Algorithmic symmetry classification with invariance. (English) Zbl 1202.35016
Authors’ abstract: Symmetry classification for a system of differential equations can be achieved algorithmically by applying a differential reduction and completion algorithm to the infinitesimal determining equations of the system. The branches of the classification should be invariant under the action of the equivalence group. We show that such invariance can be tested algorithmically knowing only the determining equations of the equivalence group. The method relies on computing the prolongation of a group operator reduced modulo these determining equations. The method is implemented in Maple: a novel pivot selection strategy is able to guide the rifsimp command towards more favourable branchings.
Reviewer: Ma Wen-Xiu (Tampa)
MSC:
| 35B06 | Symmetries, invariants, etc. in context of PDEs |
| 35A30 | Geometric theory, characteristics, transformations in context of PDEs |
| 58J70 | Invariance and symmetry properties for PDEs on manifolds |
| 37K05 | Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010) |
Keywords:
differential reduction; equivalence transformation; symmetry classification; Lie symmetries; Maple; infinitesimal determining equationsReferences:
| [1] | Bluman G, Kumei S (1989) Symmetries and differential equations. Springer, New York · Zbl 0698.35001 |
| [2] | Olver P (1993) Application of Lie groups to differential equations, 2nd edn. Springer, New York · Zbl 0785.58003 |
| [3] | Ovsiannikov LV (1982) Group analysis of differential equations. Academic Press Inc, New York (trans: Russian by Y. Chapovsky, ed., trans: Ames WF), Harcourt Brace Jovanovich Publishers, London · Zbl 0485.58002 |
| [4] | Cheviakov AF (2007) GeM software package for computation of symmetries and conservation laws of differential equations. Comput Phys Commun 176(1): 48–61 · Zbl 1196.34045 · doi:10.1016/j.cpc.2006.08.001 |
| [5] | Hereman W (1994) Review of symbolic software for the computation of Lie symmetries of differential equations. Euromath Bull 1: 45–79 · Zbl 0891.65081 |
| [6] | Wolf T, Brand A (1992) The computer algebra package CRACK for investigating PDEs. In: Pommaret J-F (ed) Proceedings ERCIM advanced course on partial differential equations and group theory. Sankt Augustin, Germany, Gesellschaft für Mathematik and Datenverarbeitung, pp 1–19 |
| [7] | Ibragimov N (ed) (1994) CRC handbook of Lie group analysis of differential equations, vol 1 symmetries, exact solutions and conservation laws. CRC Press, Boca Raton · Zbl 0864.35001 |
| [8] | Gungor F, Lahno VI, Zhdanov RZ (2004) Symmetry classification of KDV-type nonlinear evolution equations. J Math Phys 45: 2280–2313 · Zbl 1071.35112 · doi:10.1063/1.1737811 |
| [9] | Lahno V, Zhdanov R, Magda O (2006) Group classification and exact solutions of nonlinear wave equations. Acta Appl Math 91(3): 253–313 · Zbl 1113.35132 · doi:10.1007/s10440-006-9039-0 |
| [10] | Lisle I, Reid G (2006) Symmetry classification using noncommutative invariant differential operators. Found Comput Math 6: 353–386 · Zbl 1107.35011 · doi:10.1007/s10208-005-0186-x |
| [11] | Mahomed FM (2007) Symmetry group classification of ordinary differential equations: survey of some results. Math Methods Appl Sci 30(16): 1995–2012 · Zbl 1135.34029 · doi:10.1002/mma.934 |
| [12] | Lie S (1881) Über die Integration durch bestimmte Integrale von einer Klasse linearer partieller Differentialgleichungen. Arch Math VI(H) 3: 328–368 · JFM 13.0298.01 |
| [13] | Basarab-Horwath P, Lahno V, Zhdanov R (2001) The structure of Lie algebras and the classification problem for partial differential equations. Acta Appl Math 69(1): 43–94 · Zbl 1054.35002 · doi:10.1023/A:1012667617936 |
| [14] | Gagnon L, Winternitz P (1990) Non-Painlevé reductions of nonlinear Schrödinger equations. Phys Rev A 42: 5029–5030 · doi:10.1103/PhysRevA.42.5029 |
| [15] | Olver P (1995) Equivalence, invariants and symmetry. Cambridge University Press, Cambridge · Zbl 0837.58001 |
| [16] | Reid G, Wittkopf A, Boulton A (1996) Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur J Appl Math 7: 604–635 · Zbl 0892.35041 · doi:10.1017/S0956792500002618 |
| [17] | Boulier F, Lazard D, Ollivier F, Petitot M (1995) Representation for the radical of a finitely generated differential ideal. In: Proc ISSAC ’95, ACM Press, New York · Zbl 0911.13011 |
| [18] | Gandarias ML, Ibragimov NH (2008) Equivalence group of a fourth-order evolution equation unifying various non-linear models. Commun Nonlinear Sci Numer Simul 13(2): 259–268 · Zbl 1155.35410 · doi:10.1016/j.cnsns.2005.12.011 |
| [19] | Huang D-J, Ivanova NM (2007) Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations. J Math Phys 48(7): 073507, 23 · Zbl 1144.81358 |
| [20] | Ibragimov NH, Torrisi M (1994) Equivalence groups for balance equations. J Math Anal Appl 184(3): 441–452 · Zbl 0809.35053 · doi:10.1006/jmaa.1994.1213 |
| [21] | Bublik VV (2000) Group classification of the Navier–Stokes equations for compressible viscous heat-conducting gas. In: Computer algebra in scientific computing (Samarkand, 2000). Springer, Berlin, pp 61–67 · Zbl 1051.35501 |
| [22] | Wittkopf A (2004) Algorithms and implementations for differential elimination. PhD thesis, Department of Mathematics, Simon Fraser University, Canada |
| [23] | Fels M, Olver P (1998) Moving coframes: I. A practical algorithm. Acta Appl Math 51: 161–213 · Zbl 0937.53012 · doi:10.1023/A:1005878210297 |
| [24] | Fels M, Olver P (1999) Moving coframes: II. Regularization and theoretical foundations. Acta Appl Math 55: 127–208 · Zbl 0937.53013 · doi:10.1023/A:1006195823000 |
| [25] | Mansfield EL (2001) Algorithms for symmetric differential systems. Found Comput Math 1(4): 335–383 · Zbl 0986.35080 |
| [26] | Hubert E (2005) Differential algebra for derivations with nontrivial commutation rules. J Pure Appl Algebra 200: 163–190 · Zbl 1151.12003 · doi:10.1016/j.jpaa.2004.12.034 |
| [27] | Reid G (1990) A triangularization algorithm which determines the lie symmetry algebra of any system of PDEs. J Phys A 23: L853–L859 · Zbl 0724.35001 · doi:10.1088/0305-4470/23/17/001 |
| [28] | Schwarz F (1992) Reduction and completion algorithms for partial differential equations. In Proc ISSAC ’92, ACM Press, New York, pp 49–56 · Zbl 0978.65515 |
| [29] | Reid G, Lisle I, Boulton A, Wittkopf A (1992) Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs,. In: Proc ISSAC ’92, ACM Press, New York, pp 63–68 · Zbl 0978.65514 |
| [30] | Blinkov Y, Cid C, Gerdt V, Plesken W, Robertz D (2003) The MAPLE package ”Janet”: II. linear partial differential equations. In: Ganzha V, Mayr E, Vorozhtsov E (eds) CASC 2003, Garching. Institute of Informatics, Technical University of Munich |
| [31] | Schwarz F (2008) Algorithmic Lie theory for solving ordinary differential equations. Chapman and Hall/CRC, Boca Raton, FL · Zbl 1139.34003 |
| [32] | Akhatov I, Gazizov R, Ibragimov N (1991) Nonlocal symmetries: heuristic approach. J Soviet Math 55: 1401–1450 · Zbl 0760.35002 · doi:10.1007/BF01097533 |
| [33] | Ibragimov N (ed) (1994) CRC handbook of Lie group analysis of differential equations, vol 2: applications in engineering and physical sciences. CRC Press, Boca Raton · Zbl 0864.35001 |
| [34] | Cox D, Little J, O’Shea D (1997) Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra, 2nd edn. Springer, New York |
| [35] | Greuel G-M, Pfister G (2008) A singular introduction to commutative algebra, 2nd edn. Springer, Berlin · Zbl 1344.13002 |
| [36] | Bluman G, Temuerchaolu (2005) Conservation laws for nonlinear telegraph equations. J Math Anal Appl 310: 459–476 · Zbl 1077.35087 |
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