Birkhoff normal form for partial differential equations with tame modulus

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Long time existence for fully nonlinear NLS with small Cauchy data on the circle (2018, background): “3 of [6] and the references therein) for any ρ ∈ N it admits an asymptotic expansion of the form λj ∼ j(2) + c0(V ) + c1(V )j + .”
A Birkhoff normal form theorem for some semilinear PDEs (2008, methodology): “The theory presented here was developed in [Bam03, BG06 , DS06, BDGS07, Gre06].”
Branching of Cantor Manifolds of Elliptic Tori and Applications to PDEs (2011, methodology): “12 of [4] (see also Theorem 6.”
Almost global existence for Hamiltonian semilinear Klein‐Gordon equations with small Cauchy data on Zoll manifolds (2005, methodology): “In [ 2 ] this property was only veried for a quite general class of 1 d PDEs and for a particular NLS equation on the torus Td with arbitrary”
A Normal Form for the Schrödinger Equation with Analytic Non-linearities (2010, background): “[1,4,5]), by which we cancel all the terms of degree 2(q + 1) which do not Poisson commute with K .”