A KAM Theorem for Hamiltonian Partial Differential Equations with Unbounded Perturbations

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KAM for autonomous quasi-linear perturbations of mKdV (2014, methodology): “For unbounded perturbations the first KAM results have been proved by Kuksin [24] and Kappeler–Pöschel [21] for KdV (see also Bourgain [9]), and more recently by Liu–Yuan [20], Zhang–Gao–Yuan [34] for derivative NLS, and by Berti–Biasco–Procesi [4,5] for derivative NLW.”
KAM for quasi-linear and fully nonlinear forced perturbations of Airy equation (2014, unknown): “is doubtful, see [27], [16], [35], because of the possible phenomenon of formation of singularities outlined in Lax [33], Klainerman and Majda [28].”
Periodic solutions of fully nonlinear autonomous equations of Benjamin-Ono type (2012, methodology): “In [28] Liu and Yuan apply a Birkhoff normal form and KAM method to show the existence of quasiperiodic solutions of a Benjamin-Ono equation that is a Hamiltonian analytic perturbation of (2.”
Time quasi-periodic gravity water waves in finite depth (2017, methodology): “The existence of quasi-periodic solutions of PDEs (which we shall call, in a broad sense, KAM theory) with unbounded perturbations (i.e. the nonlinearity contains derivatives) has been first proved by Kuksin [44] and Kappeler-Pöschel [42] for KdV, then by Liu-Yuan [48], Zhang-Gao-Yuan [63] for deri”
KAM for Reversible Derivative Wave Equations (2012, methodology): “In turn [20] extends the approach of Kuksin [24]-[25], Kappeler-Pöschel [21], which is valid for more dispersive PDEs, like KdV.”