A Nekhoroshev type theorem for the derivative nonlinear Schrödinger equation

Exponential stability estimate for the derivative nonlinear Schrödinger equation*

“…The polynomial long time stability (or the almost global existence solution) where acquired for some quasi-linear and fully nonlinear PDEs such as derivative nonlinear Schrödinger equation, water waves equations, KdV equation. See [11,12,16,20,33,34] for example.…”

“…• It needs a stronger nonresonant conditions (see (9) for the first kind of the resonant set) than (16) to construct the Birkhoff normal form of high order. Noting that there is a factor n * 1 (l + k) in the numerator of (9).…”

Exact global control of small divisors in rational normal form *

“…To our knowledge, there is no result of sub-exponentially long time stability for equations with internal parameters in previous papers. Now, for instance, we compare (2.25) with the stability time of equations with external parameters in [CMW20,BMP20].…”

Weak Sobolev almost periodic solutions for the 1d NLS

“…Comparable difficulties in tackling the Sobolev case appear in Birkhoff Normal Form theory for PDEs leading to two different types of results. In the analytic or Gevrey case one has sub-exponential stability times (see [FG13] and [CMW20]), whereas in the Sobolev case the stability times seem to be controlled by the Sobolev exponent (see [BG06], [FI], [BD18], [BMP20a]). The counterpart of total and long time stability results is the construction of unstable trajectories, which undergo growth of the Sobolev norms, see [Bou96, CKS + 10, GK15, GHH + 18, GGMP].…”