Let's Lie: A Miraculous Haul of Fishes

M. C. Nucci

Teoretičeskaâ i matematičeskaâ fizika, Tome 144 (2005) no. 2, pp. 394-404

Résumé

In the recent literature, several equations have been studied with purported new approaches because the authors claim that these equations were not amenable to exact treatment using known methods. But we show that all these equations have sufficient Lie point symmetries to make them integrable by quadrature, if not linearizable. When one gets a “miraculous haul of fishes”, namely, exact methods of solution, first integrals, even linearization, then Lie symmetries shall be found. Lie group analysis was and should still be considered an essential tool for anyone who wants to solve equations of relevance in physics and other scientific fields.

Détail

Keywords: Lie group analysis, first integrals.
Mots-clés : Jacobi's last multiplier

Bibliographie

[1] Notices of AMS, 48 (2001), 416–417 ; {http://www.ams.org/notices/200104} | MR | Zbl

[2] T. Hawkins, The Emergence of the Theory of Lie Groups, An Essay in the History of Mathematics 1869–1926, Springer, Berlin, 2000 | MR

[3] T. Hawkins, Arch. Hist. Exact Sciences, 42 (1991), 187–278 | DOI | MR | Zbl

[4] S. Lie, Theorie der Transformationsgruppen, Erster Abschnitt, Teubner, Leipzig, 1888; Zweiter Abschnitt, Teubner, Leipzig, 1890 ; Dritter und Letzter Abschnitt, Teubner, Leipzig, 1893 | Zbl

[5] S. Lie, Geometrie der Berührungstransformationen, Teubner, Leipzig, 1896 | Zbl

[6] S. Lie, Vorlesungen über Differentialgleichungen mit bekannten infinitesimalen Transformationen, Teubner, Leipzig, 1912 | Zbl

[7] W. F. Ames, Nonlinear Partial Differential Equations in Engineering, V. 2, Academic Press, N.Y., 1972 ; G. W. Bluman, J. D. Cole, Similarity Methods for Differential Equations, Springer, Berlin, 1974 ; Л. В. Овсянников, Групповой анализ дифференциальных уравнений, Наука, М., 1978 ; П. Олвер, Приложения групп Ли к дифференциальным уравнениям, Мир, М., 1989 ; G. W. Bluman, S. Kumei, Symmetries and Differential Equations, Springer, Berlin, 1989 ; C. Rogers, W. F. Ames, Nonlinear Boundary Value Problems in Science and Engineering, Academic Press, N.Y., 1989 ; H. Stephani, Differential Equations. Their Solution Using Symmetries, Cambridge Univ. Press, Cambridge, 1989 ; J. M. Hill, Differential Equations and Group Methods for Scientists and Engineers, CRC Press, Boca Raton, 1992 ; N. H. Ibragimov (ed.), CRC Handbook of Lie Group Analysis of Differential Equations. V. I. Symmetries, Exact Solutions, and Conservation Laws, CRC Press, Boca Raton, 1994 ; V. II. Applications in Engineering and Physical Sciences, CRC Press, Boca Raton, 1995 ; N. H. Ibragimov, Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, N.Y., 1999 ; P. E. Hydon, Symmetry Methods for Differential Equations: A Beginner's Guide, Cambridge Univ. Press, Cambridge, 2000 | MR | Zbl | MR | Zbl | MR | MR | Zbl | MR | Zbl | MR | MR | Zbl | MR | Zbl | MR | Zbl | Zbl | MR | Zbl | MR

[8] F. González-Gascón, A. González-López, J. Math. Phys., 24 (1983), 2006–2021 | DOI | MR | Zbl

[9] M. C. Nucci, J. Math. Phys., 37 (1996), 1772–1775 | DOI | MR | Zbl

[10] V. Torrisi, M. C. Nucci, “Application of Lie group analysis to a mathematical model which describes HIV transmission”, The Geometrical Study of Differential Equations, eds. J. A. Leslie, T. P. Hobart, AMS, Providence, 2001, 31–40 | MR

[11] M. C. Nucci, J. Math. Phys., 44 (2003), 4107–4118 ; P. G. L. Leach, M. C. Nucci, J. Math. Phys., 45 (2004), 3590–3604 | DOI | MR | Zbl | DOI | MR | Zbl

[12] M. C. Nucci, J. Phys. A, 37 (2004), 11391–11400 | DOI | MR | Zbl

[13] M. Marcelli, M. C. Nucci, J. Math. Phys., 44 (2003), 2111–2132 | DOI | MR | Zbl

[14] E. Noether, Kgl. Ges. Wiss. Nach. Math-phys. Kl., 2 (1918), 235–269

[15] N. Kh. Ibragimov, Opyt gruppovogo analiza obyknovennykh differentsialnykh uravnenii, Znanie, M., 1991 | MR | Zbl

[16] M. C. Nucci, “Interactive REDUCE programs for calculating Lie point, non-classical, Lie-Bäcklund, and approximate symmetries of differential equations: manual and floppy disk”, CRC Handbook of Lie Group Analysis of Differential Equations. V. 3. New Trends, ed. N. H. Ibragimov, CRC Press, Boca Raton, 1996, 415–481 | MR

[17] V. K. Chandrasekar, M. Senthilvelan, M. Lakshmanan, J. Phys. A, 37 (2004), 4527–4534 | DOI | MR | Zbl

[18] M. Senthil Velan, M. Lakshmanan, J. Phys A, 28 (1995), 1929–1942 | DOI | MR | Zbl

[19] L. Hsu, N. Kamran, Lett. Math. Phys., 15 (1988), 91–99 | DOI | MR | Zbl

[20] M. C. Nucci, Lie will out. Dipartimento di Matematica e Informatica, Università di Perugia, 06123 Perugia, Italy, 2004, in preparation

[21] C. G. J. Jacobi, Giornale Arcadico di Scienze, Lettere ed Arti, 99 (1844), 129–146; J. Reine Angew. Math., 27 (1844), 199–268 ; 29 (1845), 213–279 ; 333–376 ; Vorlesungen über Dynamik, Druck und Verlag von Georg Reimer, Berlin, 1886 | DOI | MR | Zbl | DOI | MR | Zbl | Zbl

[22] L. Bianchi, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Enrico Spoerri, Pisa, 1918

[23] L. G. S. Duarte, S. E. S. Duarte, L. A. C. P. da Mota, J. E. F. Skea, J. Phys. A, 34 (2001), 3015–3024 | DOI | MR | Zbl

[24] F. Calogero, Nuovo Cimento B, 43 (1978), 177–241 | DOI | MR

[25] F. Calogero, Physica D, 152–153 (2001), 78–84 | DOI | MR | Zbl

[26] V. E. Zakharov, “On the dressing method”, Inverse Methods in Action, Proc. of Multicennials Meeting on Inverse Problems (Montpellier, 27 November – 1 December, 1989), ed. P. C. Sabatier, Springer, Berlin, 1990, 602–623 | DOI | MR

[27] A. González-López, J. Math. Phys., 29 (1988), 1097–1105 | DOI | MR | Zbl

[28] M. E. Fels, Proc. Lond. Math. Soc. III. Ser., 71 (1995), 221–240 | DOI | MR | Zbl

[29] C. W. Soh, F. M. Mahomed, Int. J. Nonlinear Mech., 36 (2001), 671–677 | DOI | MR | Zbl