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

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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.

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References

  1. Kumei, S., Bluman, G.W.: When nonlinear differential equations are equivalent to linear differential equations. SIAM J. Appl. Math. 42, 1157–1173 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  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)

    MATH  MathSciNet  Google Scholar 

  4. Bluman, G.W., Doran-Wu, P.: The use of factors to discover potential systems or linearizations. Acta Appl. Math. 41, 21–43 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  5. Anco, S.C., Bluman, G.W.: Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78, 2869–2873 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  8. Wolf, T.: A comparison of four approaches to the calculation of conservation laws. Eur. J. Appl. Math. 13, 129–152 (2002)

    Article  MATH  Google Scholar 

  9. Anco, S.C.: Conservation laws of scaling-invariant field equations. J. Phys. A: Math. Gen. 36, 8623–8638 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Chapter  Google Scholar 

  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. Bluman, G.W.: Connections between symmetries and conservation laws. SIGMA 1, 16 (2005). Paper 011

    MathSciNet  Google Scholar 

  13. Bäcklund, A.V.: Über Flächentransformationen. Math. Ann. 9, 297–320 (1876)

    Article  Google Scholar 

  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)

  15. Anco, S.C., Bluman, G.W.: Derivation of conservation laws from nonlocal symmetries of differential equations. J. Math. Phys. 37, 2361–2375 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  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)

    Article  MATH  MathSciNet  Google Scholar 

  18. Kersten, P.H.M.: Infinitesimal symmetries: a computational approach. CWI Tract No. 34, Centrum voor Wiskunde en Informatica, Amsterdam (1987)

  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)

    Article  MathSciNet  Google Scholar 

  20. 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

  21. Varley, E., Seymour, B.: Exact solutions for large amplitude waves in dispersive and dissipative systems. Stud. Appl. Math. 72, 241–262 (1985)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Brock University, St. Catharines, ON, L2S 3A1, Canada

    Stephen Anco & Thomas Wolf

  2. Department of Mathematics, University of British Columbia, Vancouver, V6T 1Z2, Canada

    George Bluman

Authors
  1. Stephen Anco
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  2. George Bluman
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  3. Thomas Wolf
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Correspondence to George Bluman.

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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

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  • DOI: https://doi.org/10.1007/s10440-008-9205-7

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