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In this paper we prove a result of almost global existence for some abstractnonlinear PDEs on flat tori and apply it to some concrete equations, namely anonlinear Schr\"odinger equation with a convolution potential, a beam equationand a quantum hydrodinamical equation. We also apply it to the stability ofplane waves in NLS. The main point is that the abstract result is based on anonresonance condition much weaker than the usual ones, which rely on thecelebrated Bourgain's Lemma which provides a partition of the "resonant sites"of the Laplace operator on irrational tori.

Almost global existence for some Hamiltonian PDEs with small Cauchy data on general tori