**abstract:**
Considering general classes of nonlinear Schrödinger equations on the circle with nontrivial cubic part and without external parameters, I will present the construction of a new type of normal forms, namely rational normal forms, on open sets surrounding the origin in high Sobolev regularity. With this new tool we prove that, given a large constant \($M$\) and a sufficiently small parameter \($\varepsilon$\), for generic initial data of size \($\varepsilon$\), the flow is conjugated to an integrable flow up to an arbitrary small remainder of order \($\varepsilon^{M+1}$\). This implies that for such initial data \($u(0)$\) we control the Sobolev norm of the solution \($u(t)$\) for time of order \($\varepsilon^{-M}$\). Furthermore this property is locally stable: if \($v(0)$\) is sufficiently close to \($u(0)$\) (of order \($\varepsilon^{3/2}$\)) then the solution \($v(t)$\) is also controled for time of order \($\varepsilon^{-M}$\). (This is a joint work with Erwan Faou and Benoît Grébert).

Wed 22 May, 10:00 - 10:50, Aula Dini

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