Threshold solutions for the focusing 3d cubic Schroedinger equation
Abstract
We study the focusing 3d cubic NLS equation with H^1 data at the massenergy threshold, namely, when M[u_0]E[u_0] = M[Q]E[Q]. In earlier works of HolmerRoudenko and DuyckaertsHolmerRoudenko, the behavior of solutions (i.e., scattering and blow up in finite time) is classified when M[u_0]E[u_0] < M[Q]E[Q]. In this paper, we first exhibit 3 special solutions: e^{it}Q and Q^+, Q^; here Q is the ground state, and Q^+, Q^ exponentially approach the ground state solution in the positive time direction, meanwhile Q^+ having finite time blow up and Q^ scattering in the negative time direction. Secondly, we classify solutions at this threshold and obtain that up to \dot{H}^{1/2} symmetries, they behave exactly as the above three special solutions, or scatter and blow up in both time directions as the solutions below the massenergy threshold. These results are obtained by studying the spectral properties of the linearized Schroedinger operator in this masssupercritical case, establishing relevant modulational stability and careful analysis of the exponentially decaying solutions to the linearized equation.
 Publication:

arXiv eprints
 Pub Date:
 June 2008
 arXiv:
 arXiv:0806.1752
 Bibcode:
 2008arXiv0806.1752D
 Keywords:

 Mathematics  Analysis of PDEs