Abstract
In this paper, one-dimensional (1D) nonlinear beam equations utt+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 parameters m>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 infinite-dimensional 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].
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