Long time stability of small finite gap solutions of the cubic Nonlinear Schr\"odinger equation on mathbb T²

In this paper we study long time stability of a class of nontrivial, quasi-periodic solutions depending on one spacial variable of the cubic defocusing non-linear Schr\"odinger equation on the two dimensional torus. We prove that these quasi-periodic solutions are orbitally stable for finite but long times, provided that their Fourier support and their frequency vector satisfy some complicated but explicit condition, which we show holds true for most solutions. The proof is based on a normal form result. More precisely we expand the Hamiltonian in a neighborhood of a quasi-periodic solution, we reduce its quadratic part to diagonal constant coefficients through a KAM scheme, and finally we remove its cubic terms with a step of nonlinear Birkhoff normal form. The main difficulty is to impose second and third order Melnikov conditions; this is done by combining the techniques of reduction in order of pseudo-differential operators with the algebraic analysis of resonant quadratic Hamiltonians.