. A P ] 2 9 A ug 2 00 5 NON - GENERIC BLOW - UP SOLUTIONS FOR THE CRITICAL FOCUSING NLS IN 1 -

We consider the critical focusing NLS in 1-d of the form (1.1) i∂ t ψ + ∂ 2 x ψ = −|ψ| 4 ψ, i = √ −1, ψ = ψ(t, x), and ψ complex valued. It is well-known that this equation permits standing wave solutions of the form φ(t, x) = e iαt φ 0 (x, α), α > 0 Indeed, requiring positivity and evenness in x for φ 0 (x, α) implies for example φ 0 (x, α) = α 1 2 (3 2) 1 4 cosh 1 2 (α 2 x) Another remarkable feature of the equation (1.1) is the large symmetry group carrying solutions into solutions: this is generated by Galilei transformations: ψ(t, x) −→ e i(γ+vx−v 2 t) e −i(2tv+µ)p ψ(t, x) = e i(γ+vx−v 2 t) ψ(t, x − 2tv − µ), p = −i d dx SL(2, R)-transformations: ψ(t, x) −→ (a + bt) − 1 2 e ibx 2 4(a+bt) ψ(c + dt a + bt , x a + bt), a b c d ∈ SL(2, R) Observe that the latter subsume re-scalings ψ(t, x) → a 1 2 ψ(a 2 t, ax) while the former subsume phase-shifts ψ(t, x) → e iγ ψ(t, x) as well as translations. We usually identify a matrix a b c d ∈ SL(2, R) with the corresponding transformation. It is the SL(2, R)-transformations that distinguish the critical NLS from the sub-and supercritical NLS, and allows us to exhibit explicit blow-up solutions: indeed, fixing a b c d ∈ SL(2, R), we have the explicit solution (1.2) f (t, x) = (a + bt) − 1 2 e i c+dt a+bt e ibx 2 4(a+bt) φ 0 (x a + bt , 1), which blows up for t = − a b. Fixing a ∼ 1, b ∼ −1, it is then a natural question to ask whether one may perturb the initial data of (1.2) at time t = 0 such that the corresponding solution exhibits the same type of blow-up behavior. More precisely, the solution should asymptotically behave like 1 T −t e iΨ(t,x) φ(x−µ(t) T −t) for a bounded function µ(t) and suitable Schwartz function φ, with blow up time T. The recent work of Merle-Raphael [MeRa] has demonstrated that this is generically impossible, i. e. there are open sets of initial data containing f (0, x) in their closure 1 and …