Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T18:02:48.183Z Has data issue: false hasContentIssue false
  • English
  • Français

Symmetry-invariant conservation laws of partial differential equations†

Published online by Cambridge University Press:  13 March 2017

STEPHEN C. ANCO  and
ABDUL H. KARA
STEPHEN C. ANCO
Affiliation:
Department of Mathematics and Statistics, Brock University St. Catharines, ON L2S3A1, Canada email: [email protected]
ABDUL H. KARA
Affiliation:
School of Mathematics, University of the Witwatersrand Wits 2050, Johannesburg, South Africa email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A simple characterization of the action of symmetries on conservation laws of partial differential equations is studied by using the general method of conservation law multipliers. This action is used to define symmetry-invariant and symmetry-homogeneous conservation laws. The main results are applied to several examples of physically interest, including the generalized Korteveg-de Vries equation, a non-Newtonian generalization of Burger's equation, the b-family of peakon equations, and the Navier–Stokes equations for compressible, viscous fluids in two dimensions.

Keywords

conservation lawsymmetrysymmetry-invariantsymmetry-homogeneousmultipliers
Type
Papers
Information
European Journal of Applied Mathematics , Volume 29 , Issue 1 , February 2018 , pp. 78 - 117
Copyright
Copyright © Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

S.C. Anco is supported by an NSERC research grant.

References

[1] Anco, S. C. & Bluman, G. (1997) Direct construction of conservation laws from field equations. Phys. Rev. Lett. 78 (15), 2869–2873.CrossRefGoogle Scholar
[2] Anco, S. C. & Bluman, G. (2002) Direct construction method for conservation laws of partial differential equations. I. Examples of conservation law classifications. Euro. J. Appl. Math. 13, 545–566.CrossRefGoogle Scholar
[3] Anco, S. C. & Bluman, G. (2002) Direct construction method for conservation laws of partial differential equations. II. General treatment. Euro. J. Appl. Math. 13, 567–585.CrossRefGoogle Scholar
[4] Olver, P. (1986) Applications of Lie Groups to Differential Equations, Springer-Verlag, New York.CrossRefGoogle Scholar
[5] Bluman, G. & Anco, S. C. (2002) Symmetry and Integration Methods for Differential Equations, Springer Applied Mathematics Series, Vol. 154, Springer-Verlag, New York.Google Scholar
[6] Bluman, G., Cheviakov, A. & Anco, S. C. (2010) Applications of Symmetry Methods to Partial Differential Equations, Springer Applied Mathematics Series, Vol. 168, Springer, New York.CrossRefGoogle Scholar
[7] Martinez Alonso, L. (1979) On the Noether map. Lett. Math. Phys. 3, 419–424.CrossRefGoogle Scholar
[8] Vinogradov, A. M. (1984) Local symmetries and conservation laws. Acta Appl. Math. 2 (1), 21–78.CrossRefGoogle Scholar
[9] Anco, S. C. (2017) Generalization of Noether's theorem in modern form to non-variational partial differential equations, In: Melnik, R., Makarov, R., Belair, J. (editors), Recent progress and Modern Challenges in Applied Mathematics, Modeling and Computational Science, Fields Institute Communications, Vol. 79.Google Scholar
[10] Bluman, G., Cheviakov, A. & Anco, S. C. (2008) Construction of conservation laws: How the direct method generalizes Noether's theorem. In: Group Analysis of Differential Equations and Integrable Systems (Proceedings of the 4th International Workshop), 2008, Cyprus, pp. 13–35.Google Scholar
[11] Ibragimov, N. H. (1985) Transformation Groups Applied to Mathematical Physics. (translated from Russian), Mathematics and its Applications (Soviet Series), Reidel, Dordrecht.CrossRefGoogle Scholar
[12] Khamitova, R. S. (1982) The structure of a group and the basis of conservation laws. (Russian) Teoret. Mat. Fiz. 52 (2), 244–251; English translation Theoret. Math. Phys. 52(2), 777–781.Google Scholar
[13] Ibragimov, N. H., Kara, A. H. & Mahomed, F. M. (1998) Lie-Bäcklund and Noether symmetries with applications. Nonlinear Dyn. 15 (2), 115–136.CrossRefGoogle Scholar
[14] Bluman, G. Temuerchaolu & Anco, S. C. (2006) New conservation laws obtained directly from symmetry action on known conservation laws. J. Math. Anal. Appl. 322, 233–250.CrossRefGoogle Scholar
[15] Kara, A. H. & Mahomed, F. M. (2002) A basis of conservation laws for partial differential equations. J. Nonlinear Math. Phys. 9, 60–72.CrossRefGoogle Scholar
[16] Kara, A. H. & Mahomed, F. M. (2000) Relationship between symmetries and conservation laws. Int. J. Theor. Phys. 39, 23–40.CrossRefGoogle Scholar
[17] Anco, S. C. (2016) Symmetry properties of conservation laws. Int. J. Mod. Phys. B 30, 1640004 (12pp).CrossRefGoogle Scholar
[18] Schwarz, F. (1998) Janet bases for symmetry groups. In: Buchberger, B. & Winkler, F. (editors), Groebner Bases and Applications, Lecture Notes Series, Vol. 251, London Mathematical Society, pp. 221–234.CrossRefGoogle Scholar
[19] Schwarz, F. (2007) Algorithmic Lie Theory for Solving Linear Ordinary Differential Equations, Chapman & Hall/CRC.CrossRefGoogle Scholar
[20] Verbotevsky, A. (1997) Notes on the horizontal cohomology. In: Henneaux, M., Krasil'shchik, J., Vinogradov, A. (editors), Secondary Calculus and Cohomological Physics, Contemporary Mathematics, Vol. 219, American Mathematical Society, Providence, pp. 211–232.Google Scholar
[21] Wolf, T. (2002) A comparison of four approaches to the calculation of conservation laws. Euro. J. Appl. Math. 13, 129–152.CrossRefGoogle Scholar
[22] Anco, S. C. (2003) Conservation laws of scaling-invariant field equations. J. Phys. A: Math. Gen. 36, 8623–8638.CrossRefGoogle Scholar
[23] Deconinck, B. & Nivala, M. (2009) Symbolic integration and summation using homotopy methods. Math. Comput. Simul. 80, 825–836.CrossRefGoogle Scholar
[24] Poole, D. & Hereman, W. (2010) The homotopy operator method for symbolic integration by parts and inversion of divergences with applications. Appl. Anal. 89, 433–455.CrossRefGoogle Scholar
[25] Caviglia, G. (1986) Symmetry transformations, isovectors, and conservation laws. J. Math. Phys. 27, 972–978.CrossRefGoogle Scholar
[26] Lunev, F. A. (1990) An analogue of the Noether theorem for non-Noether and nonlocal symmetries. (Russian) Teoret. Mat. Fiz. 84 (2), 205–210; English translation Theoret. Math. Phys. 84(2), 816–820.Google Scholar
[27] Zharinov, V. V. (1992) Lecture Notes on Geometrical Aspects of Partial Differential Equations, Series on Soviet and East European Mathematics, Vol. 9, World Scientific, River Edge, NJ.CrossRefGoogle Scholar
[28] Anco, S. C. & Bluman, G. (1996) Derivation of conservation laws from nonlocal symmetries of differential equations. J. Math. Phys. 37, 2361–2375.CrossRefGoogle Scholar
[29] Krook, M. & Wu, T. T. (1976) Formation of Maxwellian tails. Phys. Rev. Lett. 36, 1107–1109.CrossRefGoogle Scholar
[30] Euler, N., Leach, P. G. L., Mahomed, F. M. & Steeb, W.-H. (1988) Symmetry vector fields and similarity solutions of a nonlinear field equation describing the relaxation to a Maxwell distribution. Int. J. Theor. Phys. 27, 717–723.CrossRefGoogle Scholar
[31] Batchelor, G. K. (1967) An Introduction to Fluid Dynamics, Cambridge University Press.Google Scholar
[32] Dullin, H. R., Gottwald, G. A. & Holm, D. D. (2004) On asymptotically equivalent shallow water wave equations. Physica D 190, 1–14.CrossRefGoogle Scholar
[33] Camassa, R. & Holm, D. D. (1993) An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664.CrossRefGoogle ScholarPubMed
[34] Degaspersis, A. & Procesi, M. (1999) Asymptotic integrability. In: Degaspersis, A. & Gaeta, G. (editors), Symmetry and Perturbation Theory, World Scientific, pp. 23–37.Google Scholar
[35] Degaspersis, A., Hone, A. N. W. & Holm, D. D. (2003) Integrable and non-integrable equations with peakons. In: Ablowitz, M. J., Boiti, M., Pempinelli, F. & Prinari, B. (editors), Nonlinear Physics: Theory and Experiment II, World Scientific, pp. 37–43.CrossRefGoogle Scholar
[36] Singh, K., Gupta, R. K. & Kumar, S. (2011) Exact solutions of b-family equation: Classical Lie approach and direct method. Inter. J. Nonlin. Sci. 11, 59–67.Google Scholar