Hostname: page-component-77c89778f8-m42fx Total loading time: 0 Render date: 2024-07-19T04:36:01.241Z Has data issue: false hasContentIssue false
  • English
  • Français

Generalised symmetries of partial differential equations via complex transformations

Published online by Cambridge University Press:  17 April 2009

D. Catalano Ferraioli ,
G. Manno  and
F. Pugliese
D. Catalano Ferraioli
Affiliation:
Dip. di Matematica, Università di Milano, via Saldini, 20133 Milano, Italy, e-mail: catalano@mat.unimi.it
G. Manno
Affiliation:
Dip. di Matematica, Università di Lecce, via per Arnesano, 73100 Lecce, Italy, e-mail: gianni.manno@unile.it
F. Pugliese
Affiliation:
Dip. di Matematica e Informatica, Università di Salerno, via Ponte don Melillo, 84084 Fisciano (SA) Italy, e-mail: fpugliese@unisa.it
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We consider two systems of real analytic partial differential equations, related by a holomorphic contact map H. We study how the generalised symmetries of the first equation are mapped into those of the second one, and determine under which conditions on H such a map is invertible. As an application of these results, an example of physical interest is discussed.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 2 , October 2007 , pp. 243 - 262
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Bluman, G.W. and Kumei, S., Symmetries and differential equations, applied mathematical sciences 81 (Springer-Verlag, New York, 1989).CrossRefGoogle Scholar
[2]Bocharov, A. V., Chetverikov, V. N., Duzhin, S. V., Khor'kova, N. G., Krasil'shchik, I. S., Samokhin, A. V., Torkhov, Yu. N., Verbovetsky, A. M. and Vinogradov, A. M., Symmetries and conservation laws for differential equations of mathematical physics, (Krasilśhchik, I. S. and Vinogradov, A. M., Editors) (Amer. Math. Soc., Providence, R.I., 1999).CrossRefGoogle Scholar
[3]Bruhat, F. and Whitney, H., ‘Quelques propriétés fondamentales des ensembles analytiques-réels’, Comment. Math. Helv. 33 (1959), 132–160.Google Scholar
[4]Ferraioli, D. Catalano, Manno, G. and Pugliese, F., ‘Contact symmetries of the elliptic Euler-Darboux equation’, Note Mat. 23 (2004), 3–14.Google Scholar
[5]Dieudonné, J., Foundations of modern analysis (Academic Press, New York and London, 1960).Google Scholar
[6]Grauert, H. and Fritzsche, K., Several complex variables, Graduate Texts in Mathematics 38 (Springer-Verlag, New York, Heidelberg, 1976).CrossRefGoogle Scholar
[7]Krasilśhchik, I.S. and Verbovetsky, A.M., ‘Homological methods in equations of mathematical physics’, (Open Education and Sciences, Opava (Czech Rep.)). math.DG/9808130.Google Scholar
[8]Kulkarni, R.S., ‘On complexifications of differentiable manifolds’, Invent. Math. 44 (1978), 46–64.CrossRefGoogle Scholar
[9]Olver, P.J., Application of Lie groups to differential equations (Springer-Verlag, New York, 1993).CrossRefGoogle Scholar
[10]Rogers, C. and Shadwick, W. F., Baecklund transformations and their applications, Mathematics in Science and Engineering, 161 (Academic Press, New York, London, 1982).Google Scholar
[11]Rosenhaus, V., ‘The unique determination of the equation by its invariance group and field-space symmetry’, Algebras Groups Geom. 3 (1986), 148–166.Google Scholar
[12]Saunders, D. J., The geometry of jet bundles (Cambridge Univ. Press, Cambridge, 1989).CrossRefGoogle Scholar
[13]Shemarulin, V.E., ‘Higher symmetry algebra structure and local equivalences of Euler-Darboux equations’, (Russian), Dokl. Akad. Nauk. 330 (1993), 24–27. Translation in Russian Acad. Sci. Dokl. Math. 47 (1993) 383–388.Google Scholar
[14]Shemarulin, V.E., ‘Higher symmetries and conservation laws of Euler-Darboux equations’, in Geometry in partial differential equations (World Sci. Publishing, River Edge, NJ, 1994), pp. 389–422.CrossRefGoogle Scholar
[15]Shemarulin, V.E., ‘Higher symmetry algebra structures and local equivalences of Euler-Darboux equations’, in The interplay between differential geometry and differential equations, Amer. Math. Soc. Transl. Ser. 2 167 (Amer. Math. Soc., Providence, RI, 1995), pp. 217–243.Google Scholar
[16]Sparano, G., Vilasi, G. and Vinogradov, A.M., ‘Vacuum Einstein metrics with bidimensional Killing leaves I: local aspects’, Diff. Geom. Appl. 16 (2002), 95–120.CrossRefGoogle Scholar
[17]Sparano, G., Vilasi, G. and Vinogradov, A.M., ‘Vacuum Einstein metrics with bidimensional Killing leaves II: global aspects’, Diff. Geom. Appl. 17 (2002), 15–35.CrossRefGoogle Scholar
[18]Tognoli, A., ‘Sulla classificazione dei fibrati analitici reali’, Ann. Scuola Norm. Sup. Pisa 21 (1967), 709–743.Google Scholar
[19]Tomassini, G., ‘Tracce delle funzioni olomorfe sulle sottovarietà analitiche reali d'una varietà complessa’, Ann. Scuola Norm. Sup. Pisa 20 (1966), 31–43.Google Scholar
[20]Vinogradov, A.M., ‘Local symmetries and conservation laws’, Acta Appl. Math. 2 (1981), 21–78.CrossRefGoogle Scholar
[21]Vinogradov, A.M., ‘An informal introduction to the geometry of jet spaces’, Rend. Sem. Fac. Sci. Univ. Cagliari 58 (1988), 301–333.Google Scholar