论文标题
花图上的站立波
Standing waves on a flower graph
论文作者
论文摘要
花图由一个半线和$ n $对称环组成，该循环连接到一个带有$ n \ geq 2 $的单个顶点（如果$ n = 1 $）。我们在花图上考虑了立方非线性Schrodinger方程的框架上的正面单室状态。我们论文的主要新颖性是对二阶微分方程的周期函数的严格应用，以理解公制图上的驻波的对称性和分叉。我们表明，积极的单层对称状态（这是小固定质量的能量基础）完全出现一个分叉的较大质量，此时出现了其他积极的单杆状态的$（n-1）$分支：每个分支都出现：每个分支都有$ K $较大的组件和$（n-k）$（n-k）$较小的组件，其中$ 1 \ leq kk k \ leq n leq n $ leq \ leq n-1 $。我们表明，只有$ k = 1 $的分支代表大型固定质量的局部能量最小化器，但是，对于较大的固定质量，无法获得能量状态。从周期函数获得的分析结果以数值说明。
A flower graph consists of a half line and $N$ symmetric loops connected at a single vertex with $N \geq 2$ (it is called the tadpole graph if $N = 1$). We consider positive single-lobe states on the flower graph in the framework of the cubic nonlinear Schrodinger equation. The main novelty of our paper is a rigorous application of the period function for second-order differential equations towards understanding the symmetries and bifurcations of standing waves on metric graphs. We show that the positive single-lobe symmetric state (which is the ground state of energy for small fixed mass) undergoes exactly one bifurcation for larger mass, at which point $(N-1)$ branches of other positive single-lobe states appear: each branch has $K$ larger components and $(N-K)$ smaller components, where $1 \leq K \leq N-1$. We show that only the branch with $K = 1$ represents a local minimizer of energy for large fixed mass, however, the ground state of energy is not attained for large fixed mass. Analytical results obtained from the period function are illustrated numerically.
