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Symmetry-based algorithms for invertible mappings of polynomially nonlinear PDE to linear PDE. (English) Zbl 1486.12006

The paper under review is a sequel to the previous work of the authors [Z. Mohammadi et al., in: Proceedings of the 44th international symposium on symbolic and algebraic computation, ISSAC ’19, Beijing, China, July 15–18, 2019. New York, NY: Association for Computing Machinery (ACM). 331–338 (2019; Zbl 1467.35005)] where they introduced the MapDE algorithm to determine the existence of analytic invertible mappings of an input (source) differential polynomial system (DPS) to a specific target DPS, and sometimes by heuristic integration an explicit form of the mapping. In this paper the authors extend MapDE to determine if a source nonlinear DPS can be mapped to a linear differential system.
The main contribution of the paper is the presentation of an algorithmic method for determining the mapping of a nonlinear system to a linear system when it exists. Using a technique of [S. Kumei and G. W. Bluman, SIAM J. Appl. Math. 42, 1157–1173 (1982; Zbl 0506.35003)], the authors exploit the fact that the target system \(\hat{R}\), which corresponds to the source system of differential equations \(R\), must admit a sub-pseudo group corresponding to the superposition property that linear systems by definition must satisfy. Once existence is established, a second stage can determine features of the map and sometimes by integration, explicit forms of the mapping. For an algorithmic treatment using differential elimination (differential algebra), the authors limit their consideration to systems of differential polynomials, with coefficients from \(\mathbb{Q}\) or some computable extension of \(\mathbb{Q}\) in \(\mathbb{C}\). (Thus the input system \(R\) should be a system of DPS.) The paper contains examples that illustrate the approach; the authors also illustrate the powerful maximal symmetry groups facility as a natural tool to be used in conjunction with MapDE.

MSC:

12H05 Differential algebra
68W30 Symbolic computation and algebraic computation
35A30 Geometric theory, characteristics, transformations in context of PDEs
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[1] Anco, S.; Bluman, G.; Wolf, T., Invertible mappings of nonlinear PDEs to linear PDEs through admitted conservation laws, Acta Appl. Math., 101, 21-38 (2008) · Zbl 1157.35002
[2] Anderson, IM; Torre, CG, New symbolic tools for differential geometry, gravitation, and field theory, J. Math. Phys., 53, 013511 (2012) · Zbl 1273.83013
[3] Arnaldsson, O.: Involutive moving frames. Ph.D. Thesis, University of Minnesota (2017) · Zbl 1437.58004
[4] Bluman, G.; Kumei, S., Symmetries and Differential Equations (1989), Berlin: Springer, Berlin · Zbl 0698.35001
[5] Bluman, G.; Kumei, S., Symmetry based algorithms to relate partial differential equations: I. Local symmetries, Eur. J. Appl. Math., 1, 189-216 (1990) · Zbl 0718.35003
[6] Bluman, G.; Kumei, S., Symmetry based algorithms to relate partial differential equations: II. Linearization by nonlocal symmetries, Eur. J. Appl. Math., 1, 217-223 (1990) · Zbl 0718.35004
[7] Bluman, G.; Yang, Z.; Ganghoffer, JF; Mladenov, I., Some recent developments in finding systematically conservation laws and nonlocal symmetries for partial differential equations, Similarity and Symmetry Methods Applications in Elasticity and Mechanics of Materials, 1-59 (2014), Cham: Springer, Cham · Zbl 1303.76115
[8] Bluman, GW; Cheviakov, AF; Anco, SC, Applications of Symmetry Methods to Partial Differential Equations (2010), Cham: Springer, Cham · Zbl 1223.35001
[9] Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Representation for the radical of a finitely generated differential ideal. Proc. ISSAC 95, pp. 158-166 (1995) · Zbl 0911.13011
[10] Boulier, F., Lazard, D., Ollivier, F., Petitot, M.: Computing representations for radicals of finitely generated differential ideals. J. Appl. Algebra Eng. Commun. Comput. 20 (2009) · Zbl 1185.12003
[11] Boulier, F.; Lemaire, F.; Maza, MM, Computing differential characteristic sets by change of ordering, J. Symb. Comput., 45, 1, 124-149 (2010) · Zbl 1194.68264
[12] Carminati, J.; Vu, K., Symbolic computation and differential equations: Lie symmetries, J. Symb. Comput., 29, 95-116 (2000) · Zbl 0958.68543
[13] Cheviakov, AF, GeM software package for computation of symmetries and conservation laws of differential equations, Comput. Phys. Commun., 176, 1, 48-61 (2007) · Zbl 1196.34045
[14] Fels, M.; Olver, P., Moving coframes II. Regularization and theoretical foundations, Acta Appl. Math., 55, 127-208 (1999) · Zbl 0937.53013
[15] Golubitsky, O.; Kondratieva, M.; Ovchinnikov, A.; Szanto, A., A bound for orders in differential Nullstellensatz, J. Algebra, 322, 11, 3852-3877 (2009) · Zbl 1189.12002
[16] Huang, S.L.: Properties of lie algebras of vector fields from lie determining system. Ph.D. thesis, University of Canberra (2015)
[17] Hubert, E., Differential invariants of a Lie group action: syzygies on a generating set, J. Symb. Comput., 44, 4, 382-416 (2009) · Zbl 1176.12004
[18] Kruglikov, B.; The, D., Jet-determination of symmetries of parabolic geometries, Math. Ann., 371, 3, 1575-1613 (2018) · Zbl 1400.58002
[19] Kumei, S.; Bluman, G., When nonlinear differential equations are equivalent to linear differential equations, SIAM J. Appl. Math., 42, 1157-1173 (1982) · Zbl 0506.35003
[20] Lange, M., The differential counting polynomial, Found. Comput. Math., 18, 2, 291-308 (2018) · Zbl 1386.12007
[21] Lemaire, F.: Contribution à l’algorithmique en algèbre différentielle. Université des Sciences et Technologie de Lille-Lille, vol. I (2002)
[22] Lisle, I.; Huang, S-L, Algorithms calculus for Lie determining systems, J. Symb. Comput., 79, 482-498 (2017) · Zbl 1361.35012
[23] Lisle, IG; Reid, GJ, Geometry and structure of Lie pseudogroups from infinitesimal defining systems, J. Symb. Comput., 26, 3, 355-379 (1998) · Zbl 0926.58014
[24] Lisle, IG; Reid, GJ, Symmetry classification using noncommutative invariant differential operators found, Comput. Math., 6, 3, 1615-3375 (2006) · Zbl 1107.35011
[25] Lyakhov, D., Gerdt, V., Michels, D.: Algorithmic verification of linearizability for ordinary differential equations. In: Proceedings of ISSAC ’17, ACM, pp. 285-292 (2017) · Zbl 1454.34064
[26] Mahomed, FM; Leach, PGL, Symmetry Lie algebra of nth order ordinary differential equations, J. Math. Anal. Appl., 1151, 80-107 (1990) · Zbl 0719.34018
[27] Mansfield, E., A Practical Guide to the Invariant Calculus (2010), Cambridge: Cambridge University Press, Cambridge · Zbl 1203.37041
[28] Mikhalev, AV; Levin, AB; Pankratiev, EV; Kondratieva, MV, Differential and Difference Dimension Polynomials. Mathematics and Its Applications (2013), Dordrecht: Springer, Dordrecht · Zbl 0930.12005
[29] Mohammadi, Z., Reid, G., Huang, S.-L.T: Introduction of the MapDE algorithm for determination of mappings relating differential equations (2019). arXiv:1903.02180v1 [math.AP]. To appear in Proceedings of ISSAC ’19, ACM · Zbl 1467.35005
[30] Neut, S.; Petitot, M.; Dridi, R., Élie Cartan’s geometrical vision or how to avoid expression swell, J. Symb. Comput., 44, 3, 261-270 (2009) · Zbl 1161.65054
[31] Olver, P., Application of Lie groups to differential equations (1993), Berlin: Springer, Berlin
[32] Olver, P., Equivalence, Invariance, and Symmetry (1995), Cambridge: Cambridge University Press, Cambridge
[33] Reid, GJ, Finding abstract Lie symmetry algebras of differential equations without integrating determining equations, Eur. J. Appl. Math., 2, 319-340 (1991) · Zbl 0768.35002
[34] Reid, GJ; Lisle, IG; Boulton, A.; Wittkopf, AD, Algorithmic determination of commutation relations for Lie symmetry algebras of PDEs, Proc. ISSAC, 92, 63-68 (1992) · Zbl 0978.65514
[35] Reid, G.J., Wittkopf, A.D.: Determination of maximal symmetry groups of classes of differential equations. Proc. ISSAC 2000, pp. 272-280 (2000) · Zbl 1326.68367
[36] Riquier, C., Les systèmes déquations aux dérivées partielles (1910), Paris: Gauthier-Villars, Paris
[37] Robertz, D., Formal Algorithmic Elimination for PDEs (2014), Berlin: Springer, Berlin · Zbl 1339.35007
[38] Rocha, T.; Figueiredo, A., SADE: a Maple package for the symmetry analysis of differential equations, Comput. Phys. Commun., 182, 2, 467-476 (2011) · Zbl 1217.65165
[39] Rust, C.J., Reid, G.J.: Rankings of partial derivatives. Proc. ISSAC 97, pp. 9-16 (1997) · Zbl 0960.12003
[40] Rust, C.J., Reid, G.J., Wittkopf, A.D.: Existence and uniqueness theorems for formal power series solutions of analytic differential systems. Proc. ISSAC 99, pp. 105-112 (1999)
[41] Seiler, W., Involution: The Formal Theory of Differential Equations and Its Applications in Computer Algebra, Algorithms and Computation in Mathematics (2010), Berlin: Springer, Berlin · Zbl 1205.35003
[42] Thomas, JM, Differential Systems (1937), New York: AMS Colloquium Publications, New York
[43] Valiquette, F.: Solving local equivalence problems with the equivariant moving frame method. SIGMA: Symmetry Integrability and Geometry Methods and Applications, vol. 9 (2013) · Zbl 1277.58004
[44] Wolf, T., Investigating differential equations with crack, liepde, applsymm and conlaw, Handb. Comput. Algebra Found. Appl. Syst., 37, 465-468 (2002)
[45] Wolf, T.; Zheng, Z.; Wang, D., Partial and complete linearization of PDEs based on conservation laws, Differential Equations with Symbolic Computations, Trends in Mathematics, 291-306 (2005), Basel: Birkhäuser, Basel · Zbl 1157.70312
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