Abstract
An algorithmic method using conservation law multipliers is introduced that yields necessary and sufficient conditions to find invertible mappings of a given nonlinear PDE to some linear PDE and to construct such a mapping when it exists. Previous methods yielded such conditions from admitted point or contact symmetries of the nonlinear PDE. Through examples, these two linearization approaches are contrasted.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Kumei, S., Bluman, G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1982)
Bluman, G.W., Kumei, S.: Symmetry based algorithms to relate partial differential equations: I. Local symmetries. Eur. J. Appl. Math. 1, 189–216 (1990)
Bluman, G.W., Kumei, S.: Symmetry based algorithms to relate partial differential equations: II. Linearization by nonlocal symmetries. Eur. J. Appl. Math. 1, 217–223 (1990)
Bluman, G.W., Doran-Wu, P.: The use of factors to discover potential systems or linearizations. Acta Appl. Math. 41, 21–43 (1995)
Anco, S.C., Bluman, G.W.: Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78, 2869–2873 (1997)
Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations. Part I: Examples of conservation law classifications. Eur. J. Appl. Math. 13, 545–566 (2002)
Anco, S.C., Bluman, G.W.: Direct construction method for conservation laws of partial differential equations. Part II: General treatment. Eur. J. Appl. Math. 13, 567–585 (2002)
Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13, 129–152 (2002)
Anco, S.C.: Conservation laws of scaling-invariant field equations. J. Phys. A: Math. Gen. 36, 8623–8638 (2003)
Wolf, T.: Partial and complete linearization of PDEs based on conservation laws. In: Wang, D., Zheng, Z. (eds.) Trends in Mathematics: Differential Equations with Symbolic Computation, pp. 291–306. Birkhäuser, Basel (2005)
Wolf, T.: Applications of Crack in the classification of integrable systems. In: CRM Proceedings and Lecture Notes, vol. 37, pp. 283–300. AMS, Providence (2004)
Bluman, G.W.: Connections between symmetries and conservation laws. SIGMA 1, 16 (2005). Paper 011
Bäcklund, A.V.: Über Flächentransformationen. Math. Ann. 9, 297–320 (1876)
Müller, E.A., and Matschat, K.: Über das Auffinden von Ähnlichkeitslösungen partieller Differentialgleichungssysteme unter Benützung von Transformationsgruppen, mit Anwendungen auf Probleme der Strömungsphysik. In: Miszellaneen der Angewandten Mechanik, pp. 190–222. Berlin (1962)
Anco, S.C., Bluman, G.W.: Derivation of conservation laws from nonlocal symmetries of differential equations. J. Math. Phys. 37, 2361–2375 (1996)
Bluman, G.W., Temuerchaolu, Anco, S.C.: New conservation laws obtained directly from symmetry action on a known conservation law. J. Math. Anal. Appl. 322, 233–250 (2006)
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 Appl. Math. 2, 79–96 (1984)
Kersten, P.H.M.: Infinitesimal symmetries: a computational approach. CWI Tract No. 34, Centrum voor Wiskunde en Informatica, Amsterdam (1987)
Reid, G.J., Wittkopf, A.D., Boulton, A.: Reduction of systems of nonlinear partial differential equations to simplified involutive forms. Eur. J. Appl. Math. 7, 604–635 (1996)
Wittkopf, A.D.: Algorithms and implementations for differential elimination. Ph.D. Thesis, Department of Mathematics, Simon Fraser University (2004). http://ir.lib.sfu.ca:8080/retrieve/205/etd0400.pdf
Varley, E., Seymour, B.: Exact solutions for large amplitude waves in dispersive and dissipative systems. Stud. Appl. Math. 72, 241–262 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
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). https://doi.org/10.1007/s10440-008-9205-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10440-008-9205-7