Strong Confinement Limit for the Nonlinear Schrödinger Equation Constrained on a Curve

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• 作者：Florian Méhats ; Nicolas Raymond
• 刊名：Annales Henri Poincaré
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：18
• 期：1
• 页码：281-306
• 全文大小：
• 刊物类别：Physics and Astronomy
• 刊物主题：Theoretical, Mathematical and Computational Physics; Dynamical Systems and Ergodic Theory; Quantum Physics; Mathematical Methods in Physics; Classical and Quantum Gravitation, Relativity Theory; Eleme
• 出版者：Springer International Publishing
• ISSN：1424-0661
• 卷排序：18

文摘

This study is devoted to the cubic nonlinear Schrödinger equation in a two-dimensional waveguide with shrinking cross section of order \({\varepsilon}\). For the Cauchy data living essentially on the first mode of the transverse Laplacian, we provide a tensorial approximation of the solution \({\psi^{\varepsilon}}\) in the limit \({\varepsilon \to 0}\), with an estimate of the approximation error, and derive a limiting nonlinear Schrödinger equation in dimension one. If the Cauchy data \({\psi^{\varepsilon}_0}\) have a uniformly bounded energy, then it is a bounded sequence in \({\mathsf{H}^1}\), and we show that the approximation is of order \({\mathcal{O}(\sqrt{\varepsilon})}\). If we assume that \({\psi^{\varepsilon}_0}\) is bounded in the graph norm of the Hamiltonian, then it is a bounded sequence in \({\mathsf{H}^{2}}\), and we show that the approximation error is of order \({\mathcal{O}(\varepsilon)}\).