Abstract
The question of complete integrability of evolution equations associated ton×n first order isospectral operators is investigated using the inverse scattering method. It is shown that forn>2, e.g. for the three-wave interaction, additional (nonlinear) pointwise flows are necessary for the assertion of complete integrability. Their existence is demonstrated by constructing action-angle variables. This construction depends on the analysis of a natural 2-form and symplectic foliation for the groupsGL(n) andSU(n).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
[AKNS] Ablowitz, M. J., Kaup, D. J., Newell, A. C., Segur, H.: The inverse scattering transform-Fourier analysis for nonlinear problems. Stud. Appl. Math.53, 249–315 (1974)
[A] Arnold, V. I.: Mathematical methods of classical mechanics. Berlin, Heidelberg, New York: Springer 1989
[BY] Bar Yaacov, D.: Analytic properties of scattering and inverse scattering for first order systems, Dissertation, Yale 1985
[BC1] Beals, R., Coifman, R. R.: Scattering and inverse scattering for first order systems. Commun. Pure Appl. Math.37, 39–90 (1984)
[BC2] Beals, R., Coifman, R. R.: Inverse scattering and evolution equations. Commun. Pure Appl. Math.38, 29–42 (1985)
[BC3] Beals, R., Coifman, R. R.: Linear spectral problems, nonlinear equations, and the\(\bar \partial \)-method. Inverse Problems5, 87–130 (1989).
[Ca] Caudrey, P. J.: The inverse problem for a generaln×n spectral equation. PhysicaD6, 51–66 (1982)
[Co] Copson, E. T.: Theory of functions of a complex variable. Oxford: Oxford University Press 1935
[Dr] Drinfeld, V. G.: Quantum groups. In: Proc. International Congress of Mathematicians. Berkeley 1986
[Ga] Gardner, C. S.: Korteweg-de Vries equation and generalizations, IV. The Korteweg-de Vries equation as a Hamiltonian system. J. Math. Phys.12, 1548–1551 (1971)
[Ge] Gerdzhikov, V. S.: On the spectral theory of the integro-differential operator Λ generating nonlinear evolution equations. Lett. Math. Phys.6, 315–323 (1982)
[Ka] Kaup, D. J.: The three-wave interaction—a non-dispersive phenomenon. Stud. Appl. Math.55, 9–44 (1976)
[KD] Konopelchenko, B. G., Dubrovsky, V. G.: Hierarchy of Poisson brackets for elements of a scattering matrix. Lett. Math. Phys.8, 273 (1984)
[La] Lax, P. D.: Almost periodic solutions of the KdV equation. SIAM Rev.18, 351–375 (1976)
[Lu] Lu, J-h.: personal communication
[Ma] Manakov, S. V.: An example of a completely integrable nonlinear wave field with nontrivial dynamics (Lee Model). Teor. Mat. Phys.28, 172–179 (1976)
[Ne] Newell, A. C.: The general structure of integrable evolution equations. Proc. R. Soc. Lond.A365, 283–311 (1979)
[Sa] Sattinger, D. H.: Hamiltonian hierarchies on semisimple Lie algebras. Stud. Appl. Math.72, 65–86 (1985)
[Sh] Shabat, A. B.: An inverse scattering problem. Diff. Uravn.15, 1824–1834 (1978); Diff. Eq.13, 1299–1307 (1980)
[Sk] Sklyanin, E. K.: Quantum variant of the method of the inverse scattering problem. Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov95, 55–128, 161 (1980)
[Wh] Whittaker, E. T.: Analytical Mechanics. New York: Dover 1944
[ZF] Zakharov, V. E., Faddeev, L. D.: Korteweg-de Vries equation: A completely integrable Hamiltonian system. Funct. Anal. Appl.5, 280–287 (1971)
[ZM1] Zakharov, V. E., Manakov, S. V.: On resonant interaction of wave packets in nonlinear media. ZhETF Pis. Red.18, 413 (1973)
[ZM2] Zakharov, V. E., Manakov, S. V.: On the complete integrability of the nonlinear Schrödinger equation. Teor. Mat. Fyz.19, 332–343 (1974)
[ZS1] Zakharov, V. E., Shabat, A. B.: Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP34, 62–69 (1972)
[ZS2] Zakharov, V. E., Shabat, A. B.: A scheme for integrating nonlinear equations of mathematical physics by the method of the inverse scattering transform, I. Funct. Anal. Appl.8, 226–235 (1974)
Author information
Authors and Affiliations
Additional information
Communicated by N. Yu. Reshetikhin
Research supported by NSF grants DMS-8916968 and DMS 8901607
Rights and permissions
About this article
Cite this article
Beals, R., Sattinger, D.H. On the complete integrability of completely integrable systems. Commun.Math. Phys. 138, 409–436 (1991). https://doi.org/10.1007/BF02102035
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02102035