Acta Applicandae Mathematicae

, Volume 101, Issue 1–3, pp 21–38 | Cite as

Invertible Mappings of Nonlinear PDEs to Linear PDEs through Admitted Conservation Laws



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.


Conservation laws Linearization Symmetries 

Mathematics Subject Classification (2000)

35A30 58J70 35L65 35A34 22E65 70H33 


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  1. 1.
    Kumei, S., Bluman, G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1982) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bluman, G.W., Kumei, S.: Symmetry based algorithms to relate partial differential equations: I. Local symmetries. Eur. J. Appl. Math. 1, 189–216 (1990) MATHMathSciNetGoogle Scholar
  3. 3.
    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) MATHMathSciNetGoogle Scholar
  4. 4.
    Bluman, G.W., Doran-Wu, P.: The use of factors to discover potential systems or linearizations. Acta Appl. Math. 41, 21–43 (1995) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Anco, S.C., Bluman, G.W.: Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78, 2869–2873 (1997) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    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) MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    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) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13, 129–152 (2002) MATHCrossRefGoogle Scholar
  9. 9.
    Anco, S.C.: Conservation laws of scaling-invariant field equations. J. Phys. A: Math. Gen. 36, 8623–8638 (2003) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    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) CrossRefGoogle Scholar
  11. 11.
    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) Google Scholar
  12. 12.
    Bluman, G.W.: Connections between symmetries and conservation laws. SIGMA 1, 16 (2005). Paper 011 MathSciNetGoogle Scholar
  13. 13.
    Bäcklund, A.V.: Über Flächentransformationen. Math. Ann. 9, 297–320 (1876) CrossRefGoogle Scholar
  14. 14.
    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) Google Scholar
  15. 15.
    Anco, S.C., Bluman, G.W.: Derivation of conservation laws from nonlocal symmetries of differential equations. J. Math. Phys. 37, 2361–2375 (1996) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    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) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    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) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Kersten, P.H.M.: Infinitesimal symmetries: a computational approach. CWI Tract No. 34, Centrum voor Wiskunde en Informatica, Amsterdam (1987) Google Scholar
  19. 19.
    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) MathSciNetCrossRefGoogle Scholar
  20. 20.
    Wittkopf, A.D.: Algorithms and implementations for differential elimination. Ph.D. Thesis, Department of Mathematics, Simon Fraser University (2004).
  21. 21.
    Varley, E., Seymour, B.: Exact solutions for large amplitude waves in dispersive and dissipative systems. Stud. Appl. Math. 72, 241–262 (1985) MATHMathSciNetGoogle Scholar

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© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of MathematicsBrock UniversitySt. CatharinesCanada
  2. 2.Department of MathematicsUniversity of British ColumbiaVancouverCanada

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