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

Abstract

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

26 pages, 2 figures, final version

References in corpus (2)

• The double-power nonlinear Schrödinger equation and its generalizations: uniqueness, non-degeneracy and applications
• Stability of algebraic solitons for nonlinear Schrödinger equations of derivative type: variational approach