Stability of standing wave for the fractional nonlinear Schrodinger equation

In this paper, we study the stability and instability of standing waves for the fractional nonlinear Schrodinger equation i partial derivative(t)u = (-Delta)(s)u - vertical bar u vertical bar(2 sigma) u, where (t, x) is an element of R x R-N, 1/2 < s < 1, and N >= 2. Using a sharp Gagliardo-Nirenberg-type inequality and profile decomposition, we obtain that when 0 < sigma < 2s/N, the standing waves are orbitally stable; when sigma = 2s/N, the ground state solitary waves are strongly unstable to blowup. Published by AIP Publishing.