论文标题
具有双功率非线性的非线性schrödinger方程的代数立波的不稳定性
Instability of algebraic standing waves for nonlinear Schrödinger equations with double power nonlinearities
论文作者
论文摘要
我们考虑具有双重功率非线性的非线性schrödinger方程{align*} i \ partial_t u+u+Δu-| u |^{p-1} p-1} u+| U+| |^{q-1} u = 0，\ quad（t i = $ 1 <p <q <1+4/（n-2）_+$。由于较低功率阶的非线性产生偏置作用，该方程的频率为零代数衰减的站立波，我们称之为代数驻波，以及通常以正频率呈指数衰减的常静波。在本文中，我们研究了两种类型的驻波的稳定性。当$ q \ ge 1+4/n $且小频率$ q <1+4/n $时，我们证明了所有频率的强大不稳定，这尤其给出了代数驻留波的稳定性属性的第一个结果。当$ q <1+4/n $ $ q <1+4/n $时，不稳定的不稳定不仅可以在一维情况下改善先前的结果，而且还为高维情况下的不稳定性带来了第一个结果。我们方法的关键是利用代数立波。
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.
