论文标题
在麦克斯韦方程中渐近耦合模式的渐近耦合模式近似的理由
Justification of the Asymptotic Coupled Mode Approximation of Out-of-Plane Gap Solitons in Maxwell Equations
论文作者
论文摘要
在周期性的介质间隙孤子中，光谱间隙内有频率但接近光谱带的频率可以通过缓慢变化的包膜ANSATZ正式近似。 ANSATZ基于频带边缘的线性BLOCH波，以及用于信封的有效耦合模式方程（CME）。我们在Kerr非线性Maxwell系统描述的二维光子晶体中提供了这种CME渐近药的严格理由。我们使用Bloch变量中的Lyapunov-Schmidt还原过程和嵌套的固定点参数。该定理在精确解决方案和信封近似之间提供了$ H^2（\ Mathbb r^2）$中的错误估计。结果证明了[多尺度模型的Dohnal和Dörfler中的形式和数值CME Approximation。 Simul。，p。 162-191，11（2013）]。
In periodic media gap solitons with frequencies inside a spectral gap but close to a spectral band can be formally approximated by a slowly varying envelope ansatz. The ansatz is based on the linear Bloch waves at the edge of the band and on effective coupled mode equations (CMEs) for the envelopes. We provide a rigorous justification of such CME asymptotics in two-dimensional photonic crystals described by the Kerr nonlinear Maxwell system. We use a Lyapunov-Schmidt reduction procedure and a nested fixed point argument in the Bloch variables. The theorem provides an error estimate in $H^2(\mathbb R^2)$ between the exact solution and the envelope approximation. The results justify the formal and numerical CME-approximation in [Dohnal and Dörfler, Multiscale Model. Simul., p. 162-191, 11 (2013)].
