THE CANTOR MANIFOLD THEOREM WITH SYMMETRY AND APPLICATIONS TO PDEs
2014; Mathematical Society of the Republic of China (Taiwan); Volume: 18; Issue: 5 Linguagem: Inglês
10.11650/tjm.18.2014.4240
ISSN2224-6851
AutoresZhenguo Liang, Zhuoqun Yu, Min Wang,
Tópico(s)Numerical methods for differential equations
ResumoIn this paper we introduce a new Cantor manifold theorem and then apply it to one new type of one-dimensional ($1d$) beam equations $$ u_{tt}+u_{xxxx}+mu-2u^2u_{xx}-2uu_x^2=0, m\gt 0,$$ with periodic boundary conditions. We show that the above equation admits small-amplitude linearly stable quasi-periodic solutions corresponding to finite dimensional invaraint tori of an associated infinite dimensional dynamical system. The proof is based on a partial Birkhoff normal form and an infinite dimensional KAM theorem for Hamiltonians with symmetry (cf. [19]).
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