A Point Symmetry Group of a Differential Equation which cannot be Found Using Infinitesimal Methods

Reid, Gregory J.; Weih, D. T.; Wittkopf, Allan

doi:10.1007/978-94-011-2050-0_33

Cited by 16 publications

(27 citation statements)

References 4 publications

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“…This phenomenon occurs in algorithms which use pseudo-reduction (such as the rif algorithm and that of Mansfield [58]). On the other hand, algorithms which use reduction, such as Wittkopf s dijf_reduce algorithm [59], have the advantage that they do not case-split, but also the disadvantage that they may not terminate. The CartanKuranishi Theorem guarantees that, after a finite number of prolongations and projections, an analytic system satisfying the constant rank condition at each iteration will become involutive on a subset of its locus.…”

Section: Discussionmentioning

confidence: 99%

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

1996

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We describe an algorithm which uses a finite number of differentiations and algebraic operations to simplify a given analytic nonlinear system of partial differential equations to a form which includes all its integrability conditions. This form can be used to test whether a given differential expression vanishes as a consequence of such a system and may be more amenable to numerical or analytical solution techniques than the original system. It is also useful for determining consistent initial conditions for such a system. A computer implementable version of our algorithm is given for polynomially nonlinear systems of partial differential equations. This version uses Grobner basis techniques for constructing the radical of the polynomial ideal generated by the equations of such systems.

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Section: Discussionmentioning

confidence: 99%

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

1996

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“…used by Reid et al (1993) to demonstrate the application of a differential analogue of Buchberger's algorithm. This ODE has a one-dimensional Lie algebra of point symmetry generators, with a basis…”

Section: The Basic Methodsmentioning

confidence: 99%

“…It is applicable to any ODE with an N -parameter Lie group of point symmetries (N 1), for which the Lie symmetry generators can be found by the standard method. Most ODEs of physical interest fall into this category, as does the example used by Reid et al (1993) to demonstrate the Gröbner basis method.…”

Section: B)mentioning

confidence: 99%

Discrete point symmetries of ordinary differential equations

Hydon

1998

Proc. R. Soc. A.

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This paper describes a method that enables the user to construct systematically the set of all discrete point symmetries of a given ordinary differential equation (ODE) of order two or greater, provided that the ODE has at least a one-parameter Lie group of point symmetries.The method is easy to use, and is based upon Lie's method of constructing continuous symmetries. The calculations are simple, and computer algebra is not usually required. Various examples are used to illustrate the method. The paper concludes with a proof that every ODE whose Lie group of point symmetries is isomorphic to the unimodular group has at least four inequivalent real discrete point symmetries.

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“…[4]. Preliminary simplifications may be derived from the form-preserving transformations of any given class containing L[u] = 0.…”

Section: Diffeomorphism (22) Is a Symmetry Of L[u] = 0 If And Only Imentioning

confidence: 99%

How to calculate all point symmetries of linear and linearizable differential equations

Gray

2015

Proc. R. Soc. A.

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The systematic method for calculating all point symmetries of partial differential equations (PDEs) with non-trivial Lie point symmetry algebras does not currently apply to linear PDEs. Consider any given locally solvable system of linear homogeneous PDEs, each of order 2 or higher, of general Kovalevskaya form. Suppose that its Lie point symmetry generators' characteristics are linear in the dependent variables, while those that are homogeneous of degree 1 in the dependent variables contain a finite number of arbitrary parameters (this is common for systems that arise from applications). Herein, every point symmetry is shown to be projectable/fibre-preserving: the transformed independent variables depend only on the original independent variables. Moreover, two (conditional) Lie-algebraic characterizations of the generators of scalings and superpositions of solutions are derived. One of them holds if the system cannot be decoupled (with a smooth, locally invertible, linear change of dependent variables). If any such characterization holds, every point symmetry maps the dependent variables to functions that are linear in the original dependent variables. The aforementioned systematic method is adapted to calculate these symmetries. This version of the method also applies to linearizable systems.

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A Point Symmetry Group of a Differential Equation which cannot be Found Using Infinitesimal Methods

Cited by 16 publications

References 4 publications

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

Reduction of systems of nonlinear partial differential equations to simplified involutive forms

Discrete point symmetries of ordinary differential equations

How to calculate all point symmetries of linear and linearizable differential equations

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