# KAM for the nonlinear wave equation on the circle: A normal form theorem

• Moudhaffar Bouthelja
Article

## Abstract

In this paper we prove a KAM theorem in infinite dimension which treats the case of multiple eigenvalues (or frequencies) of finite order. More precisely, we consider a Hamiltonian normal form in infinite dimension:
\begin{aligned} h(\rho )=\omega (\rho )\cdot r + \frac{1}{2} \langle \zeta ,A(\rho )\zeta \rangle , \end{aligned}
where $$r \in \mathbb {R}^n$$, $$\zeta =((p_s,q_s)_{s \in \mathcal {L}})$$ and $$\mathcal {L}$$ is a subset of $$\mathbb {Z}$$ and $$\rho$$ is a parameter that belongs to a compact. We assume that the infinite matrix $$A(\rho )$$ is of the form $$A(\rho )= D(\rho )+N(\rho )$$, where $$D(\rho )$$ is a diagonal matrix with finite eigenvalue multiplicity and N is a bloc diagonal matrix. We assume that the size of each bloc of N is the multiplicity of the corresponding eigenvalue in D. In this context, if we start from a torus, then the solution of the associated Hamiltonian system remains on that torus. Under certain conditions emitted on the frequencies, we can affirm that the trajectory of the solution fills the torus. In this context, the starting torus is an invariant torus. Then, we perturb this integrable Hamiltonian and we want to prove that the starting torus is a persistent torus. We prove that, if the perturbation is small and under certain conditions of non-resonance of the frequencies, then the starting torus is a persistent torus. We use this result to prove the existence of quasi-periodic solutions for the nonlinear wave equation with convolutive potential on the circle. In Bouthelja (KAM for the nonlinear wave equation on the circle: small amplitude solutions (preprint), arXiv:1712.01597, 2017) we use this result to prove the existence of such solutions for the nonlinear wave equation on the circle without an external parameter.

## References

1. 1.
Arnold, V.I.: Proof of a theorem of A.N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18(5), 9–36 (1963)
2. 2.
Arnold, V.I.: Small denominators and problems of stability of motion in classical and celestial mechanics. Russ. Math. Surv. 18(6), 85–191 (1963)
3. 3.
Berti, M.: KAM for PDES. Bollettino dell’Unione Matematica Italiana 9(2), 115–142 (2016)
4. 4.
Berti, M., Biasco, L., Procesi, M.: KAM theory for the Hamiltonian derivative wave equation. Ann. Sci. École. Norm. Supér. (4) 46(2), 301–373 (2013)
5. 5.
Bouthelja, M.: KAM for the nonlinear wave equation on the circle: small amplitude solutions (preprint) (2017). arXiv:1712.01597
6. 6.
Chierchia, L., You, J.: KAM tori for 1D nonlinear wave equations with periodic boundary conditions. Commun. Math. Phys. 211(2), 497–525 (2000)
7. 7.
Dumas, H.S.: The KAM Story. World Scientific, Hackensack (2014). A friendly introduction to the content, history, and significance of classical Kolmogorov–Arnold–Moser theory
8. 8.
Eliasson, L.H., Kuksin, S.B.: KAM for the nonlinear Schrödinger equation. Ann. Math. 172, 371–435 (2010)
9. 9.
Eliasson, L.H., Grébert, B., Kuksin, S.B.: KAM for the nonlinear beam equation. Geom. Funct. Anal. 26(6), 1588–1715 (2016)
10. 10.
Grébert, B., Paturel, E.: KAM for the Klein Gordon equation on $$\mathbb{S}^d$$. Boll. dell’Unione Mat. Ital. 9(2), 237–288 (2016).
11. 11.
Grébert, B., Thomann, L.: KAM for the quantum harmonic oscillator. Commun. Math. Phys. 307(2), 383–427 (2011)
12. 12.
Kolmogorov, A.N.: On conservation of conditionally periodic motions for a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR 98, 527–530 (1954)
13. 13.
Kuksin, B., Pöschel, S.J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation. Ann. Math. (2) 143(1), 149–179 (1996)
14. 14.
Kuksin, B., Poschel, S.J.: Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrodinger equation. Ann. Math. 142, 149–179 (1996)
15. 15.
Kuksin, S.B.: Hamiltonian perturbations of infinite-dimensional linear systems with imaginary spectrum. Funktsional Anal. i Prilozhen. 21(3), 22–37, 95 (1987)
16. 16.
Kuksin, S.B.: Perturbation of quasiperiodic solutions of infinite-dimensional hamiltonian systems. Math. USSR Izv. 32(1), 39 (1989)
17. 17.
Kuksin, S.B.: A KAM-theorem for equations of the Korteweg–De Vries type. Rev. Math. Phys. 10, 1–64 (1998)
18. 18.
Möser, J.: On invariant curves of area-preserving mappings of an annulus. Nachr. Akad. Wiss. Göttingen II, 1–20 (1962)
19. 19.
Pöschel, J.: Quasi-periodic solutions for a nonlinear wave equation. Comment. Math. Helv. 71(2), 269–296 (1996)
20. 20.
Pöschel, J.: A KAM-theorem for some nonlinear partial differential equations. Annali della Scuola Normale Superiore di Pisa Classe di Scienze 23(1), 119–148 (1996)
21. 21.
Procesi, M., Procesi, C.: Reducible quasi-periodic solutions for the non linear schrödinger equation. Bollettino dell’Unione Matematica Italiana 9(2), 189–236 (2016)
22. 22.
Villani, C.: Théorème vivant. Grasset, Paris (2012)
23. 23.
Wayne, C.E.: Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory. Commun. Math. Phys. 127(3), 479–528 (1990)