An algorithm for the complete symmetry classification of differential equations based on Wu’s method. (English) Zbl 1204.35021

An alternative algorithm using Wu’s method (differential characteristic set algorithm) is proposed for complete symmetry classification of partial differential equations involving arbitrary parameters. Such classification is determined by decomposing the solution set of the determining equations into a union of a series of zero sets of differential characteristic sets of the corresponding differential polynomial system. Each branch of the resulting decomposition yields a class of symmetries and the corresponding parameters. The proposed algorithm makes the classification direct and systematic, which also provides a novel application of Wu’s method to differential equations. To show the efficiency of the algorithm, complete potential symmetry classifications are given for linear and nonlinear wave equations with an arbitrary function parameter and both classical and nonclassical symmetries of a parametric Burgers equation as illustrative examples.
Reviewer: Ma Wen-Xiu (Tampa)


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
35K05 Heat equation
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
35A25 Other special methods applied to PDEs


diffgrob2; SYMMGRP
Full Text: DOI


[1] Olver PJ (1993) Applications of Lie groups to differential equations, 2 edn. Springer-Verlag, New York
[2] Bluman GW, Kumei S (1991) Symmetries and differential equations. Applied Mathematical Sciences 81. Springer-Verlag/World Publishing Corp, New York · Zbl 0698.35001
[3] Ovsiannikov LV (1982) Group analysis of differential equations (trans: Ames WF). Academic Press, New York
[4] Reid GJ (1991) Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution. Eur J Appl Math 2: 293–318 · Zbl 0768.35001
[5] Reid GJ, Wittkopf AD (2000) Determination of maximal symmetry groups of classes of differential equations. In: Proceedings international symposium on symbolic and algebraic computation (ISSAC), St Andrews, Scotland, pp 272–280 · Zbl 1326.68367
[6] Reid GJ (1991) Finding abstract Lie symmetry algebras of differential equations without integrating determining equations. Eur J Appl Math 2: 319–340 · Zbl 0768.35002
[7] Reid GJ, Wittkopf AD, Boulton A (1996) Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur J Appl Math 7: 605–635 · Zbl 0892.35041
[8] Schwarz F (1992a) An algorithm for determining the size of symmetry groups. Computing 49: 95–115 · Zbl 0759.68042
[9] Schwarz F (1992b) Reduction and completion algorithm for partial differential equations. In: Wang P (eds) Proceedings international symposium on symbolic and algebraic computation (ISSAC) 1992. ACM Press, Berkeley, pp 49–56 · Zbl 0978.65515
[10] Wolf T, Brand A (1995) Investigating DEs with CRACK and related programs, SIGSAM bulletin, special issue, pp 1–8
[11] Mansfield E (1991) Differential Gröbner bases. Dissertation (PhD thesis), University of Sydney
[12] Mansfield EL (1993) Applications of the differential algebra package diffgrob2 to classical symmetries of differential equations. J Symb Comp 5–6(23): 517–533
[13] Lisle IG, Reid GJ (2006) Symmetry classification using noncommutative invariant differential operators. Found Comput Math 6(3): 353–386 · Zbl 1107.35011
[14] Boulier F, Lazard D, Ollivier F, Petitot M (1995) Representation for the radical of a finitely generated differential ideal. In: Proceedings of international symposium on symbolic and algebraic computation (ISSAC) 1995. ACM Press, New York, pp 158–166 · Zbl 0911.13011
[15] Hubert E (1999) Essential components of algebraic differential equations. J Symb Comp 28(4–5): 657–680 · Zbl 0943.34002
[16] Ibragimov NH (1994) CRC handbook of Lie group analysis of differential equations, vol 3: new trends in theoretical developments and computational methods. CRC Press, Boca Raton
[17] Hereman W (1996) Review of symbolic software for Lie symmetry analysis, vol 3. CRC Press, Boca Raton, pp 367–413
[18] Topunov VL (1989) Reducing systems of linear partial differential equations to a passive form. Acta Appl Math 16: 191–206 · Zbl 0703.35005
[19] Riquier C (1910) Les systèmes d’équations aux dérivées partielles. Gauthier-Villars, Paris
[20] Janet M (1920) Sur les systèmes d’équations aux dérivées partielles. J de Math 3: 65–151 · JFM 47.0440.04
[21] Kolchin ER (1973) Differential algebra and algebraic groups. Academic Press, New York · Zbl 0264.12102
[22] Ritt JF (1950) Differential algebra, AMS Colloquium publications. American Mathematical Society, New York
[23] Clarkson PA, Mansfield EL (2001) Open problems in symmetry analysis. In: Leslie JA, Robart T (eds) Geometrical study of differential equations. Contemporary Mathematics Series, vol 285. AMS, Providence, RI, pp 195–205 · Zbl 1172.35306
[24] Ames WF, Lohner RJ, Adams E (1981) Group properties of u tt = [f(u)u x ] x . Int J Non-Linear Mech 16: 439–447 · Zbl 0503.35058
[25] Fushchych WI, Shtelen WM, Serov NI (1993) Symmetry analysis and exact solutions of nonlinear equations of mathematical physics (transl: English). Kluwer, Dordrecht
[26] Lie S (1881) Über die Integration durch bestimmte Integrale von einer Klass linear partieller Differentialgleichungen. Arch fur Math VI(H3): 328–368
[27] Ibragimov NH, Torrisi M, Valenti A (1991) Preliminary group classification of equation v tt = f(v,v x )v xx + g(x,v x ). J Math Phys 32: 2988–2995 · Zbl 0737.35099
[28] Akhatov I, Gazizov R, Ibragimov NH (1991) Nonlocal symmetries: heuristic approach. J Soviet Math 55: 1401–1450 · Zbl 0760.35002
[29] Torrisi M, Tracina R (1998) Equivalence transformation and symmetries for a heat conduction model. Int J Non-linear Mech 33: 473–487 · Zbl 0911.35005
[30] Zhdanov R, Lahno V (1990) Group classification of heat conductivity equations with a nonlinear source. J Phys A Math Gen 32: 7405–7418 · Zbl 0990.35009
[31] Nikitin AG, Popovych RO (2001) Group classification of nonlinear Schrödinger equations. Ukr Math J 53: 1053–1060 · Zbl 0993.58020
[32] Popovych RO, Ivanova NM (2004) New results on group classification of nonlinear diffusion-convection equations. J Phys A Math Gen 37: 7547–7565 · Zbl 1067.35006
[33] Huang DJ, Ivanova NM (2007) Group analysis and exact solutions of a class of variable coefficient nonlinear telegraph equations. J Math Phys 48(7): 1–23 (Article No 073507)
[34] Wittkopf AD (2004) Algorithms and implementations for differential elimination. PhD thesis, Department of Mathematics, SFU, Canada
[35] Wu WT (1984) Basic principles of mechanical theorem–proving in elementary geometry. J Syst Sci Math Sci 4: 207–235
[36] Wu WT (2000) Mathematics mechanization. Science Press, Beijing
[37] Wu WT (1989) On the foundation of algebraic differential geometry. J Syst Sci Math Sci 2: 289–312 · Zbl 0739.14001
[38] Cox D, Little J, O’Shea D (1992) Ideals, varieties, and algorithms: an introduction to computational algebraic geometry and commutative algebra. Springer-Verlag, New York · Zbl 0756.13017
[39] Chaolu T (1999) Wu-d-characteristic algorithm of symmetry vectors for partial differential equations. Acta Math Sci (in Chinese) 19(3): 326–332
[40] Chaolu T (2003) An algorithmic theory of reduction of a differential polynomial system. Adv Math (China) 32(2): 208–220
[41] Chaolu T, Gao XS (2002) Nearly d-char-set for a differential polynomial system. Acta Math Sci (in Chinese) 45(6): 1041–1050 · Zbl 1033.68049
[42] Gao XS, Wang DK, Liao Q, Yang H (2006) Equation solving and machine proving–problem solving with MMP (in Chinese). Science Press, Beijing (The Package can be downloaded from http://www.mmrc.iss.ac.cn/xgao/software.html )
[43] Clarkson PA, Mansfield EL (1994) Algorithms for the non-classical method of symmetry reductions. SIAM J Appl Math 54: 1693–1719 · Zbl 0823.58036
[44] Collins GE (1967) Subresultants and reduced polynomial sequences. J ACM 14: 128–142 · Zbl 0152.35403
[45] Li ZM (1987) A new proof of Collin’s theorem, MM-Res preprints. MMRC 1: 33–37
[46] Baikov VA, Gazizov RA, Ibragimov NH (1988) Approximate group analysis of the nonlinear equation \({u_{tt}-(f(u)u_x)_x+\epsilon\phi(u)u_t=0}\) . Differents Uravn 24(7): 1127
[47] Bluman GW, Chaolu T (2005) Local and nonlocal symmetries for nonlinear telegraph equations. J Math Phys 46: 1–9 (Article No 023505)
[48] Bluman GW, Chaolu T (2005) Conservation laws for nonlinear telegraph-type equations. J Math Anal Appl 310: 459–476 · Zbl 1077.35087
[49] Bluman GW, Chaolu T (2005) Comparing symmetries and conservation laws of nonlinear telegraph equations. J Math Phys 46: 1–14 (Article No 073513)
[50] Bluman GW, Chaolu T, Anco SC (2006) New conservation laws obtained directly from symmetry action on a known conservation law. J Math Anal Appl 322(1): 233–250 · Zbl 1129.35067
[51] Seidenberg A (1956) An elimination theory for differential algebra. Univ California Publ Math (NS) 3(2): 31–38 · Zbl 0072.26502
[52] Rosenfeld A (1959) Specializations in differential algebra. Trans Am Math Soc 90: 394–407 · Zbl 0192.14001
[53] Li ZM, Wang DM (1999) Coherent, regular and simple systems in zero decompositions of partial differential systems. J Syst Sci Math Sci 12(Suppl): 43–60 · Zbl 1094.13545
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.