波导中非平面边界下的变量精确半分离
Exact semi-separation of variables in waveguides with nonplanar boundaries
AI总结：
本文提出了一种新的方法，通过改进的级数展开实现波导中非平面边界下的精确快速收敛变量分离，适用于任意光滑领域。
摘要：
在长形波导中，未知场Φ=∑φ_n Z_n的级数展开常用于声学、光学、地球物理、水波等应用，属于耦合模理论(CMTs)的范畴。横切函数Z_n通过求解局部Sturm-Liouville问题（参考波导）确定。通常情况下，分配给Z_n的边界条件无法与Φ的物理边界条件兼容，导致缓慢收敛的级数，使CMTs成为近似方法。本文将Athanassoulis和Belibassakis（1999）引入的启发式方法推广并证明。证明了适当增强的级数展开成为Φ场的精确快速收敛表示，适用于任何光滑非平面边界和任何光滑的Φ。该级数展开可以在域内任何地方逐项求导，包括边界，实现非分离域的精确半分离变量。通过求解拉普拉斯方程的边界值问题和计算相应的狄利克雷-诺依曼算子（涉及非线性水波的哈密顿方程）来展示该方法的效率。本文方法仅需少量模式即可对相当一般的领域提供准确结果。还讨论了对一般波导的扩展。
英文摘要：
Series expansions of unknown fields $Φ=\sumφ_n Z_n$ in elongated waveguides are commonly used in acoustics, optics, geophysics, water waves and other applications, in the context of coupled-mode theories (CMTs). The transverse functions $Z_n$ are determined by solving local Sturm-Liouville problems (reference waveguides). In most cases, the boundary conditions assigned to $Z_n$ cannot be compatible with the physical boundary conditions of $Φ$, leading to slowly convergent series, and rendering CMTs mild-slope approximations. In the present paper, the heuristic approach introduced in (Athanassoulis & Belibassakis 1999, J. Fluid Mech. 389, 275-301) is generalized and justified. It is proved that an appropriately enhanced series expansion becomes an exact, rapidly-convergent representation of the field $Φ$, valid for any smooth, nonplanar boundaries and any smooth enough $Φ$. This series expansion can be differentiated termwise everywhere in the domain, including the boundaries, implementing an exact semi-separation of variables for non-separable domains. The efficiency of the method is illustrated by solving a boundary value problem for the Laplace equation, and computing the corresponding Dirichlet-to-Neumann operator, involved in Hamiltonian equations for nonlinear water waves. The present method provides accurate results with only a few modes for quite general domains. Extensions to general waveguides are also discussed.
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