Symbolic Computation and Differential Equations: Lie Symmetries

Carminati, John; Vu, Khai T.

doi:10.1006/jsco.1999.0299

Cited by 63 publications

(63 citation statements)

References 20 publications

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“…Many applications of group analysis to PDEs are collected in [8]. The classical Lie method is an algorithmic procedure for which many symbolic manipulation programs were designed [9,10]. This software became imperative in finding symmetries associated with large systems of PDEs.…”

Section: Introductionmentioning

confidence: 99%

Symmetries and exact solutions of the rotating shallow-water equations

Чесноков

2009

Eur. J. Appl. Math

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Abstract. Lie symmetry analysis is applied to study the nonlinear rotating shallow water equations. The 9-dimensional Lie algebra of point symmetries admitted by the model is found. It is shown that the rotating shallow water equations are related with the classical shallow water model with the change of variables. The derived symmetries are used to generate new exact solutions of the rotating shallow equations. In particular, a new class of time-periodic solutions with quasi-closed particle trajectories is constructed and studied. The symmetry reduction method is also used to obtain some invariant solutions of the model. Examples of these solutions are presented with a brief physical interpretation.

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Section: Introductionmentioning

confidence: 99%

Symmetries and exact solutions of the rotating shallow-water equations

Чесноков

2009

Eur. J. Appl. Math

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“…For this purpose, we recompute the symmetries of (1) for the case F = 0 and β arbitrary. This is done upon using the computer algebra packages MuLie [9] and DESOLV [6]. For this combination, equation (1) admits the six-dimensional Lie symmetry algebra a β generated by the operators…”

Section: The Lie Symmetriesmentioning

confidence: 99%

“…Consequently, some of the ansätze presented in [10], which lead to a reduction of the number of independent variables of (1), are overly intricate. More precisely, they could be realised by means of considering reduction using one of the inequivalent subalgebras (6) or (7) and subsequently acting on the resulting invariant solutions by finite symmetry transformations.…”

Section: Group-invariant Solutionsmentioning

confidence: 99%

Symmetry Analysis of Barotropic Potential Vorticity Equation

Bihlo

Popovych

2009

Commun. Theor. Phys.

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Recently F. Huang [Commun. Theor. Phys. 42 (2004) 903] and X. Tang and P.K. Shukla [Commun. Theor. Phys. 49 (2008) 229] investigated symmetry properties of the barotropic potential vorticity equation without forcing and dissipation on the beta-plane. This equation is governed by two dimensionless parameters, F and β, representing the ratio of the characteristic length scale to the Rossby radius of deformation and the variation of earth' angular rotation, respectively. In the present paper it is shown that in the case F = 0 there exists a well-defined point transformation to set β = 0. The classification of one-and two-dimensional Lie subalgebras of the Lie symmetry algebra of the potential vorticity equation is given for the parameter combination F = 0 and β = 0. Based upon this classification, distinct classes of group-invariant solutions is obtained and extended to the case β = 0.PACS numbers: 02.20.Sv

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“…The last four columns in this table indicate the scope of the programs: point symmetries, generalized symmetries, non-classical symmetries and whether the determining system can be solved automatically. Recent Maple programs for generating classical symmetries are DESOLV by Carminati and Vu [10], RIF by Reid and Wittkopf and SYMMETRIE by Hickman.…”

Section: Literature and Implementationsmentioning

confidence: 99%