Strong Instability of Standing Waves for the Nonlinear Schrodinger Equation in Trapped Dipolar Quantum Gases

In this paper, we study the strong instability of standing waves for the nonlinear Schrodinger equation arising in trapped dipolar quantum gases. Two cases are considered: the first when the system is free, the second when a partial/complete harmonic potential is added. In the free case, we present a new argument to prove that the ground state standing waves are strongly unstable by blow-up. In the second case, if partial derivative S-2(mu)omega(Q(omega)(mu))vertical bar(mu=1) <= 0, we deduce that the ground state standing wave u(t, x) = e(i omega t) Q(omega)(x) is strongly unstable by blow-up, where S-omega is the action, and Q(omega)(mu) = mu(3/2)Q(omega)(mu x) is the L-2-invariant scaling.