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We prove that small, semi-linear Hamiltonian perturbations of the defocusingnonlinear Schr\"odinger (dNLS) equation on the circle have an abundance ofinvariant tori of any size and (finite) dimension which support quasi-periodicsolutions. When compared with previous results the novelty consists inconsidering perturbations which do not satisfy any symmetry condition (they maydepend on $x$ in an arbitrary way) and need not be analytic. The maindifficulty is posed by pairs of almost resonant dNLS frequencies. The proof isbased on the integrability of the dNLS equation, in particular the fact thatthe nonlinear part of the Birkhoff coordinates is one smoothing. We implement aNewton-Nash-Moser iteration scheme to construct the invariant tori. The keypoint is the reduction of linearized operators, coming up in the iterationscheme, to $ 2 \times 2 $ block diagonal ones with constant coefficientstogether with sharp asymptotic estimates of their eigenvalues.

Large KAM tori for perturbations of the dNLS equation