Abstract
Two countable sets of integrable dynamical systems which turn into the Korteweg-de Vries equation in a continous limit are constructed. The integrability of the dynamics of the scattering matrix entries for these systems is proved and an integrable reduction in the finitedimensional case is pointed out. A construction of the integrable dynamical systems connected with the simple Lie algebras and generalizing the discrete kdV equation is presented. Two general constructions of differential and integro-differential equations (with respect to time t) possessing a countable set of first integrals are found. These equations admit the Lax representation in some infinite-dimensional subalgebras of the Lie algebra of integral operators on an arbitrary manifold M n with measure μ. A construction of matrix equations having a set of attractors in the space of all matrix entries is given.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Ablowitz, M. J., Kaup, D. J., Newell, A. C., and Segur, H.: Nonlinear evolution equation of physical significance, Phys. Rev. Lett. 32 (1973), 125–130.
Bogoyavlensky, O. I.: On perturbations of the periodic Toda lattice, Commun. Math. Phys. 51 (1976), 201–209.
Bogoyavlensky, O. I.: New integrable problem of classical mechanics, Commun. Math. Phys. 94 (1984), 255–269.
Bogoyavlensky, O. I.: Some constructions of integrable dynamical systems, Sov. Izvest. Acad. Sci. 51 (1987). 737–766.
Bogoyavlensky, O. I.: Integrable dynamical systems connected with the KdV equation, Sov. Izvest. Acad. Sci. 51 (1987), 1123–1141.
Bourbaki, N.: Groupes et algébres de Lie, Hermann, Paris, 1968.
Jamilov, R. I.: On classification of the discrete equations, in Integrable Systems, Ufa Univ. Press, 1982, pp. 95–114.
Kac, M. and Van Moerbeke, P.: On an explicitly soluble system of nonlinear differential equations related to certain Toda lattices, Adv. Math. 16 (1975), 160–169.
Lax, P.: Almost periodic behavior of nonlinear waves, Adv. Math. 16 (1975), 368–379.
Manakov, S. V.: On complete integrability and stochastization in discrete dynamical systems, Sov. ZETPh. 67 (1974), 543–555.
Manakov, S. V.: Note on the integration of the Euler equations of an n-dimensional rigid body dynamics, Sov. Funk. Anal. Appl. 10 (1976), 93–94.
Moser, J.: Three integrable systems connected with isospectral deformation, Adv. Math. 16 (1975), 197–220.
Toda, M.: Theory of nonlinear lattices, Springer, Berlin, Heidelberg, New York, 1981.
Vladimirov, V. S.: Generalized Functions in Mathematical Physics, Nauka, Moscow, 1978.
Zakharov, V. E., Manakov, S. V., Novikov, S. P., and Pitaevskij, L. E.: Theory of Solitons, Nauka, Moscow, 1980.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bogoyavlensky, O.I. Five constructions of integrable dynamical systems connected with the Korteweg-de Vries equation. Acta Appl Math 13, 227–266 (1988). https://doi.org/10.1007/BF00046965
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00046965