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

TL;DR

This work analyzes the instability of the algebraically decaying stationary solution $\phi_0$ for the one-dimensional nonlinear Schrödinger equation with double power nonlinearity. By introducing the one-sided derivative limit $\eta_0=\lim_{\omega\downarrow 0}\partial_\omega\phi_\omega$ and proving $L_0\eta_0=-\phi_0$, the authors connect the sign of $M'(0)=\int_{\mathbb{R}}\phi_0\eta_0\,dx$ to instability. They construct unstable directions using cutoff perturbations $\psi_R=\phi_0+\beta_R\chi_R\eta_0$ and establish precise asymptotics, leading to a variational instability criterion analogous to the GSS framework but applicable at the zero-frequency limit. The main result shows that in 1D, if $M'(0)\in[-\infty,0)$ (in particular when $2p+q>7$ with $1<p<q<5$), the stationary solution $\phi_0$ is unstable. This extends instability analysis to stationary states where conventional spectral coercivity fails and highlights the sharp role of $M'(0)$ in the double-power NLS dynamics.

Abstract

We consider a double power nonlinear Schrödinger equation which possesses the algebraically decaying stationary solution $φ_0$ as well as exponentially decaying standing waves $e^{iωt}φ_ω(x)$ with $ω>0$. It is well-known from the general theory that stability properties of standing waves are determined by the derivative of $ω\mapsto M(ω):=\frac{1}{2}\|φ_ω\|_{L^2}^2$; namely $e^{iωt}φ_ω$ with $ω>0$ is stable if $M'(ω)>0$ and unstable if $M'(ω)<0$. 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 $φ_0$ in one dimension under the condition $M'(0):=\lim_{ω\downarrow 0}M'(ω)\in[-\infty, 0)$. The key in the proof is the construction of the one-sided derivative of $ω\mapstoφ_ω$ at $ω=0$, which is effectively used to construct the unstable direction of $φ_0$.

Figures (2)

• Figure 1: Comparison of three conditions
• Figure 2: Stability/instability with small $\omega>0$

Theorems & Definitions (47)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• Lemma 2.3
• proof