Abstract
The algebra of higher symmetries and the space of conservation laws for Zakharov's nonlinear equations of the interaction between long and short waves are completely described. The scheme of computations due to Vinogradov is used. As a result, the local nonintegrability of these equations is proved.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
VinogradovA. M.: Symmetries and conservation laws of partial differential equations. Basic notions and results,Acta Appl. Math. 15 (1989), 3–21.
ZakharovV. E.: Collapse of Langmuir waves,Zh. Exper. i Teor. Phiz. 62 (1972), 1745–1759 (in Russian).
Zakharov, V. E. and Rubenchick, A. M.: On the nonlinear interaction of high-frequency and low-frequency waves,Zh. Prikl. Mech. i Techn. Phiz. No. 5 (1972), 84–98 (in Russian).
DavydovA. S.:Solitons in Molecular Systems, D. Reidel, Dordrecht, 1985.
PetrovV. V.: Interaction of internal waves and small-scale surface turbulence in ocean,Izvestia AN SSSR. Phiz. Atmosph. i Okeana. 14 (1978), 342–347 (in Russian).
ShulmanE. I.: On the integrability of the equations of the short wave-long wave resonant interaction,Dokl. AN SSSR 259 (1981), 579–581 (in Russian).
ZakharovV. E. and ShulmanE. I.: Degenerate dispersion laws, motion invariance, and kinetic equations,Physica D 1 (1980), 192–202.
VinogradoyA. M.: Integrability and symmetries, in A. V.Gaponov-Grekhov and M. I.Rabinovich (eds.),Nonlinear Waves. Structures and Bifurcations, Nauka, Moscow, 1987, pp. 279–290 (in Russian).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Verbovetsky, A.M. Local nonintegrability of long-short wave interaction equations. Acta Appl Math 15, 121–136 (1989). https://doi.org/10.1007/BF00131932
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00131932