Abstract :
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
