Traveling waves of the quintic focusing NLS-Szegő equation

Overview & Synopsis

We study the influence of Szegő projector Π on the L 2 −critical one-dimensional non linear focusing Schrödinger equation, leading to the quintic focusing NLS-Szegő equation i∂ t u + ∂ 2 x u + Π(|u| 4 u) = 0, (t, x) ∈ R × R, u(0, ·) = u 0. This equation is globally well-posed in H 1 + = Π(H 1 (R)), for every initial datum u 0. The solution L 2-scatters both forward and backward in time if u 0 has sufficiently small mass. We prove the orbital stability with scaling of the traveling wave : u ω,c (t, x) = e iωt Q(x + ct), for some ω, c > 0, where Q is a ground state associated to Gagliardo-Nirenberg type functional I (γ) (f) = ∂ x f 2 L 2 f 4 L 2 + γ −i∂ x f, f 2 L 2 f 2 L 2 f 6 L 6 , ∀f ∈ H 1 + \{0}, for some γ ≥ 0. The ground states are completely classified in the case γ = 2, leading to the actual orbital stability without scaling for appropriate traveling waves. As a consequence, the scattering mass threshold of the focusing quintic NLS-Szegő equation is strictly below the mass of ground state associated to the functional I (0) , unlike the recent result by Dodson [6] on the usual quintic focusing non linear Schrödinger equation.