Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs

Reid, Gregory J.; Lisle, I. G.; Boulton, Alan; Wittkopf, Allan

doi:10.1145/143242.143269

Cited by 15 publications

(38 citation statements)

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“…From these non-commutative rif-forms, one can apply the existence and uniqueness result [18,Theorem 9.11] to get counts of both the degree of arbitrariness in the classifying system C, and the dimension of the symmetry algebra. Moreover the method of [30] can be adapted to find structure constants for the symmetry algebra associated with the leaf (see §7).…”

Section: Classification Proceduresmentioning

confidence: 99%

“…In this case, the noncommutative rif-form S ∪ C is given by (30,27). The system S ∪ C is linear in its highest derivations so M consists of all equations from (30,27), while N = ∅.…”

Section: Computation Of Dimension and Initial Datamentioning

confidence: 99%

“…An existence and uniqueness theorem is available for the output form of some of the programs [32,33] and allows determination of the size of the group. In addition the formal series solution for ξ i can be constructed up a given order, allowing computation of the structure of the unknown group without explicitly determining ξ [29,30]. Some programs focus on explicit determination of the ξ i by employing integration [34,35] in addition to differentiations and eliminations.…”

Section: Introductionmentioning

confidence: 99%

“…Our method is based on differential elimination procedures developed by Reid et al [28,29,30,31] and which complete the symmetry system by adjoining compatibility conditions [5]. After a finite number of steps, the system reaches a reduced form where a local existence-uniqueness theorem can be used to deduce properties of the defining system, such as the dimension of its solution space.…”

Section: Introductionmentioning

confidence: 99%

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Symmetry Classification Using Noncommutative Invariant Differential Operators

Lisle

Reid

2006

Found Comput Math

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Given a class F of differential equations, the symmetry classification problem is to determine for each member f ∈ F the structure of its Lie symmetry group G f , or equivalently of its Lie symmetry algebra. The components of the symmetry vector fields of the Lie algebra are solutions of an associated over-determined 'defining system' of differential equations. The usual computer classification method which applies a sequence of total derivative operators and eliminations to this associated system often fails on problems of interest due to the excessive size of expressions generated in intermediate computations.We provide an alternative classification method which exploits the knowledge of an equivalence group G preserving the class. A noncommutative differential elimination procedure due to Lemaire, Reid and Zhang, where each step of the procedure is invariant under G, can be applied and an existence and uniqueness theorem for the output used to classify the structure of symmetry groups for each f ∈ F .The method is applied to a class of nonlinear diffusion convection equations v x = u, v t = B(u)u x − K(u) which is invariant under a large but easily determined equivalence group G. In this example the complexity of the calculations is much reduced by the use of G-invariant differential operators.

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Section: Classification Proceduresmentioning

confidence: 99%

“…In this case, the noncommutative rif-form S ∪ C is given by (30,27). The system S ∪ C is linear in its highest derivations so M consists of all equations from (30,27), while N = ∅.…”

Section: Computation Of Dimension and Initial Datamentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetry Classification Using Noncommutative Invariant Differential Operators

Lisle

Reid

2006

Found Comput Math

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“…Both Lie's attempt to use his infinitesimal method based on the infinitesimal determining system obtained by linearizing the determining system, and Cartan's method using intricate recursive prolongation of exterior differential systems are either limited in scope or impractical from the standpoint of applications. Along this Date: April 13, 2005. line of research in the last decade, G. Reid, et al, [15,16,17,26,27], developed methods for determining Cartan structure equations of Lie pseudo-groups, which depended only on algebraic and differential manipulation, without any integration, of infinitesimal determining systems, hence increasing the feasibility of their computer algebra implementation. Their algorithms were successfully applied to certain types of Lie symmetry pseudo-groups of differential equations.…”

Section: Introductionmentioning

confidence: 99%

Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

Cheh

Olver

Pohjanpelto

2005

Journal of Mathematical Physics

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Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems

Lisle

Reid

1998

Journal of Symbolic Computation

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An algorithm is described which uses a finite number of differentiations and linear operations to determine the Cartan structure of a transitive Lie pseudogroup from its infinitesimal defining equations. In addition, an algorithm is presented for determining from the infinitesimal defining system whether a Lie pseudogroup has essential invariants. If such invariants exist, the pseudogroup is intransitive. These methods make feasible the calculation of the Cartan structure of infinite Lie pseudogroups of symmetries of differential equations. The structure of the symmetry pseudogroup of the KP equation is presented.c 1998 Academic Press 0747-7171/98/090355 + 25 $30.00/0 c 1998 Academic Press the remainder of this section, the indices i, j, l range from 1 to n. We use the notation ξ i J to represent the Jth derivative ∂ k ξ i /∂x j1 · · · ∂x j k of ξ i , where J = (j 1 j 2 · · · j k ) is a symmetric multi-index, whose order will be denoted by k = #(J).Reduction of a qth order involutive PDE system to a first-order involutive form with equivalent symbol is achieved via an algorithm described by Pommaret (1978, pp. 109, 161). The derivatives ξ i J , 1 ≤ #(J) ≤ q − 1 are relabelled as new dependent variables, and the given system expressed as a first-order system in these variables. Certain firstorder differential relations between the ξ i J are then appended; the composite system is first-order involutive. Now the system must be adjusted so that it is the defining system of a Lie algebra. Following Cartan, we lift the vector field X on M to one on M × M , by giving it trivial action on the second copy of M . Prolonging the vector field q − 1 times giveswhere x i are coordinates on the first copy of M , and X i on the second. Here the derivatives of the Xs are denoted by X i J = ∂ k X i /∂x j1 ∂x j2 · · · ∂x j k We have set ψ i = 0; the ψ i J are given by the standard prolongation formula (Olver, 1993), and there is summation on the repeated index i and the repeated multi-index J, 1 ≤ #(J) ≤ q − 1. The main result is Theorem 2.9. Let L be a Lie algebra system of vector fields X = ξ i ∂ x i whose infinitesimal defining system is qth order involutive. Then the prolongation X (q−1) of X to (x i , X i , X i J )-space is a Lie algebra system with an infinitesimal defining system for ξ i , ψ i , ψ i J which is first-order involutive, and which is constructively determined.Note that the independent variables in the infinitesimal defining system of the original Lie algebra are x i . For the prolonged Lie algebra the independent variables in the firstorder defining system are (x i , X i , X i J ), and the corresponding dependent variables ξ i , ψ i ,Proof. Let S denote the first-order involutive system with independent variables x i obtained by the transformation of Pommaret from the qth order involutive infinitesimal defining system. Let T denote S augmented with the equations(2.6) (Thus the system T has independent variables x i , X i J , and dependent variables ξ i J , with 0 ≤ #(J) ≤ q − 1.) Since the system S has no X i J...

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Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs

Cited by 15 publications

References 3 publications

Symmetry Classification Using Noncommutative Invariant Differential Operators

Symmetry Classification Using Noncommutative Invariant Differential Operators

Maurer–Cartan equations for Lie symmetry pseudogroups of differential equations

Geometry and Structure of Lie Pseudogroups from Infinitesimal Defining Systems

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