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Birkhoff normal form for partial differential equations with tame modulus. (English) Zbl 1110.37057
The infinite-dimensional Hamiltonian system
$dq_j/dt=\omega_j p_j+\partial P/ \partial p_j,\;dp_j/dt=-\omega_jq_j-\partial P/\partial q_j,\quad j=1,2,\dots,$ with the Hamilton function $H(p,q)=\tfrac 12\sum \omega_l(p_l^2+q_l^2)+P(p,q),\;p=(p_1,p_2,\dots),q=(q_1,q_2, \dots),$ where $$P$$ has a zero of order at least three at the origin are thoroughly analyzed in the Hilbert space of sequences $$(p,q)= \ell_s^2(\mathbb{R})\oplus\ell_s^2(\mathbb{R})$$ with the norm $$\|x\|^2_s= \sum l^{2s}|x_l|^2$$ in the components $$\ell_s^2(\mathbb{R})$$. Assuming certain “tame modulus” property of the summand $$P$$, then for any $$r\geq 1$$, there exists an analytic canonical transformation $$T$$ such that the Hamiltonian $$H\circ T$$ is of the Birkhoff normal form, $H\circ T=\tfrac 12\sum\omega_l(p_l^2+q_l^2)+Z+R,$ where $$Z$$ is a polynomial of degree at most $$r+2$$ which is a normal form with respect to $$\omega$$ and $$R$$ is small. It follows that in the nonresonant case, any small amplitude solution remains close to a torus for very long times. Advaced applications to the nonlinear wave equation $$u_{tt}-u_{xx}+V(x)u= g(x,u)$$ and to the Schrödinger equation $$-i\psi=-\psi_{xx}+V(x)\psi+ \partial g(x,\psi, \psi^*)/\partial\psi^*$$ on a torus are given. Estimates in hight Sobolev norms and lower bounds on the existence time of solutions are obtained.

##### MSC:
 37K55 Perturbations, KAM for infinite-dimensional Hamiltonian and Lagrangian systems 34C20 Transformation and reduction of ordinary differential equations and systems, normal forms 34D10 Perturbations of ordinary differential equations 35Q55 NLS equations (nonlinear Schrödinger equations) 37K05 Hamiltonian structures, symmetries, variational principles, conservation laws (MSC2010)
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##### References:
 [1] D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations , Math. Z. 230 (1999), 345–387. · Zbl 0928.35160 · doi:10.1007/PL00004696 [2] -, On long time stability in Hamiltonian perturbations of non-resonant linear PDE s, Nonlinearity 12 (1999), 823–850. · Zbl 0989.37073 · doi:10.1088/0951-7715/12/4/305 [3] -, An averaging theorem for quasilinear Hamiltonian PDE s, Ann. Henri Poincaré 4 (2003), 685–712. · Zbl 1031.37056 · doi:10.1007/s00023-003-0144-6 [4] -, Birkhoff normal form for some nonlinear PDE s, Comm. Math. Phys. 234 (2003), 253–285. · Zbl 1032.37051 · doi:10.1007/s00220-002-0774-4 [5] -, “Birkhoff normal form for some quasilinear Hamiltonian PDEs” in XIVth International Congress on Mathematical Physics (Lisbon, 2003) , World Sci., Hackensack, N.J., 2005, 273–280. · Zbl 1108.37048 [6] D. Bambusi and B. GréBert, Forme normale pour NLS en dimension quelconque, C. R. Math. Acad. Sci. Paris 337 (2003), 409–414. · Zbl 1030.35143 · doi:10.1016/S1631-073X(03)00368-6 [7] G. Benettin, L. Galgani, and A. Giorgilli, A proof of Nekhoroshev’s theorem for the stability times in nearly integrable Hamiltonian systems , Celestial Mech. 37 (1985), 1–25. · Zbl 0602.58022 · doi:10.1007/BF01230338 [8] J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE , Internat. Math. Res. Notices 1994 , no. 11, 475–497. · Zbl 0817.35102 · doi:10.1155/S1073792894000516 [9] -, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations , Geom. Funct. Anal. 6 (1996), 201–230. · Zbl 0872.35007 · doi:10.1007/BF02247885 · eudml:58224 [10] -, Quasi-periodic solutions of Hamiltonian perturbations of $$2D$$ linear Schrödinger equations , Ann. of Math. (2) 148 (1998), 363–439. JSTOR: · Zbl 0928.35161 · doi:10.2307/121001 · www.math.princeton.edu [11] -, On diffusion in high-dimensional Hamiltonian systems and PDE , J. Anal. Math. 80 (2000), 1–35. · Zbl 0964.35143 · doi:10.1007/BF02791532 [12] -, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations , Ergodic Theory Dynam. Systems 24 (2004), 1331–1357. · Zbl 1087.37056 · doi:10.1017/S0143385703000750 [13] -, Green’s Function Estimates for Lattice Schrödinger Operators and Applications , Ann. of Math. Stud. 158 , Princeton Univ. Press, Princeton, 2005. · Zbl 1137.35001 [14] W. Craig, Problèmes de petits diviseurs dans les équations aux dérivées partielles , Panor. et Synthèses 9 , Soc. Math. France, Montrouge, 2000. · Zbl 0977.35014 [15] W. Craig and C. E. Wayne, Newton’s method and periodic solutions of nonlinear wave equations , Comm. Pure Appl. Math. 46 (1993), 1409–1498. · Zbl 0794.35104 · doi:10.1002/cpa.3160461102 [16] J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres , Int. Math. Res. Not. 2004 , no. 37, 1897–1966. · Zbl 1079.35070 · doi:10.1155/S1073792804133321 [17] -, Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds , to appear in Amer. J. Math. · Zbl 1108.58023 · doi:10.1353/ajm.2006.0038 [18] T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator , SIAM J. Math. Anal. 33 (2001), 113–152. · Zbl 1097.34553 · doi:10.1137/S0036141099365753 [19] T. Kappeler and J. PöSchel, KdV & KAM , Ergeb. Math. Grenzgeb. (3) 45 , Springer, Berlin, 2003. [20] S. Klainerman, On “almost global” solutions to quasilinear wave equations in three space dimensions , Comm. Pure Appl. Math. 36 (1983), 325–344. · Zbl 0522.35063 · doi:10.1002/cpa.3160360304 [21] S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum , Funct. Anal. Appl. 21 (1987), 192–205. · Zbl 0716.34083 · doi:10.1007/BF02577134 [22] -, Nearly Integrable Infinite-Dimensional Hamiltonian Systems , Lecture Notes in Math. 1556 , Springer, Berlin, 1993. · Zbl 0784.58028 · doi:10.1007/BFb0092243 [23] -, Analysis of Hamiltonian PDEs , Oxford Lecture Ser. Math. Appl. 19 , Oxford Univ. Press, New York, 2000. [24] S. B. Kuksin and J. PöSchel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation , Ann. of Math. (2) 143 (1996), 149–179. JSTOR: · Zbl 0847.35130 · doi:10.2307/2118656 · links.jstor.org [25] V. A. Marchenko, Sturm-Liouville Operators and Applications , Oper. Theory Adv. Appl. 22 , Birkhäuser, Basel, 1986. [26] N. V. Nikolenko, The method of Poincaré normal forms in problems of integrability of equations of evolution type , Russian Math. Surveys 41 , no. 5 (1986), 63–114. · Zbl 0632.35026 · doi:10.1070/RM1986v041n05ABEH003423 [27] J. PöSchel and E. Trubowitz, Inverse Spectral Theory , Pure Appl. Math. 130 , Academic Press, Boston, 1987. · Zbl 0623.34001 [28] C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479–528. · Zbl 0708.35087 · doi:10.1007/BF02104499 [29] J. Xu, J. You, and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy , Math. Z. 226 (1997), 375–387. · Zbl 0899.34030 · doi:10.1007/PL00004344
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