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Namely, we show that, for generic $m$, many of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{G=0}$, written as the system $$ u_t=-v,\\quad v_t=\\Delta^2 u+mu, $$ persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. 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Hakan Eliasson, Sergei B. Kuksin","submitted_at":"2016-04-06T15:16:04Z","abstract_excerpt":"In this paper we prove a KAM theorem for small-amplitude solutions of the non linear beam equation on the d-dimensional torus $$u_{tt}+\\Delta^2 u+m u + \\partial_u G(x,u)=0\\ ,\\quad t\\in { \\mathbb{R}} , \\; x\\in \\ { \\mathbb{T}}^d, \\qquad \\qquad (*) $$ where $G(x,u)=u^4+ O(u^5)$. Namely, we show that, for generic $m$, many of the small amplitude invariant finite dimensional tori of the linear equation $(*)_{G=0}$, written as the system $$ u_t=-v,\\quad v_t=\\Delta^2 u+mu, $$ persist as invariant tori of the nonlinear equation $(*)$, re-written similarly. 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