The cubic fourth-order Schrodinger equation

We investigate the cubic defocusing fourth order Schr\"odinger equation $iu_t + \Delta^2u + |u|^2u=0$ in arbitrary space dimension $\mathbb{R}^n$ for arbitrary $H^2$ initial data. We prove that the equation is globally well-posed when $n \le 8$ and ill-posed when $n \ge 9$, with the additional important information that scattering holds true when $5 \le n \le 8$.Comment: 38 pages, references adde

(1) GEOMETRIC AND PROJECTIVE INSTABILITY FOR THE GROSS-PITAEVSKI EQUATION

Abstract. — Using variational methods, we construct approximate solutions for the Gross-Pitaevski equation which concentrate on circles in R 3. These solutions will help to show that the L 2 flow is unstable for the usual topology and for the projective distance. 1