Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case
Instability of the solitary wave solutions for the generalized derivative nonlinear Schrödinger equation in the critical frequency case
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DOI：
10.4310/mrl.2020.v27.n2.a2
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发表时间：
2018-03
影响因子：
1
通讯作者：
Zihua Guo;Cui Ning;Yifei Wu
中科院分区：
文献类型：
--
作者：
Zihua Guo;Cui Ning;Yifei Wu
我们研究了广义导数非线性 Schr\"odinger 方程 $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0 的孤立波解的稳定性理论。 $$ 该方程具有以下形式的孤立波解的二参数族 \begin{align*} \phi_{\omega,c}(x)=\varphi_{\omega,c}(x)\exp{\big\{ i\frac c2 x-\frac{i}{2\sigma+2}\int_{-\infty}^{x}\varphi^{2\sigma}_{\omega,c}(y)dy\big\}}。 \end{align*} 这里 $ \varphi_{\omega,c}$ 是 一些实值函数。在\cite{LiSiSu1}中证明，如果$-2\sqrt{\omega }<c <2z_0\sqrt{\omega }$，孤立波解是稳定的，如果$2z_0\sqrt{\omega }<c，则孤立波解是不稳定的 <2\sqrt{\omega }$ 对于某些 $z_0\in(0,1)$。我们证明了边界情况 $c =2z_0\sqrt{\omega }$ 对于 $1<\sigma<2$ 的不稳定性，改进了 \cite{Fu-16-DNLS} 中 $3/2<\sigma<2$ 的先前结果。
We study the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0. $$ The equation has a two-parameter family of solitary wave solutions of the form \begin{align*} \phi_{\omega,c}(x)=\varphi_{\omega,c}(x)\exp{\big\{ i\frac c2 x-\frac{i}{2\sigma+2}\int_{-\infty}^{x}\varphi^{2\sigma}_{\omega,c}(y)dy\big\}}. \end{align*} Here $ \varphi_{\omega,c}$ is some real-valued function. It was proved in \cite{LiSiSu1} that the solitary wave solutions are stable if $-2\sqrt{\omega }<c <2z_0\sqrt{\omega }$, and unstable if $2z_0\sqrt{\omega }<c <2\sqrt{\omega }$ for some $z_0\in(0,1)$. We prove the instability at the borderline case $c =2z_0\sqrt{\omega }$ for $1<\sigma<2$, improving the previous results in \cite{Fu-16-DNLS} where $3/2<\sigma<2$.