Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
Justification of the coupled-mode approximation for a nonlinear elliptic problem with a periodic potential
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具有周期性势的非线性椭圆问题的耦合模式近似的论证
DOI：
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发表时间：
2007
期刊：
影响因子：
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通讯作者：
G. Schneider
中科院分区：
文献类型：
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作者：
D. Pelinovsky;G. Schneider
在物理文献中，耦合模系统用于简化具有小周期势的非线性Maxwell方程和Gross-Pitaevskii方程，并通过包含双曲函数的解析表达式来近似称为间隙孤子的局域解。我们通过在傅里叶空间中采用Lyapunov-Schmidt约简的方法，证明了在相关椭圆问题中使用一维平稳耦合模系统是正确的。特别地，证明了周期解/反周期解和衰减解的存在性，并将误差项控制在合适的范数内。利用多维平稳耦合模系统分析了小多维周期势下周期/反周期解的分岔问题。
Coupled-mode systems are used in physical literature to simplify the nonlinear Maxwell and Gross–Pitaevskii equations with a small periodic potential and to approximate localized solutions called gap solitons by analytical expressions involving hyperbolic functions. We justify the use of the 1D stationary coupled-mode system for a relevant elliptic problem by employing the method of Lyapunov–Schmidt reductions in Fourier space. In particular, existence of periodic/anti-periodic and decaying solutions is proved and the error terms are controlled in suitable norms. The use of multi-dimensional stationary coupled-mode systems is justified for analysis of bifurcations of periodic/anti-periodic solutions in a small multi-dimensional periodic potential.