Invariant tori for Benjamin-Ono equation with unbounded quasi-periodically forced perturbation (Q379747)

The authors consider the non-autonomous Benjamin-Ono equation NEWLINE\[NEWLINEu_t+\mathcal Hu_{xx}uu_x-(F(\omega t,x,u))_x=0NEWLINE\]NEWLINE under periodic boundary conditions NEWLINE\[NEWLINEu(t, x + 2\pi) = u(t, x),\quad -\infty<t<\infty,NEWLINE\]NEWLINE where \(u\) is real-valued and \(\mathcal H\) is the Hilbert transform defined for \(2\pi\)-periodic functions with mean value zero. Using an abstract infinite-dimensional KAM theorem dealing with unbounded perturbation vector fields and the partial Birkhoff normal form, the authors prove that there exists a Cantorian branch of KAM tori and thus many time quasi-periodic solutions for the above equation.