Instability of stationary solutions for double power nonlinear Schrodinger equations in one dimension

We consider a double power nonlinear Schrodinger equation possessing the algebraically decaying stationary solution phi(0) as well as exponentially decaying standing waves ei omega t phi omega(x) with omega>0. According to the general theory, stability properties of standing waves are determined by the derivative of omega bar right arrow M(omega):=1/2||phi omega||(2)(L2); namely ei omega t phi omega with omega>0 is stable if M '(omega)>0 and unstable if M '(omega)<0. However, the stability/instability of stationary solutions is outside the general theory from the viewpoint of spectral properties of linearized operators. In this paper we prove the instability of the stationary solution phi(0) in one dimension under the condition lim omega down arrow 0M '(omega)is an element of[-infinity,0). The key in the proof is the construction of the one-sided derivative of omega bar right arrow phi(omega) at omega=0, which is effectively used to construct the unstable direction.