We study instability of a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ to $$iu_t+\Delta u+|u|^{p-1}u=0,\quad\text{for $x\in \mathbb{R}^n$, $t>0$,}$$ where $n=2$, $m\in \mathbb{N}$ and $(r,\theta)$ are polar coordinates in $\mathbb{R}^2$. Grillakis \cite{Gr} proved that every radially symmetric standing wave solution is unstable if $p>1+4/n$. However, we do not have any examples of unstable standing wave solutions in the subcritical case $(p <1+n/4)$. Suppose $\phi_{\omega,m}$ is nonnegative. We investigate a limiting profile of $\phi_{\omega,m}$ as $m\to\infty$ and prove that, for every $p>1$, there exists an $m_*\in \mathbb{N}$ such that, for $m\ge m_*$, a vortex soliton $e^{i(m\theta+\omega t)}\phi_{\omega,m}(r)$ becomes unstable under perturbations of the form $e^{i(m+j)\theta}v(r)$ with $1\ll j\ll m$.

In this article, we investigate a way to analyze and approximate singularly perturbed convection-diffusion equations in a channel domain when a nonlinear reaction term with polynomial growth is present. We verify that the boundary layer structures are governed by certain simple recursive linear equations and this simplicity implies explicit pointwise and norm estimates. Furthermore, we can utilize the boundary layer structures (elements) in the finite elements discretizations which lead to the stability in the approximating systems and accurate approximation solutions with an economical mesh design, i.e., uniform mesh.

We establish the existence, the uniqueness, and the stability of the solution to the Cauchy problem associated to a general class of systems of first-order linear partial differential equations under minimal regularity assumptions on their coefficients.