Use and construction of potential symmetries.

*(English)*Zbl 0792.35008Summary: Group-theoretic methods based on local symmetries are useful to construct invariant solutions of PDEs and to linearize nonlinear PDEs by invertible mappings. Local symmetries include point symmetries, contact symmetries and, more generally, Lie-Bäcklund symmetries. An obvious limitation in their utility for particular PDEs is the non-existence of local symmetries.

A given system of PDEs with a conserved form can be embedded in a related auxiliary system of PDEs. A local symmetry of the auxiliary system can yield a nonlocal symmetry (potential symmetry) of the given system. The existence of potential symmetries leads to the construction of corresponding invariant solutions as well as to the linearization of nonlinear PDEs by non-invertible mappings.

Recent work considers the problem of finding all potential symmetries of given systems of PDEs. Examples include linear wave equations with variable wave speeds as well as nonlinear diffusion, reaction-diffusion, and gas dynamics equations.

A given system of PDEs with a conserved form can be embedded in a related auxiliary system of PDEs. A local symmetry of the auxiliary system can yield a nonlocal symmetry (potential symmetry) of the given system. The existence of potential symmetries leads to the construction of corresponding invariant solutions as well as to the linearization of nonlinear PDEs by non-invertible mappings.

Recent work considers the problem of finding all potential symmetries of given systems of PDEs. Examples include linear wave equations with variable wave speeds as well as nonlinear diffusion, reaction-diffusion, and gas dynamics equations.

##### MSC:

35A30 | Geometric theory, characteristics, transformations in context of PDEs |

58J70 | Invariance and symmetry properties for PDEs on manifolds |

58J72 | Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds |

##### Keywords:

local symmetries; Lie-Bäcklund symmetries; nonlocal symmetry (potential symmetry); invariant solutions
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\textit{G. Bluman}, Math. Comput. Modelling 18, No. 10, 1--14 (1993; Zbl 0792.35008)

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##### References:

[1] | Barenblatt, G.I., Similarity, self-similarity, and intermediate asymptotics, (1979), Consultants Bureau New York · Zbl 0467.76005 |

[2] | Bluman, G.W.; Cole, J.D., Similarity methods for differential equations, (1974), Springer New York, (Appl. Math. Sci. No. 13) · Zbl 0292.35001 |

[3] | Bluman, G.W.; Kumei, S., Symmetries and differential equations, (1989), Springer New York, (Appl. Math. Sci. No. 81) · Zbl 0718.35004 |

[4] | Hill, J.M., Solution of differential equations by means of one-parameter groups, (1982), Pitman Boston · Zbl 0497.34002 |

[5] | Ibragimov, N.H., Transformation groups applied to mathematical physics, (1985), Reidel Boston |

[6] | Olver, P.J., Applications of Lie groups to differential equations, (1986), Springer New York, (GTM, No. 107) · Zbl 0656.58039 |

[7] | Ovsiannikov, L.V., Group properties of differential equations, (1962), (in Russian), Novosibirsk · Zbl 0485.58002 |

[8] | Ovsiannikov, L.V., Group analysis of differential equations, (1982), Academic Press New York · Zbl 0485.58002 |

[9] | Rogers, C.; Ames, W.F., Nonlinear boundary value problems in science and engineering, (1989), Academic Press Boston · Zbl 0699.35004 |

[10] | Stephani, H., Differential equations—their solution using symmetries, (1989), Cambridge U.P Cambridge, U.K · Zbl 0704.34001 |

[11] | () |

[12] | Kumei, S.; Bluman, G.W., When nonlinear differential equations are equivalent to linear differential equations, SIAM J. appl. math., 42, 1157-1173, (1982) · Zbl 0506.35003 |

[13] | Bluman, G.W.; Kumei, S., Symmetry-based algorithms to relate partial differential equations: I. local symmetries, Euro. J. appl. math., 1, 189-216, (1990) · Zbl 0718.35003 |

[14] | Bluman, G.W.; Kumei, S., On invariance properties of the wave equation, J. math. phys., 28, 307-318, (1987) · Zbl 0662.35065 |

[15] | Bluman, G.W.; Kumei, S.; Reid, G.J.; Bluman, G.W.; Kumei, S.; Reid, G.J., New classes of symmetries for partial differential equations, J. math. phys., J. math. phys., 29, 2320-811, (1988) · Zbl 0684.58046 |

[16] | Bluman, G.W., Potential symmetries, (), 85-100 · Zbl 0791.58114 |

[17] | Bluman, G.W.; Kumei, S., Symmetry-based algorithms to relate partial differential equations: II. linearization by nonlocal symmetries, Euro. J. appl. math, 1, 217-223, (1990) · Zbl 0718.35004 |

[18] | Head, A.K.; Head, A.K., LIE: A mumath program for the calculation of the LIE algebra of differential equations, (1992), CSIRO Division of Material Sciences Clayton, Australia · Zbl 0921.65080 |

[19] | Schwarz, F., Automatically determining symmetries of partial differential equations, Computing, 34, 91-106, (1985) · Zbl 0555.65076 |

[20] | Wolf, T.; Brand, A., The computer algebra package CRACK for investigating pdes, ERCIM advanced course on partial differential equations and group theory, (1992), Bonn |

[21] | Champagne, B.; Hereman, W.; Winternitz, P., The computer calculation of Lie point symmetries of large systems of differential equations, Comp. phys. comm., 66, 319-340, (1991) · Zbl 0875.65079 |

[22] | Kersten, P.H.M., Infinitesimal symmetries: A computational approach, (1987), Centrum voor Wiskunde en Informatica Amsterdam, (CWI Tract No. 34) · Zbl 0648.68052 |

[23] | Reid, G.J., Algorithms for reducing a system of PDEs to standard form, determining the dimension of its solution space and calculating its Taylor series solution, Euro. J. appl. math., 2, 293-318, (1991) · Zbl 0768.35001 |

[24] | Reid, G.J., Finding abstract Lie symmetry algebras of differential equations without integrating determining equations, Euro. J. appl. math., 2, 319-340, (1991) · Zbl 0768.35002 |

[25] | Topunov, V.L., Reducing systems of linear differential equations to passive form, Acta applic. math., 16, 191-206, (1989) · Zbl 0703.35005 |

[26] | Lisle, I., Equivalence transformations for classes of differential equations, Ph.D. thesis, (1992), University of British Columbia |

[27] | Akhatov, I.S.; Gazizov, R.K.; Ibragimov, N.K., Nonlocal symmetries. heuristic approach, J. sov. math, 1401-1450, (1991) · Zbl 0760.35002 |

[28] | Noether, E., Invariante variationsprobleme. nachr. König. gesell. wissen. Göttingen, Math-phys., Kl, 235-257, (1918) |

[29] | Anderson, R.L.; Kumei, S.; Wulfman, C.E., Generalization of the concept of invariance of differential equations, Phys. rev. lett., 28, 988-991, (1972) |

[30] | Vinogradov, A.M., Symmetries and conservation laws of partial differential equations: basic notions and results, Acta applic. math., 15, 3-21, (1989) · Zbl 0692.35002 |

[31] | Krasil’shchik, I.S.; Vinogradov, A.M., Nonlocal symmetries and the theory of coverings: an addendum to A.M. Vinogradov’s local symmetries and conservation laws, Acta applic. math., 2, 79-96, (1984) · Zbl 0547.58043 |

[32] | Krasil’shchik, I.S.; Vinogradov, A.M., Nonlocal trends in the geometry of differential equations: symmetries, conservation laws, and Bäcklund transformations, Acta applic. math., 15, 161-209, (1989) · Zbl 0692.35003 |

[33] | G.W. Bluman, Invariance of conserved forms under contact transformations, (Preprint) (1992) |

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