Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities

TL;DR

The paper analyzes the nonlinear Schrödinger equation with double power nonlinearity $i\partial_t u+\Delta u-a|u|^{p-1}u+b|u|^{q-1}u=0$ and establishes instability phenomena for algebraic standing waves, including strong instability for all frequencies when $q\ge 1+4/N$ and instability for small frequencies when $1<p<q<1+4/N$ under a gamma condition. A key contribution is linking zero-mass ground states $\phi_0$ with positive-frequency ground states $\phi_\omega$, proving $\phi_\omega\to\phi_0$ as $\omega\downarrow0$, and showing $d(\omega)\to d(0)$, which enables a zero-frequency instability analysis via variational and virial methods. The authors develop a robust variational framework on the Nehari manifold, derive uniform decay bounds, and establish precise zero-mass limits, enabling a unified treatment of algebraic and exponential standing waves. They introduce a new instability criterion based on the second derivative of the action at scaled profiles and leverage algebraic standing waves to derive instability results in higher dimensions, marking the first such results for algebraic standing waves in this context. Overall, the work advances understanding of orbital instability for mixed-power NLS and highlights the pivotal role of zero-mass solitons in determining the dynamics near the threshold between stability and instability.

Abstract

We consider a nonlinear Schrödinger equation with double power nonlinearity \begin{align*} i\partial_t u+Δu-|u|^{p-1}u+|u|^{q-1}u=0,\quad (t,x)\in\mathbb{R}\times\mathbb{R}^N, \end{align*} where $1<p<q<1+4/(N-2)_+$. Due to the defocusing effect from the lower power order nonlinearity, the equation has algebraically decaying standing waves with zero frequency, which we call algebraic standing waves, as well as usual standing waves decaying exponentially with positive frequency. In this paper we study stability properties of two types of standing waves. We prove strong instability for all frequencies when $q\ge 1+4/N$ and instability for small frequencies when $q<1+4/N$, which especially give the first results on stability properties of algebraic standing waves. The instability result for small positive frequency when $q<1+4/N$ not only improves previous results in one-dimensional case but also gives a first result on instability in higher-dimensional case. The key point in our approach is to take advantage of algebraic standing waves.

Figures (3)

• Figure 1: $N=1$
• Figure 2: $N=2$
• Figure 3: $N\geq 3$

Theorems & Definitions (74)

• Proposition 1.1
• Proposition 1.2: V81
• Remark 1.3
• Corollary 1.4
• proof
• Definition 1.5
• Theorem 1.6
• Remark 1.7
• Theorem 1.8: Strong instability
• Remark 1.9