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Ruxi Shi 《Journal of Functional Analysis》2019,276(12):3767-3794
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We study standing waves of NLS equation posed on the double-bridge graph: two semi-infinite half-lines attached at a circle. At the two vertices Kirchhoff boundary conditions are imposed. The configuration of the graph is characterized by two lengths, and . We study the solutions with possibly nontrivial components on the half-lines and a cnoidal component on the circle. The problem is equivalent to a nonlinear boundary value problem in which the boundary condition depends on the spectral parameter ω. After classifying the solutions with rational , we turn to irrational showing that there exist standing waves only in correspondence to a countable set of negative frequencies . Moreover we show that the frequency sequence admits cluster points and any negative real number can be a limit point of frequencies choosing a suitable irrational geometry . These results depend on basic properties of diophantine approximation of real numbers. 相似文献