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Additional resources for Cantor families of periodic solutions for completely resonant nonlinear wave equations

Example text

Proof. We have to prove that Φn (v) := Φ0 (Hn v), where Φ0 is defined in (19), possesses non-degenerate critical points at least for n large. 8 in [6]): for v(t, x) = η(t + x) − η(t − x) Φn (v) = 2πn2 η˙ 2 (t)dt − T Hence we can write √ 12n 48n4 1 Φn √ v = 2 a2 2 πa2 π 2 a22 12 η 2 (t) dt T Ψ(η) = + T η˙ 2 (s) ds − where 2 1 2 1 4 η 2 (s) ds T η˙ 2 (s) ds − T 1 4 a22 2n2 2 + v 2 L−1 v 2 + Ω π2 6 η 2 (t) dt 2 1 48n4 1 R(η) = 2 Ψ(η) + 2 R(η) 2 n a2 n η 2 (s) ds . T (111) 2 T and R : E → R is a smooth functional defined on E := {η ∈ H 1 (T) | η odd}.

V. I. Shulman, Periodic solutions of the equation utt − uxx + u3 = 0, Funct. Anal. Appl. 22, 332–333, 1988. [19] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum, Funktsional. Anal. i Prilozhen. 21, no. 3, 22–37, 95, 1987. [20] S. B. Kuksin, Analysis of Hamiltonian PDEs Oxford Lecture Series in Mathematics and its Applications. 19. Oxford University Press, 2000. [21] S. Kuksin, J. P¨ oschel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schr¨ odinger equation, Ann.

P¨ oschel, A KAM-Theorem for some nonlinear PDEs, Ann. Scuola Norm. Sup. Pisa, Cl. , 23, 119-148, 1996. [24] J. P¨ oschel, Quasi-periodic solutions for a nonlinear wave equation, Comment. Math. , 71, no. 2, 269-296, 1996. W. Su, Persistence of periodic solutions for the nonlinear wave equation: a case of finite regularity, PhD Thesis, Brown University, 1998. [26] E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Commun. Math. Phys. 3, 479-528, 1990. [27] A.