Linear Maxwell equations for transverse magnetic (TM) polarized fields support single frequency surface plasmon polaritons (SPPs) localized at the interface of a metal and a dielectric. Metals are typically dispersive, i.e. the dielectric function depends on the frequency. We prove the bifurcation of localized SPPs in dispersive media in the presence of a cubic nonlinearity and provide an asymptotic expansion of the solution and the frequency. The problem is reduced to a system of nonlinear differential equations in one spatial dimension by assuming a plane wave dependence in the direction tangential to the (flat) interfaces. The number of interfaces is arbitrary and the nonlinear system is solved in a subspace of functions with the -Sobolev regularity in each material layer. The corresponding linear system is an operator pencil in the frequency parameter due to the material dispersion. The studied bifurcation occurs at a simple isolated eigenvalue of the pencil. For geometries consisting of two or three homogeneous layers we provide explicit conditions on the existence of eigenvalues and on their simpleness and isolatedness. Real frequencies are shown to exist in the nonlinear setting in the case of -symmetric materials. We also apply a finite difference numerical method to the nonlinear system and compute bifurcating curves.

中文翻译：

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立方非线性横磁表面等离子体激元的分岔和渐近

横向磁场 (TM) 极化场的线性麦克斯韦方程支持位于金属和电介质界面处的单频表面等离子体激元 (SPP)。金属通常是色散的，即介电函数取决于频率。我们证明了在存在三次非线性的情况下色散介质中局域表面等离子体激元的分岔，并提供了解和频率的渐近展开。通过假设与（平坦）界面相切的方向上的平面波相关性，该问题被简化为一个空间维度中的非线性微分方程组。界面的数量是任意的，非线性系统在每个材料层中具有-Sobolev 正则的函数子空间中求解。由于材料色散，相应的线性系统是频率参数中的操作员铅笔。所研究的分岔发生在铅笔的简单孤立特征值处。对于由两个或三个同质层组成的几何形状，我们提供了关于特征值的存在及其简单性和孤立性的明确条件。在 对称材料的情况下，真实频率存在于非线性设置中。我们还将有限差分数值方法应用于非线性系统并计算分叉曲线。