We consider a class of generally non-self-adjoint eigenvalue problems which are nonlinear in the solution as well as in the eigenvalue parameter (“doubly” nonlinear). We prove a bifurcation result from simple isolated eigenvalues of the linear problem using a Lyapunov–Schmidt reduction and provide an expansion of both the nonlinear eigenvalue and the solution. We further prove that if the linear eigenvalue is real and the nonlinear problem \({\mathcal {PT}}\)-symmetric, then the bifurcating nonlinear eigenvalue remains real. These general results are then applied in the context of surface plasmon polaritons (SPPs), i.e. localized solutions for the nonlinear Maxwell’s equations in the presence of one or more interfaces between dielectric and metal layers. We obtain the existence of transverse electric SPPs in certain \({\mathcal {PT}}\)-symmetric configurations.

我们考虑一类通常非自伴特征值问题，这些问题在解中以及在特征值参数（“双重”非线性）中都是非线性的。我们使用Lyapunov–Schmidt约简从线性问题的简单孤立特征值证明了分支结果，并提供了非线性特征值和解的扩展。我们进一步证明，如果线性特征值是实数并且是非线性问题\（{\ mathcal {PT}} \）对称，则分叉非线性特征值仍然是实数。然后将这些一般结果应用于表面等离振子极化子（SPP），即在介电层和金属层之间存在一个或多个界面的情况下，非线性麦克斯韦方程组的局部解。我们获得了某些（\ {{\ mathcal {PT}} \}）对称配置中的横向电SPP的存在。