Threshold odd solutions to NLS on the line (Harmonic Analysis and Nonlinear Partial Differential Equations)

This is a resume of our previous paper [10] for the global dynamics of the threshold odd solutions to the nonlinear Schrodinger equation on the line.§ 1. IntroductionWe consider the following nonlinear Schrodinger equation on the real line:where u = u(t, x): I x R 一 C, u° is a given function in H1(R), and p > 5.It is well known that (NLS) is locally well-posed in H 1(R).Moreover, the energy E and the mass M, which are defined by E(u)：=1WIH2 -时L++1and M(u) ：= 11训务, 2 P十丄 are conserved.Here, uf := 3xu.A blow-up alternative holds, that is, if the forward maximal existence time T^ax is finite, then lim^-T -0 |"(圳|か=oo.See e.g.[4].For more general NLS(1.1) idtu + Au 十 |u|p-1 u = 0, (t,x) e I x Rd