Birkhoff normal form for some nonlinear PDEs
Infinite-dimensional Hamiltonian systems are considered, with Hamiltonian given by \(H=h_0+f\). Here \(h_0(p, q)=\sum_j\omega_j (p_j^2+q_j^2)/2\), and \(f\) is of order at least three in \((p, q)\). Under certain assumptions on the frequencies (described in more detail below) it is shown that there is a canonical transformation \({\mathcal T}\) defined in a neighborhood of the origin \((p, q)=(0, 0)\) such that \(H\circ {\mathcal T}=h_0+Z+R\), where \(Z\) is in Birkhoff normal form and \(R\) is a remainder term of an arbitrary high order that can be prescribed. The proof uses an inductive argument which goes roughly as follows: If a transformation \({\mathcal T}^{(k)}\) is already known that puts \(H\) into the form \(H\circ {\mathcal T}^{(k)}=h_0+Z^{(k)}+R^{(k)}\), then basically speaking the remainder term \(R^{(k)}\) is split as \(R^{(k)}=R^{(k)}_{\text{low}, \leq 2}+R^{(k)}_{\text{low}, \geq 3} +R^{(k)}_{\text{high}}\). Here \(R^{(k)}_{\text{high}}\) is of sufficiently high order in \((p, q)\), \(R^{(k)}_{\text{low}, \leq 2}\) contains the lower order terms that are of order \(\leq 2\) in \((p_j, q_j)\) with \(|j|\geq N+1\) (\(N\) fixed), whereas \(R^{(k)}_{\text{low}, \geq 3}\) denotes the lower order terms that are of order \(\geq 3\) in \((p_j, q_j)\) with \(|j|\geq N+1\). This spatial and frequency cutoff at order \(N\) allows to prove that the contribution of \(R^{(k)}_{\text{low}, \geq 3}\) to the flow is `negligible' if all parameters (including \(N\)) are combined in an appropriate way. The term \(R^{(k)}_{\text{low}, \leq 2}\) can be put into normal form by introducing a suitable transformation \({\mathcal T}_k\) via a vector field \(\chi_k\) for whose definition small denominators occur in a natural way. It turns out that then, defining \({\mathcal T}^{(k+1)} ={\mathcal T}^{(k)}\circ {\mathcal T}_k\), the Hamiltonian can be transformed as \(H\circ {\mathcal T}^{(k+1)}=h_0+Z^{(k+1)}+R^{(k+1)}\) which is at least one order better than \(H\circ {\mathcal T}^{(k)}\). For the frequencies \(\omega_j\) it is assumed that they are `strongly non-vanishing', which requires that for every \(r\in {\mathbb N}\) with \(r\geq 3\) there exist \(\gamma\) and \(\alpha\) such that, if \(N\in {\mathbb N}\) is large enough, \(|i_1|\leq |i_2|\leq\ldots\leq |i_r|\leq N<|j|\leq |k|\), and \(\omega_{i_1}\pm\ldots\pm\omega_{i_r}\pm\omega_j\pm\omega_k\neq 0\), then \(|\omega_{i_1}\pm\ldots\pm\omega_{i_r}\pm\omega_j\pm\omega_k|\geq\gamma N^{-\alpha}\) for all combinations of plus and minus signs. Note that the indices \(j\) and \(k\) correspond to \(p_l\) and/or \(q_m\) that might occur in the series expansion of \(R^{(k)}_{\text{low}, \leq 2}\) from above; by definition, there are at most two such indices \(j, k\). The paper also contains an application of the abstract result to the wave equation \(u_{tt}-u_{xx}+(m+V(x))u+h(x, u)=0\) on \([0, \pi]\) with Dirichlet boundary conditions and analytic \(h\) of order at least three. It is then shown that in every interval \([m_1, m_2]\) the set of masses such that the resulting frequencies are strongly non-vanishing is of full measure.
scientific article; zbMATH DE number 1964918