Estimates for solutions of KdV on the phase space of periodic distributions in terms of action variables (Q536129)

The KdV equation \[ \partial_t\psi =-\psi^{'''}_{xxx}+6\psi\psi'_x \] on the Sobolev space of zero-meanvalue 1-periodic distributions \(H_{-1}=\{\psi=q':q\in H\}\) is considered, where the real Hilbert space \(H=H_0\) consists of zero-meanvalue functions \(q\in L^2(\mathbb{T})\), where \(\mathbb{T}=\mathbb{R}/\mathbb{Z}\), \(\|\psi\|^2_{-1}=\|q\|^2=\int\limits_0^1q^2(x)dx\). The action-angle variables for the case \(\psi\in H_{-1}\) were introduced in [\textit{T. Kappeler, C. Möhr} and \textit{P. Topalov}``, Birkhoff coordinates for KdV on phase spaces of distributions,'' Sel. Math., New Ser. 11, No. 1, 37--98 (2005; Zbl 1089.37042)]. Here the estimates of the solution of the KdV in terms of action variables are obtained.