论文标题
nls- $δ'_s$方程的动力和变异属性在星形图上
Dynamical and variational properties of the NLS-$δ'_s$ equation on the star graph
论文作者
论文摘要
我们研究了$nlinearschrödinger方程，并在$ n $ n $ half-lines组成的星形图$γ$上使用\ mathbb {r} \ setMinus \ {0 \} $ in \ mathbb {r} \ setMinus \ {0 \} $的$δ'_s$耦合。非线性具有$ g（u）= | u |^{p-1} u，p> 1的形式。在本文的第一部分，在对$β$的一定限制下，我们证明了基态解决方案的存在，作为动作功能$s_Ω$的最小化。看来包含基态的关键点家族由$ n $ profiles（一个对称和$ n-1 $不对称）组成。特别是，对于有吸引力的$δ'_s$耦合（$β<0 $）和高于一定阈值的频率$ω$，我们设法指定了基态。 第二部分致力于研究临界点的轨道不稳定性。我们使用Grillakis/Jones不稳定定理证明了临界点的光谱不稳定。然后，$ P> 2 $的轨道不稳定是从与方程相关的数据解决映射的事实，是$ h^1（γ）$的类$ c^2 $。此外，对于$ p> 5 $，我们完成并为特定关键点的不稳定结果（通过有限的时间吹来）。
We study the nonlinear Schrödinger equation with $δ'_s$ coupling of intensity $β\in\mathbb{R}\setminus\{0\}$ on the star graph $Γ$ consisting of $N$ half-lines. The nonlinearity has the form $g(u)=|u|^{p-1}u, p>1.$ In the first part of the paper, under certain restriction on $β$, we prove the existence of the ground state solution as a minimizer of the action functional $S_ω$ on the Nehari manifold. It appears that the family of critical points which contains a ground state consists of $N$ profiles (one symmetric and $N-1$ asymmetric). In particular, for the attractive $δ'_s$ coupling ($β<0$) and the frequency $ω$ above a certain threshold, we managed to specify the ground state. The second part is devoted to the study of orbital instability of the critical points. We prove spectral instability of the critical points using Grillakis/Jones Instability Theorem. Then orbital instability for $p>2$ follows from the fact that data-solution mapping associated with the equation is of class $C^2$ in $H^1(Γ)$. Moreover, for $p>5$ we complete and concertize instability results showing strong instability (by blow up in finite time) for the particular critical points.
