For Hamiltonians with a finite number of zero normal frequencies, a KAM-type theorem states that for most frequencies the existence of invariant tori is decided by a single leftover constant, and this yields quasi-periodic solutions for a nonlinear Schrödinger equation with a zero mode.
Large KAM tori for quasi-linear perturbations of KdV
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In this paper we prove the persistence of space periodic multi-solitons of arbitrary size under any quasi-linear Hamiltonian perturbation, which is smooth and sufficiently small. This answers positively a longstanding question whether KAM techniques can be further developed to prove the existence of quasi-periodic solutions of arbitrary size of strongly nonlinear perturbations of integrable PDEs.
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A KAM theorem for the Hamiltonian with finite zero normal frequencies and its applications
For Hamiltonians with a finite number of zero normal frequencies, a KAM-type theorem states that for most frequencies the existence of invariant tori is decided by a single leftover constant, and this yields quasi-periodic solutions for a nonlinear Schrödinger equation with a zero mode.