On the variational approach to the stability of standing waves for the nonlinear Schrodinger equation

We consider the orbital stability of standing waves of the nonlinear Schrdinger equation i(partial derivativePhi)/(partial derivativet) (t,x) + DeltaPhi(t, x) + g(x, \Phi\(2))Phi(t, x) = 0 by the approach that was laid down by Cazenave and Lions in 1992. Our work covers several situations that do not seem to be included in previous treatments, namely, (i) g(x, s) - g(x, 0) --> 0 as \x\ --> infinity for all s greater than or equal to 0. This includes linear problems. (ii) g(x, s) is a periodic function of x is an element of R-N for all s greater than or equal to 0. (iii) g(x, s) is asymptotically periodic in the sense that g(x, s) - g(infinity)(x, s) --> 0 as \x\ --> infinity for some function g(infinity) that is periodic with respect to x is an element of R-N for all s greater than or equal to 0. Furthermore, we focus attention on the form of the set that is shown to be stable and may be bigger than what is usually known as the orbit of the standing wave.