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

We consider a double power nonlinear Schrödinger equation which possesses the algebraically decaying stationary solution ${\phi }_0$ as well as exponentially decaying standing waves $e^{i\omega t}{\phi }_{\omega }(x)$ with $\omega >0$. It is well-known from the general theory that stability properties of standing waves are determined by the derivative of $\omega \mapsto M(\omega ):=\frac{1}{2}\mathrm{\parallel }{\phi }_{\omega }{\mathrm{\parallel }}_{L^2}^2$; namely $e^{i\omega t}{\phi }_{\omega }$ with $\omega >0$ is stable if $M^{\mathrm{\prime }}(\omega )>0$ and unstable if $M^{\mathrm{\prime }}(\omega )<0$. 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 $M^{\mathrm{\prime }}(0):={\lim }_{\omega \downarrow 0}{M}^{\mathrm{\prime }}(\omega )\in [-\mathrm{\infty },0)$. The key in the proof is the construction of the one-sided derivative of $\omega \mapsto {\phi }_{\omega }$ at $\omega =0$, which is effectively used to construct the unstable direction of ${\phi }_0$.