KAM tori of Hamiltonian perturbations of 1D linear beam equations ✩

In this paper, one-dimensional (1D) nonlinear beam equations ut t +uxxxx + mu = f (u), with hinged boundary conditions are considered; the nonlinearity f is an analytic, odd function and f (u) = O(u3). It is proved that for all real parametersm>0 but a set of small Lebesgue measure, the above equation admits small-amplitude quasi-periodic solutions corresponding to finite-dimensional invariant tori of an associated infinite-dimensional dynamical system. The proof is based on infinitedimensional KAM theory developed by Kuksin [Lecture Notes in Mathematics, Vol. 1556, Springer, Berlin, 1993], Wayne [Commun. Math. Phys. 127 (1990) 479–528], Pöschel [Ann. Sc. Norm. Sup. Pisa, Cl. Sci. 23 (1996) 119–148].  2002 Elsevier Science (USA). All rights reserved