Open Access. Powered by Scholars. Published by Universities.®
Articles 1 - 1 of 1
Full-Text Articles in Mathematics
Schrödinger Dispersive Dstimates For A Dcaling-Critical Class Of Potentials, Marius Beceanu, Michael Goldberg
Schrödinger Dispersive Dstimates For A Dcaling-Critical Class Of Potentials, Marius Beceanu, Michael Goldberg
Mathematics and Statistics Faculty Scholarship
Consider the focussing cubic nonlinear Schr\"odinger equation in R 3 :
iψ t +Δψ=−|ψ| 2 ψ.
It admits special solutions of the form e itα ϕ , whereϕ is a Schwartz function and a positive (ϕ>0 ) solution of
−Δϕ+αϕ=ϕ 3 .
The space of all such solutions, together with those obtained from them by rescaling and applying phase and Galilean coordinate changes, called standing waves, is the eight-dimensional manifold that consists of functions of the form e i(v⋅+Γ) ϕ(⋅−y,α) . We prove that any solution starting sufficiently close to a standing wave in the Σ=W 1,2 (R 3 …