FILTERING THE L2 CRITICAL FOCUSING SCHRODINGER EQUATION

We study the influence of Szego projector on the L-2-critical non linear focusing Schrodinger equation, leading to the quintic focusing NLS-Szego equation on the line i partial derivative(t)u + partial derivative(2)(x)u + Pi(vertical bar u vertical bar(4)u) = o, (t, x) is an element of R x R, u(0, center dot) = u(0). It has no Galilean invariance but the momentum P(u) = <-i partial derivative(x)u, u >(L2) becomes the (H)over dot(1/2)-norm. Thus this equation is globally well-posed in H-+(1) = Pi(H-1(R)), for every initial datum uo. The solution L-2 scatters both forward and backward in time if u(0) has sufficiently small mass. By using the concentration compactness principle, we prove the orbital stability of some weak type of the traveling wave : u(omega,c)(t, x) = e(i omega t)(x + ct), for some omega, c > 0, where Q is a ground state associated to Gagliardo Nirenberg type functional. I-(gamma) (f) = parallel to partial derivative(x)f parallel to(2)(L2)parallel to f parallel to(4)(L2) + gamma <-i partial derivative(x)f, f >(2)(L2)parallel to f parallel to(2)(L2)/parallel to f parallel to(6)(L6), for all f is an element of H-+(1)\{0}, for some gamma >= 0. Its Euler Lagrange equation is a non local elliptic equation. The ground states are completely classified in the case gamma = 2, leading to the actual orbital stability for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS Szego equation is strictly below the mass of ground state associated to the functional I-0),I- unlike the recent result by Dodson [x] on the usual quintic focusing non linear Schrodinger equation.