We found a match
Your institution may have access to this item. Find your institution then sign in to continue.
- Title
Filtering the L<sup>2</sup>-critical focusing Schrödinger equation.
- Authors
Sun, Ruoci
- Abstract
We study the influence of Szegő projector on the L2−critical non linear focusing Schrödinger equation, leading to the quintic focusing NLS–Szegő equation on the line i∂tu + ∂2xu + Π(|u|4u) = 0,(t,x)∈R × R,u(0,⋅) = u0. It has no Galilean invariance but the momentum P(u) = ⟨−i∂xu,u⟩L2 becomes the H½−norm. Thus this equation is globally well-posed in H1+ = Π(H1(R)), for every initial datum u0. The solution L2−scatters both forward and backward in time if u0 has sufficiently small mass. By using the concentration–compactness principle, we prove the orbital stability of some weak type of the traveling wave : uω,c(t,x) = eiωtQ(x + ct), for some ω,c > 0, where Q is a ground state associated to Gagliardo–Nirenberg type functional I(γ)(ƒ) = ∥∂xƒ∥2L2∥ƒ∥4L2 + γ⟨−i∂xƒ,ƒ⟩2L2∥fƒ2L2/∥ƒ∥6L6, ∀f∈H1+{0}, for some γ ≥ 0. Its Euler–Lagrange equation is a non local elliptic equation. The ground states are completely classified in the case γ = 2, leading to the actual orbital stability 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 [8] on the usual quintic focusing non linear Schrödinger equation.
- Subjects
SCHRODINGER equation; ELLIPTIC equations; LAGRANGE equations; GALILEAN relativity; QUINTIC equations; LINEAR equations; DENSITY functionals; HARDY spaces
- Publication
Discrete & Continuous Dynamical Systems: Series A, 2020, Vol 40, Issue 10, p5973
- ISSN
1078-0947
- Publication type
Article
- DOI
10.3934/dcds.2020255