双桥图上NLS的驻波和有理-无理二分法
Standing waves for the NLS on the double-bridge graph and a rational-irrational dichotomy
摘要：
研究了图上非线性薛定谔方程（NLS）驻波搜索的边值问题。精确地说，我们感兴趣的是描述{ \it双桥图}上的NLS驻波，其中两个半无限的半直线连接在不同顶点的圆上。在两个顶点处施加所谓的基尔霍夫边界条件。图的结构用两个长度$L_1$和$L_2$来表征，我们感兴趣的是给定频率$ω$的驻波的存在性和性质.对于每个$ω>0$，仅存在圆上支持的解（椭圆余弦解），并且仅针对$L_1/L_2$的有理值;它们可以扩展到每个$ω\in \mathbb{R}$。我们研究了$ω<0$的解在圆上周期，但在半直线上具有非平凡分量。该问题被证明是等价的非线性边值问题，其中的边界条件依赖于谱参数$ω$。在用有理$L_1/L_2$对解进行分类之后，我们转向$L_1/L_2$无理，表明存在驻波仅对应于可计数的一组频率$ω_n$。证明了频率序列$\{ω_n\}_{n \geq 1}$在$-\infty$上有一个聚点，并且它至少有一个有限的极限点，一般不为零.最后，任何负实数都可以是一组允许频率的极限点，直到为图选择合适的无理几何$L_1/L_2$。这些结果依赖于实数的丢番图逼近的基本性质。
英文摘要：
We study a boundary value problem related to the search of standing waves for the nonlinear Schrödinger equation (NLS) on graphs. Precisely we are interested in characterizing the standing waves of NLS posed on the {\it double-bridge graph}, in which two semi-infinite half-lines are attached at a circle at different vertices. At the two vertices the so-called Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, $L_1$ and $L_2$, and we are interested in the existence and properties of standing waves of given frequency $ω$. For every $ω>0$ only solutions supported on the circle exist (cnoidal solutions), and only for a rational value of $L_1/L_2$; they can be extended to every $ω\in \mathbb{R}$. We study, for $ω<0$, the solutions periodic on the circle but with nontrivial components on the half-lines. The problem turns out to be equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter $ω$. After classifying the solutions with rational $L_1/L_2$, we turn to $L_1/L_2$ irrational showing that there exist standing waves only in correspondence to a countable set of frequencies $ω_n$. Moreover we show that the frequency sequence $\{ω_n\}_{n \geq 1}$ has a cluster point at $-\infty$ and it admits at least a finite limit point, in general non-zero. Finally, any negative real number can be a limit point of a set of admitted frequencies up to the choice of a suitable irrational geometry $L_1/L_2$ for the graph. These results depend on basic properties of diophantine approximation of real numbers.
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