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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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