Lagrangian Approach to Evolution Equations: Symmetries and Conservation Laws
- 283 Downloads
We show that one can apply a Lagrangian approach to certain evolution equations by considering them together with their associated equations. Consequently, one can employ Noether's theorem and derive conservation laws from symmetries of coupled systems of evolution equations. We discuss in detail the linear and non-linear heat equations as well as the Burgers equation and obtain new non-local conservation laws for the non-linear heat and the Burgers equations by extending their symmetries to the associated equations. We also provide Lagrangians for non-linear Schrödinger and Korteweg—de Vries type systems.
Unable to display preview. Download preview PDF.
- 1.Lie, S., Collected Works, Vols. 1–6, B. G. Teubner-H. Aschehoug, Leipzig-Oslo, 1922–1937 (in German).Google Scholar
- 2.Ovsiannikov, L. V., Group Analysis of Differential Equations, Nauka, Moscow, 1978 [English translation by Academic Press, 1982].Google Scholar
- 3.Ibragimov, N. H., Transformation Groups Applied to Mathematical Physics, Nauka, Moscow, 1983 [English translation D. Reidel, Dordrecht, 1985].Google Scholar
- 4.Olver, P. J., Applications of Lie Groups to Differential Equations, Springer, New York, 1986. (second ed., 1993).Google Scholar
- 5.Bluman, G. and Kumei, S., Symmetries and Differential Equations, Springer, New York, 1989.Google Scholar
- 6.Ibragimov, N. H., Elementary Lie Group Analysis and Ordinary Differential Equations, Wiley, Chichester, UK, 1999.Google Scholar
- 7.Ibragimov, N. H. (editor), CRC Handbook of Lie Group Analysis of Differential Equations, Vol. 1: Symmetries, Exact Solutions and Conservation Laws, CRC Press, Boca Raton, Florida, 1994.Google Scholar
- 8.Noether, E., ‘Invariante Variationsprobleme’, König. Gesell. Wissen., Göttingen, Math.-Phys. Kl., 1918, 235–257.Google Scholar
- 9.Dorodnitsyn, V. A. and Schvirshchevskii, S. R., ‘On Lie-Bäcklund groups admitted by the heat equation with a source’. Preprint 101, Keldysh Institute of Applied Mathematics, USSR Academy of Sciences, Moscow, 1983.Google Scholar
- 10.Brandão, A. and Kolsrud T., Time-dependent conservation laws and symmetries for classical mechanics and heat equations. Harmonic Morphisms, Harmonic Maps, and Related Topics (Brest, 1997), Chapman & Hall/CRC Press, Boca Raton, Florida, 2000, pp 113–125, Notes Mathematics, Vol. 413.Google Scholar
- 11.Danilov, Yu. A., ‘Group analysis of Turing systems and their analogs’, Preprint 3287, Institute of Atomic Energy, USSR Academy of Sciences, Moscow, 1980.Google Scholar
- 12.Anderson, R. L. and Ibragimov, N. H., Lie-Bäcklund Transformations in Applications, SIAM, Philadelphia, Pennsylvania, 1979.Google Scholar