and is especially close to the analysis of the water-wave problem in [@GerMasSha2].

The proof relies on two distinct components. The first component, [*[the energy method]{}*]{}, exploits the existence of a conserved physical energy given in , which comes from the subtle structure of the nonlinearity. This implies almost conservation of higher order energies, up to commutators that are [*lower order*]{} in derivative and are at least of [*cubic order*]{} and therefore can be controlled if the solution decays sufficiently fast. In dimension $3$, these error terms are integrable and the higher order energies remain bounded (see e.g. [@Guo; @Sha]). In dimension $2$, all the norms are only almost integrable and the highest order energy is allowed to grow slowly.

The second component of the proof is concerned with proving the decay estimate needed to control the commutators above. It exploits the fact that we consider a perturbation of a [*constant*]{} solution, which allows us to use the [*[Fourier transform method]{}*]{}. The analysis here depends on the structure of the linearized problem (in particular the [*dispersion relation*]{} $\omega(k)$) and on delicate cancellation properties of the nonlinearity (the [*[null structure]{}*]{}), which are particularly important to bound the constribution of very low 