Quasi-periodic solutions for 1D resonant beam equation

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In this paper, one--dimensional ($1D$) resonant beam equation $u_{t t} +u_{x x x x} +u^3=0,$ with hinged boundary conditions is considered. It is proved that 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 theorem, partial normal form and scaling skills.