechanics 28, no. 3 (August 9, 2012): 589–96. http://dx.doi.org/10.1017/jmech.2012.67.

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AbstractIn this paper, the conformal invariance and conserved quantities for higher-order holonomic systems are studied. Firstly, by establishing the differential equation of motion for the systems and introducing a one-parameter infinitesimal transformation group together with its infinitesimal generator vector, the determining equation of conformal invariance for the systems are provided, and the conformal factors expression are deduced. Secondly, the relation between conformal invariance and the Lie symmetry by the infinitesimal one-parameter point transformation group for the higher-order holonomic systems are deduced. Thirdly, the conserved quantities of the systems are derived using the structure equation satisfied by the gauge function. Lastly, an example of a higher-order holonomic mechanical system is discussed to illustrate these results.

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Gaur, Manoj, and K. Singh. "Symmetry Classification and Exact Solutions of a Variable Coefficient Space-Time Fractional Potential Burgers’ Equation." International Journal of Differential Equations 2016 (2016): 1–8. http://dx.doi.org/10.1155/2016/4270724.

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We investigate the symmetry properties of a variable coefficient space-time fractional potential Burgers’ equation. Fractional Lie symmetries and corresponding infinitesimal generators are obtained. With the help of the infinitesimal generators, some group invariant solutions are deduced. Further, some exact solutions of fractional potential Burgers’ equation are generated by the invariant subspace method.

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Ndogmo, J. C. "Some Results on Equivalence Groups." Journal of Applied Mathematics 2012 (2012): 1–11. http://dx.doi.org/10.1155/2012/484805.

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The comparison of two common types of equivalence groups of differential equations is discussed, and it is shown that one type can be identified with a subgroup of the other type, and a case where the two groups are isomorphic is exhibited. A result on the determination of the finite transformations of the infinitesimal generator of the larger group, which is useful for the determination of the invariant functions of the differential equation, is also given. In addition, the Levidecomposition of the Lie algebra associated with the larger group is found; the Levi factor of which is shown to be equal, up to a constant factor, to the Lie algebra associated with the smaller group.

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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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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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Tam, Honwah, Yufeng Zhang, and Xiangzhi Zhang. "New Applications of a Kind of Infinitesimal-Operator Lie Algebra." Advances in Mathematical Physics 2016 (2016): 1–12. http://dx.doi.org/10.1155/2016/7639013.

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Applying some reduced Lie algebras of Lie symmetry operators of a Lie transformation group, we obtain an invariant of a second-order differential equation which can be generated by a Euler-Lagrange formulism. A corresponding discrete equation approximating it is given as well. Finally, we make use of the Lie algebras to generate some new integrable systems including (1+1) and (2+1) dimensions.

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Smulders, C. M. P. A. "Heat Kernels on homogeneous spaces." Journal of the Australian Mathematical Society 78, no. 1 (February 2005): 109–47. http://dx.doi.org/10.1017/s1446788700015597.

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AbstractLet a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

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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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Ray, S. Saha. "Optimal subalgebra, conservations laws and symmetry reductions with analytical solutions using Lie symmetry analysis and geometric approach for the (1+1)-dimensional Manakov model." Physica Scripta 98, no. 4 (March 15, 2023): 045214. http://dx.doi.org/10.1088/1402-4896/acb7d2.

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Abstract In this article, the (1+1)-dimensional Manakov model has been examined for finding its exact closed form solitonic solutions with the help of symmetry generators. These symmetry generators are explored using the Lie symmetry analysis, commonly known as the classical Lie group approach and the geometric approach. In a geometric approach, the extended Harrison and Estabrook’s differential forms have been used for obtaining the infinitesimal generators of the Manakov model. As there are infinite possibilities for the linear combination of infinitesimal generators, so by using Olver’s standard approach a one-dimensional optimal system of subalgebra has been established. Additionally, the ‘new conservation theorem’ put forth by Ibragimov has been utilized in order to devise the conservation laws for the (1+1)-dimensional Manakov model. Finally, the exact closed form solutions are obtained with the help of Lie symmetries corresponding to the defined model.

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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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Abdigapparovich, Narmanov Otabek. "Lie Algebra of Infinitesimal Generators of the Symmetry Group of the Heat Equation." Journal of Applied Mathematics and Physics 06, no. 02 (2018): 373–81. http://dx.doi.org/10.4236/jamp.2018.62035.

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Xia, Lili, and Xinsheng Ge. "Lie Symmetry Analysis and Conservation Laws of the Axially Loaded Euler Beam." Mathematics 10, no. 15 (August 3, 2022): 2759. http://dx.doi.org/10.3390/math10152759.

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By applying the Lie symmetry method, group-invariant solutions are constructed for axially loaded Euler beams. The corresponding mathematical models of the beams are formulated. After introducing the infinitesimal transformations, the determining equations of Lie symmetry are proposed via Lie point transformations acting on the original equations. The infinitesimal generators of symmetries of the systems are presented with Maple. The corresponding vector fields are given to span the subalgebra of the systems. Conserved vectors are derived by using two methods, namely, the multipliers method and Noether’s theorem. Noether conserved quantities are obtained using the structure equation, satisfied by the gauge functions. The fluxes of the conservation laws could also be proposed with the multipliers. The relations between them are discussed. Furthermore, the original equations of the systems could be transformed into ODEs and the exact explicit solutions are provided.

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Zhou, Xuan, Wenrui Shan, Zhilei Niu, Pengcheng Xiao, and Ying Wang. "Lie symmetry analysis and some exact solutions for modified Zakharov–Kuznetsov equation." Modern Physics Letters B 32, no. 31 (November 10, 2018): 1850383. http://dx.doi.org/10.1142/s0217984918503839.

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In this study, the Lie symmetry method is used to perform detailed analysis on the modified Zakharov–Kuznetsov equation. We have obtained the infinitesimal generators, commutator table of Lie algebra and symmetry group. In addition to that, optimal system of one-dimensional subalgebras up to conjugacy is derived and used to construct distinct exact solutions. These solutions describe the dynamics of nonlinear waves in isothermal multicomponent magnetized plasmas.

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Laouini, Ghaylen, Amr M. Amin, and Mohamed Moustafa. "Lie Group Method for Solving the Negative-Order Kadomtsev–Petviashvili Equation (nKP)." Symmetry 13, no. 2 (January 29, 2021): 224. http://dx.doi.org/10.3390/sym13020224.

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A comprehensive study of the negative-order Kadomtsev–Petviashvili (nKP) partial differential equation by Lie group method has been presented. Initially the infinitesimal generators and symmetry reduction, which were obtained by applying the Lie group method on the negative-order Kadomtsev–Petviashvili equation, have been used for constructing the reduced equations. In particular, the traveling wave solutions for the negative-order KP equation have been derived from the reduced equations as an invariant solution. Finally, the extended improved (G′/G) method and the extended tanh method are described and applied in constructing new explicit expressions for the traveling wave solutions. Many new and more general exact solutions are obtained.

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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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Kaibe, Bosiu C., and John G. O’Hara. "Symmetry Analysis of an Interest Rate Derivatives PDE Model in Financial Mathematics." Symmetry 11, no. 8 (August 16, 2019): 1056. http://dx.doi.org/10.3390/sym11081056.

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We perform Lie symmetry analysis to a zero-coupon bond pricing equation whose price evolution is described in terms of a partial differential equation (PDE). As a result, using the computer software package SYM, run in conjunction with Mathematica, a new family of Lie symmetry group and generators of the aforementioned pricing equation are derived. We furthermore compute the exact invariant solutions which constitute the pricing models for the bond by making use of the derived infinitesimal generators and the associated similarity reduction equations. Using known solutions, we again compute more solutions via group point transformations.

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Bates, Larry, Richard Cushman, and Jędrzej Śniatycki. "Vector Fields and Differential Forms on the Orbit Space of a Proper Action." Axioms 10, no. 2 (June 10, 2021): 118. http://dx.doi.org/10.3390/axioms10020118.

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In this paper, we study differential forms and vector fields on the orbit space of a proper action of a Lie group on a smooth manifold, defining them as multilinear maps on the generators of infinitesimal diffeomorphisms, respectively. This yields an intrinsic view of vector fields and differential forms on the orbit space.

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Linares, Pablo, Felix Otto, and Markus Tempelmayr. "The structure group for quasi-linear equations via universal enveloping algebras." Communications of the American Mathematical Society 3, no. 1 (January 20, 2023): 1–64. http://dx.doi.org/10.1090/cams/16.

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We replace trees by multi-indices as an index set of the abstract model space to tackle quasi-linear singular stochastic partial differential equations. We show that this approach is consistent with the postulates of regularity structures when it comes to the structure group, which arises from a Hopf algebra and a comodule. Our approach, where the dual of the abstract model space naturally embeds into a formal power series algebra, allows to interpret the structure group as a Lie group arising from a Lie algebra consisting of derivations on this power series algebra. These derivations in turn are the infinitesimal generators of two actions on the space of pairs (non-linearities, functions of space-time mod constants). We also argue that there exist pre-Lie algebra and Hopf algebra morphisms between our structure and the tree-based one in the cases of branched rough paths (Grossman-Larson, Connes-Kreimer) and of the stochastic heat equation.

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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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Jiwari, Ram, Vikas Kumar, Ram Karan, and Ali Saleh Alshomrani. "Haar wavelet quasilinearization approach for MHD Falkner–Skan flow over permeable wall via Lie group method." International Journal of Numerical Methods for Heat & Fluid Flow 27, no. 6 (June 5, 2017): 1332–50. http://dx.doi.org/10.1108/hff-04-2016-0145.

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Purpose This paper aims to deal with two-dimensional magneto-hydrodynamic (MHD) Falkner–Skan boundary layer flow of an incompressible viscous electrically conducting fluid over a permeable wall in the presence of a magnetic field. Design/methodology/approach Using the Lie group approach, the Lie algebra of infinitesimal generators of equivalence transformations is constructed for the equation under consideration. Using these suitable similarity transformations, the governing partial differential equations are reduced to linear and nonlinear ordinary differential equations (ODEs). Further, Haar wavelet approach is applied to the reduced ODE under the subalgebra 4.1 for constructing numerical solutions of the flow problem. Findings A new type of solutions was obtained of the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis. Originality/value To find a solution for the MHD Falkner–Skan boundary layer flow problem using the Haar wavelet quasilinearization approach via Lie symmetric analysis is a new approach for fluid problems.

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Sinkala, Winter, and Molahlehi Charles Kakuli. "On the Method of Differential Invariants for Solving Higher Order Ordinary Differential Equations." Axioms 11, no. 10 (October 14, 2022): 555. http://dx.doi.org/10.3390/axioms11100555.

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There are many routines developed for solving ordinary differential Equations (ODEs) of different types. In the case of an nth-order ODE that admits an r-parameter Lie group (3≤r≤n), there is a powerful method of Lie symmetry analysis by which the ODE is reduced to an (n−r)th-order ODE plus r quadratures provided that the Lie algebra formed by the infinitesimal generators of the group is solvable. It would seem this method is not widely appreciated and/or used as it is not mentioned in many related articles centred around integration of higher order ODEs. In the interest of mainstreaming the method, we describe the method in detail and provide four illustrative examples. We use the case of a third-order ODE that admits a three-dimensional solvable Lie algebra to present the gist of the integration algorithm.

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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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Zhang, Feng, Yuru Hu, and Xiangpeng Xin. "Lie Symmetry Analysis, Exact Solutions, and Conservation Laws of Variable-Coefficients Boiti-Leon-Pempinelli Equation." Advances in Mathematical Physics 2021 (November 29, 2021): 1–14. http://dx.doi.org/10.1155/2021/6227384.

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In this article, we study the generalized ( 2 + 1 )-dimensional variable-coefficients Boiti-Leon-Pempinelli (vcBLP) equation. Using Lie’s invariance infinitesimal criterion, equivalence transformations and differential invariants are derived. Applying differential invariants to construct an explicit transformation that makes vcBLP transform to the constant coefficient form, then transform to the well-known Burgers equation. The infinitesimal generators of vcBLP are obtained using the Lie group method; then, the optimal system of one-dimensional subalgebras is determined. According to the optimal system, the ( 1 + 1 )-dimensional reduced partial differential equations (PDEs) are obtained by similarity reductions. Through G ′ / G -expansion method leads to exact solutions of vcBLP and plots the corresponding 3-dimensional figures. Subsequently, the conservation laws of vcBLP are determined using the multiplier method.

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Ortíz-Álvarez, Hugo Hernán, Francy Nelly Jiménez-García, and Abel Enrique Posso-Agudelo. "Some exact solutions for a unidimensional fokker-planck equation by using lie symmetries." Revista de Matemática: Teoría y Aplicaciones 22, no. 1 (January 1, 2015): 1. http://dx.doi.org/10.15517/rmta.v22i1.17499.

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The Fokker Planck equation appears in the study of diffusion phenomena, stochastics processes and quantum and classical mechanics. A particular case fromthis equation, ut − uxx − xux − u=0, is examined by the Lie group method approach. From the invariant condition it was possible to obtain the infinitesimal generators or vectors associated to this equation, identifying the corresponding symmetry groups. Exact solution were found for each one of this generators and new solution were constructed by using symmetry properties.

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Kumar, Sachin, and Vishakha Jadaun. "Symmetry analysis and some new exact solutions of Born–Infeld equation." International Journal of Geometric Methods in Modern Physics 15, no. 11 (November 2018): 1850183. http://dx.doi.org/10.1142/s0219887818501839.

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This paper propounds the Lie group analysis method for finding exact solutions of Born–Infeld (BI) equation arising in nonlinear electrodynamics. We obtain generators of infinitesimal transformations, commutator table of Lie algebra, the complete geometric vector field, group symmetries and reduction equations. For the set of geometric vector field, we find an optimal system of the vector fields. Each element in this system helps to reduce the main equation into an ordinary differential equation, which provides analytical solution to the BI equation. We perform numerical simulation to obtain an appropriate visual appearance and dynamic behavior of the traced solutions. The nature of the solutions is investigated both analytically and physically through their evolutionary profile by considering appropriate choices of arbitrary constants.

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Majhi, Pratik, Sharda Kumari, and Madan Mohan Panja. "Symmetry Analysis, One-dimensional Optimal System, and Group Invariant Solutions for the Shallow Water Waves of Finite Amplitude in (1+1)-dimensions." Journal of Advances in Mathematics and Computer Science 40, no. 1 (January 22, 2025): 85–106. https://doi.org/10.9734/jamcs/2025/v40i11964.

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The Lie group theoretic approach is employed here to obtain inequivalent group invariant solutions of a system of nonlinear partial differential equations (SNLPDEs) in (1+1)-dimensions appearing in the mathematical analysis of shallow water waves of finite amplitude, tsunami waves in particular. It is found that the system of equations admits a five-parameter Lie group of symmetry transformations with a Lie algebra \(\mathcal{G}\) of infinitesimal generators. An optimal set \(\mathcal{OG}_1\) of eighteen one-dimensional subalgebras of \(\mathcal{G}\) has been obtained. Similarity variables corresponding to each member of \(\mathcal{OG}_1\) have been determined and used to reduce the SNLPDEs to the system of ordinary differential equations. Some constants of the motion and inequivalent group invariant solutions have been provided whenever the system of reduced equations is solvable. The solutions to some of the reduced equations and the constants of motion provided here seem novel. Results obtained here may be used to check the accuracy of approximate solutions (grid formation) in the approximation (numerical) scheme.

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Lukashchuk, Stanislav Yu. "On the Property of Linear Autonomy for Symmetries of Fractional Differential Equations and Systems." Mathematics 10, no. 13 (July 2, 2022): 2319. http://dx.doi.org/10.3390/math10132319.

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The problem of finding Lie point symmetries for a certain class of multi-dimensional nonlinear partial fractional differential equations and their systems is studied. It is assumed that considered equations involve fractional derivatives with respect to only one independent variable, and each equation contains a single fractional derivative. The most significant examples of such equations are time-fractional models of processes with memory of power-law type. Two different types of fractional derivatives, namely Riemann–Liouville and Caputo, are used in this study. It is proved that any Lie point symmetry group admitted by equations or systems belonging to considered class consists of only linearly-autonomous point symmetries. Representations for the coordinates of corresponding infinitesimal group generators, as well as simplified determining equations are given in explicit form. The obtained results significantly facilitate finding Lie point symmetries for multi-dimensional time-fractional differential equations and their systems. Three physical examples illustrate this point.

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Vinita and S. Saha Ray. "On the invariant analysis, symmetry reduction with group-invariant solution and the conservation laws for (2 + 1)-dimensional modified Heisenberg ferromagnetic system." International Journal of Modern Physics B 34, no. 31 (November 6, 2020): 2050305. http://dx.doi.org/10.1142/s0217979220503051.

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In this paper, a [Formula: see text]-dimensional modified Heisenberg ferromagnetic system, which appears in the biological pattern formation and in the motion of magnetization vector of the isotropic ferromagnet, is being investigated with the aim of exploring its similarity solutions. With the aid of Lie symmetry analysis, this system of partial differential equations has been reduced to a new system of ordinary differential equations, which brings an analytical solution of the main system. Infinitesimal generators, commutator table, and the group-invariant solutions have been carried out by using Lie symmetry approach. Moreover, conservation laws of the above mentioned system have been obtained by utilizing the new conservation theorem proposed by Ibragimov. By applying this analysis, the obtained results might be helpful to understand the physical structure of this model and show the authenticity and effectiveness of the proposed method.

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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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Cui, Juya, and Ben Gao. "Symmetry analysis of an acid-mediated cancer invasion model." AIMS Mathematics 7, no. 9 (2022): 16949–61. http://dx.doi.org/10.3934/math.2022930.

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<abstract><p>Under investigation in this paper is a reaction-diffusion system, which describes acid-mediated tumor growth. First, in view of Lie group analysis, infinitesimal generators of the considered system are presented. At the same time, some group invariant solutions are computed using reduced equations. In particular, we construct explicit solutions by applying the power-series method. Furthermore, the convergence of the solutions of the power-series is certificated. Finally, the stability behavior of the model can be understood by analyzing the solutions of different parameters.</p></abstract>

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WILSON, RAJ, and ELIZABETH TANNER. "IRREDUCIBLE UNITARY REPRESENTATIONS OF SUp,q I: THE DISCRETE SERIES." International Journal of Mathematics 12, no. 01 (February 2001): 1–36. http://dx.doi.org/10.1142/s0129167x01000599.

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A class of irreducible unitary representations in the discrete series of SUp,q is explicitly determined in a space of holomorphic functions of three complex matrices. The discrete series, which is the set of all square integrable representations, corresponds to a compact subgroup of SUp,q. The relevant algebraic properties of the group SUp,q are discussed in detail. For a degenerate irreducible unitary representation an explicit construction of the infinitesimal generators of the Lie algebra [Formula: see text] in terms of differential operators is given.

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Bhutani, O. P., and K. Vijayakumar. "On certain new and exact solutions of the Emden-Fowler equation and Emden equation via invariant variational principles and group invariance." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 32, no. 4 (April 1991): 457–68. http://dx.doi.org/10.1017/s0334270000008535.

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AbstractAfter formulating the alternate potential principle for the nonlinear differential equation corresponding to the generalised Emden-Fowler equation, the invariance identities of Rund [14] involving the Lagrangian and the generators of the infinitesimal Lie group are used for writing down the first integrals of the said equation via the Noether theorem. Further, for physical realisable forms of the parameters involved and through repeated application of invariance under the transformation obtained, a number of exact solutions are arrived at both for the Emden-Fowler equation and classical Emden equations. A comparative study with Bluman-Cole and scale-invariant techniques reveals quite a number of remarkable features of the techniques used here.

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Nadjafikhah, Mehdi, and Hind Al-Bderi. "Lie symmetry analysis of a TFD system for transmission of HTVL virus type I to human CD4+ T-cells." Journal of Interdisciplinary Mathematics 27, no. 4 (2024): 975–90. http://dx.doi.org/10.47974/jim-1918.

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The paper examines a numerical show of a human T-cell lymphotropic infection sort I HTLV transmission of CD4+ T-cells. “The model allows for CD4+ T cell subsets of susceptible, latently infected, and actively infected cells and leukemia cells. In fact, this mathematical model in the fractional differential system form describes the interaction between susceptible CD4+ cells, latently HIV-infected cells, actively HIV-infected cells, latently HTLV-infected cells, Tax-expressing HTLV-infected cells, free HIV particles, HIVspecific Cytotoxic T lymphocytes and HTLV-specific” Cytotoxic T lymphocytes. In this paper, we analyze the human HTLV-I virus model with the help of the Lie symmetry method. The Lie group technique is an effective analytic algorithm that derives an exact solution. A Lie point symmetry of a differential system is a group of transformations that maps every solution of the system to another solution of the same system thus this method gives a set of solutions. The Lie group symmetry method has many advantages, including integrating ordinary differential equations, classifying differential equations, linearizing partial differential equations, obtaining the general solution of differential equations, obtaining conservation laws, and many others. The infinitesimal generators of the time fractional human HTLV-I virus system by the Lie method are derived in what follows. Next, the similarity reduction method is used to establish precise solutions for the invariant by using the Lie point symmetries. The intended system’s conservation laws are retrieved using Ibragimov’s approach and Noether’s theorem.

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NIEMI, ANTTI J., and KAUPO PALO. "ON QUANTUM INTEGRABILITY AND THE LEFSCHETZ NUMBER." Modern Physics Letters A 08, no. 24 (August 10, 1993): 2311–21. http://dx.doi.org/10.1142/s0217732393003615.

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Certain phase space path integrals can be evaluated exactly using equivalent cohomology and localization in the canonical loop space. Here we extend this to a general class of models. We consider Hamiltonians which are a priori arbitrary functions of the Cartan subalgebra generators of a Lie group which is defined on the phase space. We evaluate the corresponding path integral and find that it is closely related to the infinitesimal Lefschetz number of a Dirac operator on the phase space. Our results indicate that equivariant characteristic classes could provide a natural geometric framework for understanding quantum integrability.

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Li, Hongjie, Dandan Wang, and Mingliang Zheng. "Disturbance decoupling for biological fermentation systems with single input single output." Molecular & Cellular Biomechanics 21, no. 4 (December 10, 2024): 847. https://doi.org/10.62617/mcb847.

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The biological fermentation process has the characteristics of nonlinearity and multivariable coupling. To improve the performance of decoupling control in the fermentation process, a disturbance decoupling control based on the Lie symmetry method is proposed to obtain the analytical feedback control for a class of biological fermentation systems with single input single output (SISO). Firstly, the state-space equations and the disturbance decoupling model for a class of SISO biological fermentation systems are defined; Secondly, the key technologies and algorithm approaches of Lie symmetry theory for differential equations are introduced, and the conditions and the properties of Lie symmetry for nonlinear control systems under group action are given in detail; Finally, the derived distribution of Lie symmetric infinitesimal generators is used to prove the sufficient conditions for local disturbance decoupling in the system, and the closed-loop state feedback analytical law of the system is constructed. The proposed control method is applied to the disturbance decoupling control of mycelium concentration and substrate concentration in the biological fermentation process. Numerical simulation results show that the proposed control method can effectively improve the system decoupling control performance. Meanwhile, using Lie symmetry, the cascade decoupling standard form and the static state feedback law of biological fermentation systems can be constructed.

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Kumar, S., and D. Kumar. "Lie symmetry analysis and dynamical structures of soliton solutions for the (2 + 1)-dimensional modified CBS equation." International Journal of Modern Physics B 34, no. 25 (September 15, 2020): 2050221. http://dx.doi.org/10.1142/s0217979220502215.

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In this present article, the new [Formula: see text]-dimensional modified Calogero-Bogoyavlenskii-Schiff (mCBS) equation is studied. Using the Lie group of transformation method, all of the vector fields, commutation table, invariant surface condition, Lie symmetry reductions, infinitesimal generators and explicit solutions are constructed. As we all know, an optimal system contains constructively important information about the various types of exact solutions and it also offers clear understandings into the exact solutions and its features. The symmetry reductions of [Formula: see text]-dimensional mCBS equation is derived from an optimal system of one-dimensional subalgebra of the Lie invariance algebra. Then, the mCBS equation can further be reduced into a number of nonlinear ODEs. The generated explicit solutions have different wave structures of solitons and they are analyzed graphically and physically in order to exhibit their dynamical behavior through 3D, 2D-shapes and respective contour plots. All the produced solutions are definitely new and totally different from the earlier study of the Manukure and Zhou (Int. J. Mod. Phys. B 33, (2019)). Some of these solutions are demonstrated by the means of solitary wave profiles like traveling wave, multi-solitons, doubly solitons, parabolic waves and singular soliton. The calculations show that this Lie symmetry method is highly powerful, productive and useful to study analytically other nonlinear evolution equations in acoustics physics, plasma physics, fluid dynamics, mathematical biology, mathematical physics and many other related fields of physical sciences.

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Nath, G., and Arti Devi. "Cylindrical shock wave in a self-gravitating perfect gas with azimuthal magnetic field via Lie group invariance method." International Journal of Geometric Methods in Modern Physics 17, no. 10 (August 18, 2020): 2050148. http://dx.doi.org/10.1142/s0219887820501480.

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In this paper, we have studied the propagation of cylindrical shock waves in a self-gravitating perfect gas under the influence of azimuthal magnetic field. The method of Lie group invariance is used to construct some special class of self-similar solutions in the presence of the azimuthal magnetic field. The different cases of solutions with a power law and exponential law shock paths are obtained with the choice of arbitrary constants appearing in the expressions for the infinitesimal generators. The similarity solution for cylindrical shock wave with power law shock path is discussed in detail. The effects of variation of Alfven-Mach number, gravitation parameter, initial density variation index and adiabatic exponent on the flow variables are analyzed graphically. It is obtained that the increase in the values of Alfven-Mach number, gravitation parameter and adiabatic exponent have decaying effect on the shock strength. Also, the shock strength increases with an increase in the values of initial density variation index. A comparison is also made between the solutions in gravitating and non-gravitating cases in the presence of magnetic field.

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Tiwari, Ashish, Kajal Sharma, and Rajan Arora. "Lie symmetry analysis, optimal system, and new exact solutions of a (3 + 1) dimensional nonlinear evolution equation." Nonlinear Engineering 10, no. 1 (January 1, 2021): 132–45. http://dx.doi.org/10.1515/nleng-2021-0010.

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Abstract Studies on Non-linear evolutionary equations have become more critical as time evolves. Such equations are not far-fetched in fluid mechanics, plasma physics, optical fibers, and other scientific applications. It should be an essential aim to find exact solutions of these equations. In this work, the Lie group theory is used to apply the similarity reduction and to find some exact solutions of a (3+1) dimensional nonlinear evolution equation. In this report, the groups of symmetries, Tables for commutation, and adjoints with infinitesimal generators were established. The subalgebra and its optimal system is obtained with the aid of the adjoint Table. Moreover, the equation has been reduced into a new PDE having less number of independent variables and at last into an ODE, using subalgebras and their optimal system, which gives similarity solutions that can represent the dynamics of nonlinear waves.

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Chauhan, Astha, and Rajan Arora. "Similarity Solutions of Strong Shock Waves for Isothermal Flow in an Ideal Gas." International Journal of Mathematical, Engineering and Management Sciences 4, no. 5 (October 1, 2019): 1094–107. http://dx.doi.org/10.33889/ijmems.2019.4.5-087.

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Self-similar solutions of the system of non-linear partial differential equations are obtained using the Lie group of invariance technique. The system of equations governs the one dimensional and unsteady motion for the isothermal flow of an ideal gas. The medium has been taken the uniform. From the expressions of infinitesimal generators involving arbitrary constants, different cases arise as per the choice of the arbitrary constants. In this paper, the case of a collapse of an implosion of a cylindrical shock wave is shown in detail along with the comparison between the similarity exponent obtained by Guderley's method and by Crammer's rule. Also, the effects of the adiabatic index and the ambient density exponent on the flow variables are illustrated through the figures. The flow variables are computed behind the leading shock and are shown graphically.

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Jamal, Sameerah. "Potentials and point symmetries of Klein–Gordon equations in space-time homogenous Gödel-type metrics." International Journal of Geometric Methods in Modern Physics 14, no. 05 (April 13, 2017): 1750070. http://dx.doi.org/10.1142/s0219887817500700.

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In this paper, we study the geometric properties of generators for the Klein–Gordon equation on classes of space-time homogeneous Gödel-type metrics. Our analysis complements the study involving the “Symmetries of geodesic motion in Gödel-type spacetimes” by U. Camci (J. Cosmol. Astropart. Phys., doi: 10.1088/1475-7516/2014/07/002 ). These symmetries or Killing vectors (KVs) are used to construct potential functions admitted by the Klein–Gordon equation. The criteria for the potential function originates from three primary sources, viz. through generators that are identically the Killing algebra, or with the KV fields that are recast into linear combinations and third, real subalgebras within the Killing algebra. This leads to a classification of the [Formula: see text] Klein–Gordon equation according to the catalogue of infinitesimal Lie and Noether point symmetries admitted. A comprehensive list of group invariant functions is provided and their application to analytic solutions is discussed.

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Riaz, Muhammad Bilal, Jan Awrejcewicz, Adil Jhangeer, and Muhammad Junaid-U-Rehman. "A Variety of New Traveling Wave Packets and Conservation Laws to the Nonlinear Low-Pass Electrical Transmission Lines via Lie Analysis." Fractal and Fractional 5, no. 4 (October 18, 2021): 170. http://dx.doi.org/10.3390/fractalfract5040170.

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This research is based on computing the new wave packets and conserved quantities to the nonlinear low-pass electrical transmission lines (NLETLs) via the group-theoretic method. By using the group-theoretic technique, we analyse the NLETLs and compute infinitesimal generators. The resulting equations concede two-dimensional Lie algebra. Then, we have to find the commutation relation of the entire vector field and observe that the obtained generators make an abelian algebra. The optimal system is computed by using the entire vector field and using the concept of abelian algebra. With the help of an optimal system, NLETLs convert into nonlinear ODE. The modified Khater method (MKM) is used to find the wave packets by using the resulting ODEs for a supposed model. To represent the physical importance of the considered model, some 3D, 2D, and density diagrams of acquired results are plotted by using Mathematica under the suitable choice of involving parameter values. Furthermore, all derived results were verified by putting them back into the assumed equation with the aid of Maple software. Further, the conservation laws of NLETLs are computed by the multiplier method.

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Devi, Munesh, Rajan Arora, Mustafa M. Rahman, and Mohd Junaid Siddiqui. "Converging Cylindrical Symmetric Shock Waves in a Real Medium with a Magnetic Field." Symmetry 11, no. 9 (September 17, 2019): 1177. http://dx.doi.org/10.3390/sym11091177.

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The topic “converging shock waves” is quite useful in Inertial Confinement Fusion (ICF). Most of the earlier studies have assumed that the medium of propagation is ideal. However, due to very high temperature at the axis of convergence, the effect of medium on shock waves should be taken in account. We have considered a problem of propagation of cylindrical shock waves in real medium. Magnetic field has been assumed in axial direction. It has been assumed that electrical resistance is zero. The problem can be represented by a system of hyperbolic Partial Differential Equations (PDEs) with jump conditions at the shock as the boundary conditions. The Lie group theoretic method has been used to find solutions to the problem. Lie’s symmetric method is quite useful as it reduces one-dimensional flow represented by a system of hyperbolic PDEs to a system of Ordinary Differential Equations (ODEs) by means of a similarity variable. Infinitesimal generators of Lie’s group transformation have been obtained by invariant conditions of the governing and boundary conditions. These generators involves arbitrary constants that give rise to different possible cases. One of the cases has been discussed in detail by writing reduced system of ODEs in matrix form. Cramer’s rule has been used to find the solution of system in matrix form. The results are presented in terms of figures for different values of parameters. The effect of non-ideal medium on the flow has been studied. Guderley’s rule is used to compute similarity exponents for cylindrical shock waves, in gasdynamics and in magnetogasdynamics (ideal medium), in order to set up a comparison with the published work. The computed values are very close to the values in published articles.

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Zhang, Zhi-Yong. "Symmetry determination and nonlinearization of a nonlinear time-fractional partial differential equation." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 476, no. 2233 (January 2020): 20190564. http://dx.doi.org/10.1098/rspa.2019.0564.

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We first show that the infinitesimal generator of Lie symmetry of a time-fractional partial differential equation (PDE) takes a unified and simple form, and then separate the Lie symmetry condition into two distinct parts, where one is a linear time-fractional PDE and the other is an integer-order PDE that dominates the leading position, even completely determining the symmetry for a particular type of time-fractional PDE. Moreover, we show that a linear time-fractional PDE always admits an infinite-dimensional Lie algebra of an infinitesimal generator, just as the case for a linear PDE and a nonlinear time-fractional PDE admits, at most, finite-dimensional Lie algebra. Thus, there exists no invertible mapping that converts a nonlinear time-fractional PDE to a linear one. We illustrate the results by considering two examples.

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Rajeev, S. G. "Infinitesimal quantum group from helicity in fluid mechanics." Modern Physics Letters A 35, no. 30 (July 6, 2020): 2050245. http://dx.doi.org/10.1142/s0217732320502454.

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Arnold showed that the Euler equations of an ideal fluid describe geodesics in the Lie algebra of incompressible vector fields. We will show that helicity induces a splitting of the Lie algebra into two isotropic subspaces, forming a Manin triple. Viewed another way, this shows that there is an infinitesimal quantum group (a.k.a. Lie bi-algebra) underlying classical fluid mechanics.

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GOARD, JOANNA. "Finding symmetries by incorporating initial conditions as side conditions." European Journal of Applied Mathematics 19, no. 6 (December 2008): 701–15. http://dx.doi.org/10.1017/s0956792508007705.

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It is generally believed that in order to solve initial value problems using Lie symmetry methods, the initial condition needs to be left invariant by the infinitesimal symmetry generator that admits the invariant solution. This is not so. In this paper we incorporate the imposed initial value as a side condition to find ‘infinitesimals’ from which solutions satisfying the initial value can be recovered, along with the corresponding symmetry generator.

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Berarducci, Alessandro. "Cohomology of groups in o-minimal structures: acyclicity of the infinitesimal subgroup." Journal of Symbolic Logic 74, no. 3 (September 2009): 891–900. http://dx.doi.org/10.2178/jsl/1245158089.

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AbstractBy recent work on some conjectures of Pillay, each definably compact group in a saturated o-minimal structure is an extension of a compact Lie group by a torsion free normal divisible subgroup, called its infinitesimal subgroup. We show that the infinitesimal subgroup is cohomologically acyclic. This implies that the functorial correspondence between definably compact groups and Lie groups preserves the cohomology.

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Journal articles on the topic 'Infinitesimal generator of a Lie group'

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