Abstract
IfL is a positive self-adjoint operator on a Hubert spaceH, with compact inverse, the second-order evolution equation int,u″+Lu+∥u∥ 2H u=0 has an infinite number of first integrals, pairwise in involution. It follows from this that no nontrivial solution tends weakly to 0 inH ast→∞. Under an additional separation assumption on the eigenvalues ofL, all trajectories (u,u′) are relatively compact inD(L 1/2)×H. Finally, if all the eigenvalues are simple, the set of initial values of quasi-periodic solutions is dense in the ball Bε=(u 0,u ′0 )εD(L 1/2)×H; ∥L1/2 u 0∥ 2 H +∥u 2′ <ɛ forɛ sufficiently small.
Similar content being viewed by others
References
Arnold, V. (1989).Mathematical Methods of Classical Mechanics, Graduate Texts in Mathematics 60, 2nd ed., Springer, New York.
Brezis, H., Coron, J.-.M., and Nirenberg, L. (1980). Free vibration for a nonlinear wave equation and a theorem of P. Rabinowitz.Commun. Pure Appl. Math. 33, 667–689.
Cabannes, H., and Haraux, A. (1981). Mouvements presque-périodiques d'une corde vibrante en présence d'un obstacle fixe, rectiligne ou ponctuel.Int. J. Nonlin. Mech. 16, 449–458.
Cazenave, T., and Haraux, A. (1987). Oscillatory phenomena associated to semilinear wave equations in one spatial dimension.Trans. Am. Math. Soc. 300, 207–233.
Cazenave, T., and Haraux, A. (1988). Some oscillation properties of the wave equation in several space dimensions.J. Fund. Anal. 76, 87–109.
Cazenave, T., Haraux, A., Vázquez, L., and Weissler, F. B. (1989). Nonlinear effects in the wave equation with a cubic restoring force.Comp. Mech. 5, 49–72.
Cazenave, T., Haraux, A., and Weissler, F. B. (1991). Une équation des ondes complètement intégrable avec non-linéarité homogène de degré trois.C.R. Acad. Sci. Paris 313, 237–241.
Cazenave, T., Haraux, A., and Weissler, F. B. (1993). Detailed asymptotics for a convex Hamiltonian system with two degrees of freedom.J. Dynam. Diff. Eq. 5, 155–187.
Haraux, A. (1983). Remarks on Hamiltonian systems.Chinese J. Math. 11, 5–32.
Lax, P. D. (1975). Periodic solutions of the Korteweg-De Vries equation.Commun. Pure Appl. Math. 28, 141–188.
Rabinowitz, P. H. (1978). Free vibrations for a semi-linear wave equation.Commun. Pure Appl. Math. 31, 31–68.
Siegel, C. L., and Moser, J. K. (1971).Lectures on Celestial Mechanics, Springer, New York.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cazenave, T., Haraux, A. & Weissler, F.B. A class of nonlinear, completely integrable abstract wave equations. J Dyn Diff Equat 5, 129–154 (1993). https://doi.org/10.1007/BF01063738
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01063738