Abstract
A theorem providing necessary conditions enabling one to map a nonlinear system of first order partial differential equations to an equivalent first order autonomous and homogeneous quasilinear system is given. The reduction to quasilinear form is performed by constructing the canonical variables associated to the Lie point symmetries admitted by the nonlinear system. Some applications to relevant partial differential equations are given.
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References
Ovsiannikov, L.V.: Group Analysis of Differential Equations. Academic Press, New York (1982)
Ibragimov, N.H.: Transformation Groups Applied to Mathematical Physics. D. Reidel Publishing Company, Dordrecht (1985)
Olver, P.J., Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1986)
Baumann, G.: Symmetry Analysis of Differential Equations with Mathematica. Springer, New York (2000)
Bluman, G.W., Anco, S.C.: Symmetry and Integration Methods for Differential Equations. Springer, New York (2002)
Bluman, G.W., Cheviakov, A.F., Anco, S.C.: Applications of Symmetry Methods to Partial Differential Equations. Springer, New York (2009)
Donato, A., Oliveri, F.: Linearization procedure of nonlinear first order systems of PDE’s by means of canonical variables related to Lie groups of point transformations. J. Math. Anal. Appl. 188, 552–568 (1994)
Donato, A., Oliveri, F.: When nonautonomous equations are equivalent to autonomous ones. Appl. Anal. 58, 313–323 (1995)
Donato, A., Oliveri, F.: How to build up variable transformations allowing one to map nonlinear hyperbolic equations into autonomous or linear ones. Transp. Th. Stat. Phys. 25, 303–322 (1996)
Currò, C., Oliveri, F.: Reduction of nonhomogeneous quasilinear \(2\times 2\) systems to homogeneous and autonomous form. J. Math. Phys 49, 103504-1–103504-11 (2008)
Oliveri, F.: Lie symmetries of differential equations: classical results and recent contributions. Symmetry 2, 658–706 (2010)
Oliveri, F.: General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form. Int. J. Non-linear Mech. 47, 53–60 (2012)
Oliveri, F.: Linearizable second order Monge–Ampère equations. J. Math. Anal. Appl. 218, 329–345 (1998)
Boillat, G.: Le champ scalaire de Monge–Ampère. Det. Kg. Norske Vid. Selsk. Forth. 41, 78–81 (1968)
Von Kármán, T.: Compressibility effects in aerodynamycs. J. Aeron. Sci. 8, 337–356 (1941)
Ruggeri, T.: Su una naturale estensione a tre variabili dell’equazione di Monge–Ampère. Rend. Accad. Naz. Lincei 55, 445–449 (1973)
Donato, A., Ramgulam, U., Rogers, C.: The \((3+1)\)-dimensional Monge–Ampère equation in discontinuity wave theory: application of a reciprocal transformation. Meccanica 27, 257–262 (1992)
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Work supported by “Gruppo Nazionale per la Fisica Matematica” (G.N.F.M.) of the “Istituto Nazionale di Alta Matematica” (I.N.d.A.M.).
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Gorgone, M., Oliveri, F. Nonlinear first order partial differential equations reducible to first order homogeneous and autonomous quasilinear ones. Ricerche mat 66, 51–63 (2017). https://doi.org/10.1007/s11587-016-0286-8
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DOI: https://doi.org/10.1007/s11587-016-0286-8