A KAM theorem for Hamiltonian partial differential equations with unbounded perturbations
The authors prove a KAM theorem in infinite dimension for parameter-depending Hamiltonian maps of the form \(H= N_\xi+ P_\xi\), where \[ N= \sum_{1\leq j\leq n} \omega_j(\xi) y_j+{1\over 2} \sum_{j\geq 1} \Omega_j(\xi)(u^2_j+ v^2_j), \] and \(P_\xi\) is a small perturbation. Here \(x_j\in\mathbb{T}= (\mathbb{R}/2\pi\mathbb{Z})\), \(y_j, u_j, v_j\in \mathbb{R}\), the parameter \(\xi\) is in \(\mathbb{R}^n\), and the symplectic form is \[ \sum_{1\leq j\leq n} dy_i\wedge dx_j+ \sum_{j\geq 1} du_j\wedge dv_j. \] \(P_\xi\) is an unbounded perturbation, inducing a loss of regularity of order \(\widetilde d\): the Hamiltonian vector-field \(X_P\) sends \(\mathbb{T}^n\times \mathbb{R}^n\times (l^{a,p})^2\) to \(\mathbb{R}^n\times \mathbb{R}^n\times (l^{a,p-\widetilde d})^2\), where \(l^{a,p}\) is the set of the sequences \((w_j)_{j\geq 1}\) of real numbers for which \[ \sum_{j\geq 1} e^{2aj} j^{2p}|w_j|^2< \infty. \] For all \(\xi\), the torus defined by \(y_j= 0\), \(u_j= v_j= 0\) is invariant for the Hamiltonian vector-field \(X_N\), and the flow on this torus is linear, of frequency vector \(\omega(\xi)= (\omega_1(\xi),\dots, \omega_n(\xi))\). The authors assume that the map \(\xi\mapsto\omega(\xi)\) satisfies a usual nondegeneracy condition and that, roughly, \(\Omega_j(\xi)\sim j^d\). In particular, the normal frequencies \(\Omega_j(\xi)\) are all distinct. Under some additional assumptions on \(N_\xi\), implying the Melnikov non-resonance conditions for almost all \(\xi\), the authors prove that if \(P_\xi\) is a small enough real analytic perturbation such that the order \(\widetilde d\) of the Hamiltonian field \(X_P\) is not greater than \(d-1\), then, for most values of the parameter \(\xi\), the invariant torus is preserved through the perturbation, the perturbed torus having a frequency vector close to \(\Omega(\xi)\). This extends a, result of \textit{S. B. Kuksin} [Analysis of Hamiltonian PDE's. Oxford: Oxford University Press (2000; Zbl 0960.35001)], which was restricted to the case \(\widetilde d< d- 1\). When \(\widetilde d= d- 1\), the authors find \(C^\infty\) (not real analytic) embeddings of invariant tori. To prove their result, they follow the approach of Kuksin, letting the normal form provided around the invariant torus depend partially on the angle variables. They obtain new estimates when solving the homological equations at each iteration; these estimates enable them to deal with the case \(\widetilde d= d- 1\). The authors derive from their abstract KAM theorem new existence results for the quasi-periodic solutions of two Hamiltonian PDE: the derivative nonlinear Schrödinger equation in one spatial dimension, with Dirichlet boundary conditions, and the perturbed Benjamin-Ono equation, with periodic boundary conditions.
scientific article; zbMATH DE number 5977583
scientific article