Abstract
Kadomtsev-Pogutse equations are of great interest from the viewpoint of the theory of symmetries and conservation laws and, in particular, enable us to demonstrate their potentials ‘in action’. This paper presents, firstly, the results of computations of symmetries and conservation laws for these equations and the methods of obtaining these results. Apparently, all the local symmetries and conservation laws admitted by the considered equations are exhausted by those enumerated in this paper. Secondly, we point out some reductions of Kadomtsev-Pogutse equations to more simpler forms which have less independent variables and which, in some cases, allow us to construct exact solutions. Finally, the technique of solution deformation by symmetries and their physical interpretation are demonstrated.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
KadomtsevB. B. and PogutseO. P.:Zh. Eksper. Teoret. Fiz. 65 (1973), 2 (in Russian).
WhiteR., MonticelloD., RosenbluthM. N., StraussH., and KadomtsevB. B.: Numerical study of nonlinear evolution of kink and tearing modes in tokamaks,International Conference of Plasma Physics, Tokyo, 1974, IAEA Vienna, 1975, Vol. 1, pp. 495–504 3. Strauss, H. R.: Nonlinear three-dimensional magnetohydrodynamics of noncircular tokamaks,Phys. Fluids 19 (1976), 134–140.
WhiteR., MonticelloD. and RosenbluthM. N.: Simulation of large magnetic islands: A possible mechanism for major tokamak disruption,Phys. Rev. Lett. 39 (1977), 1618–1620.
VinogradovA. M.: Local symmetries and conservation laws,Acta Appl. Math. 2, (1984), 21–78.
SamokhinA. V.: Nonlinear MHD-equation: Symmetries, solutions and conservation laws,Dokl. Acad. Nauk SSSR 65 (1985), 1101–1106.
MarsdenJ. E. and MorrisonP. J.: Noncanonical Hamiltonian field theory and reduced MHD,Contemporary Mathematics, Fluids and Plasmas: Geometry and Dynamics. Proceedings of the AMS-IMS-SIAM Conference, Vol. 28, AMS, 1984, pp. 133–150.
KrasilshchikI. S., LychaginV. V., and VinogradovA. M.:Geometry of Jet Spaces and Nonlinear Partial Differential Equations, Gordon and Breach, New York, 1986.
Bocharov, A. V. and Bronshtein, M. L.: Efficiently implementing two method of the geometric theory of differential equations—an experience of algorithm and software design,Acta Appl. Math. to appear.
VinogradovA. M., Symmetries and conservation laws of partial differential equations: Basic notations and results,Acta. Appl. Math. 15 (1989), 3–21.
LiouvilleJ.: Sur la equation aux deriveés partielles\((\partial ^2 \ln \kappa )/(\partial u{\text{ }}\partial y) + (\kappa )/(2gz) = 0\),J. Math. Pures. Appl. 18 (1853), 71–72.
KostenkoV. G.: Integrating of a some differential equations in partial derivatives by group method, Lvov University, 1959 (Ukrainian).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Gusyatnikova, V.N., Samokhin, A.V., Titov, V.S. et al. Symmetries and conservation laws of Kadomtsev-Pogutse equations. Acta Appl Math 15, 23–64 (1989). https://doi.org/10.1007/BF00131929
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00131929