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arXiv 1910.01482math-phmath.APmath.CAmath.MPnlin.PS
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一维晶格上的Chern-Simons-Schrodinger理论

Chern-Simons-Schrodinger theory on a one-dimensional lattice

Hyungjin Huh, Swaleh Hussain, Dmitry E. Pelinovsky

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21 pages; 5 figures
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摘要：

我们提出了一个一维格点上的Chern-Simons-Schrodinger型规范不变系统。利用空间规范条件，证明了标量场平方可和序列空间中初值问题的局部适定性和整体适定性。我们还研究了定态束缚态的存在区域，它依赖于晶格间距和非线性功率。存在性问题的一个主要困难是缺乏固定方程的变分公式。我们的方法是基于反连续极限中的隐函数定理和连续极限中的可解性约束。

英文摘要：

We propose a gauge-invariant system of the Chern-Simons-Schrodinger type on a one-dimensional lattice. By using the spatial gauge condition, we prove local and global well-posedness of the initial-value problem in the space of square summable sequences for the scalar field. We also study the existence region of the stationary bound states, which depends on the lattice spacing and the nonlinearity power. A major difficulty in the existence problem is related to the lack of variational formulation of the stationary equations. Our approach is based on the implicit function theorem in the anti-continuum limit and the solvability constraint in the continuum limit.

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