arXiv subjects

Cui Ning

Publications and source records attributed to Cui Ning.

4 recordsLinked to original sources

arXiv2026

Orbital instability of solitary waves for the generalized BBM equation at the negative critical endpoint

We study the orbital stability of solitary waves to the generalized Benjamin--Bona--Mahony equation $$ u_t+u_x+\kappa\frac{p+1}{2}(u^p)_x-u_{txx}=0, \qquad (t,x)\in\mathbb R^+\times\mathbb R, $$ where $\kappa\in\mathbb R\setminus\{0\}$ and $p\geq2$ is an integer. This equation admits solitary waves of the form $$ u(t,x)=\phi_c(x-ct). $$ At the negative critical endpoint speed $$ c=c_p^-= \frac{p-1}{2(p+1)} \left(1-\sqrt{\frac{p+3}{2}}\right)<0, $$ we prove that the corresponding solitary wave is orbitally unstable whenever either $\kappa<0$, or $\kappa>0$ and $p$ is even. The proof combines a codimension-two coercivity estimate, two-parameter modulation, and a corrected localized virial functional adapted to the degeneracy of the momentum slope at the endpoint.

math.AP↗

arXiv2025

On blowup solution in NLS equation under dispersion or nonlinearity management

In this paper, we study the dispersion-managed nonlinear Schr\"odinger (DM-NLS) equation $$ i\partial_t u(t,x)+\gamma(t)\Delta u(t,x)=|u(t,x)|^{\frac4d}u(t,x),\quad x\in\R^d, $$ and the nonlinearity-managed NLS (NM-NLS) equation: $$ i\partial_t u(t,x)+\Delta u(t,x)=\gamma(t)|u(t,x)|^{\frac4d}u(t,x), \quad x\in\R^d, $$ where $\gamma(t)$ is a periodic function which is equal to $-1$ when $t\in (0,1]$ and is equal to $1$ when $t\in (1,2]$. The two models share the feature that the focusing and defocusing effects convert periodically. For the classical focusing NLS, it is known that the initial data $$ u_0(x)=T^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4T} -i\frac{\omega^2}{T}}Q_\omega\left(\frac{x}{T}\right) $$ leads to a blowup solution $$(T-t)^{-\frac{d}{2}}\fe^{i\frac{|x|^2}{4(T-t)} -i\frac{\omega^2}{T-t}}Q_\omega\left(\frac{x}{T-t}\right), $$ so when $T\leq1$, this is also a blowup solution for DM-NLS and NM-NLS which blows up in the first focusing layer. For DM-NLS, we prove that when $T>1$, the initial data $u_0$ above does not lead to a finite-time blowup and the corresponding solution is globally well-posed. For NM-NLS, we prove the global well-posedness for $T\in(1,2)$ and we construct solution that can blow up at any focusing layer. The theoretical studies are complemented by extensive numerical explorations towards understanding the stabilization effects in the two models and addressing their difference.

math.AP↗

arXiv2018

Instability of the solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation in the endpoint case

We consider the stability theory of solitary wave solutions for the generalized derivative nonlinear Schr\"odinger equation $$ i\partial_{t}u+\partial_{x}^{2}u+i|u|^{2\sigma}\partial_x u=0, $$ where $1<\sigma<2$. The equation has a two-parameter family of solitary wave solutions of the form $$ u_{\omega,c}(t,x)=e^{i\omega t+i\frac c2(x-ct)-\frac{i}{2\sigma+2}\int_{-\infty}^{x-ct}\varphi^{2\sigma}_{\omega,c}(y)dy}\varphi_{\omega,c}(x-ct). $$ The stability theory in the frequency region of $|c|<2\sqrt{\omega}$ was studied previously. In this paper, we prove the instability of the solitary wave solutions in the endpoint case $c=2\sqrt{\omega}$.

math.AP↗

arXiv2018

Instability of the solitary wave solutions for the genenalized derivative Nonlinear Schr\"odinger equation in the critical frequency case

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 $3/2<\sigma<2$.

math.AP↗