Related papers: Non-linear eigenvalue problems and applications to…
Studies aimed at understanding the global properties of the hyperpolarizabilities have focused on identifying universal properties when the hyperpolarizabilities are at the fundamental limit. These studies have taken two complimentary…
Photonic hyper-crystals combine the most interesting features of hyperbolic metamaterials and photonic crystals. Since the dispersion law of extraordinary photons in hyperbolic metamaterials does not exhibit the usual diffraction limit,…
Using sum rules and a new dipole-free sum-over-states expression, we calculate the fundamental limits of the dispersion of the real and imaginary parts of all electronic nonlinear-optical susceptibilities. As such, these general results can…
We apply a rigorous eigenmode analysis to study the electromagnetic properties of linear and weakly nonlinear metamaterials. The nonlinear response can be totally described by the linear eigenmodes when weak nonlinearities are attributed to…
We derive novel guaranteed lower bounds for eigenvalues of the Euler-Bernoulli beam with variable bending stiffness. While the standard finite element Rayleigh-Ritz method automatically yields upper bounds, we obtain lower bounds by…
We present numerical upscaling techniques for a class of linear second-order self-adjoint elliptic partial differential operators (or their high-resolution finite element discretization). As prototypes for the application of our theory we…
In this paper, we are concerned with a shape design problem, in which our target is to design, up to rigid transformations and scaling, the shape of an object given either its polarization tensor at multiple contrasts or the partial…
We present an efficient procedure for computing resonances and resonant modes of Helmholtz problems posed in exterior domains. The problem is formulated as a nonlinear eigenvalue problem (NEP), where the nonlinearity arises from the use of…
A simple and efficient variational method is introduced to accelerate the convergence of the eigenenergy computations for a Hamiltonian H with singular potentials. Closed-form analytic expressions in N dimensions are obtained for the matrix…
We consider the bifurcation problem $u'' + \lambda u = N(u)$ with two point boundary conditions where $N(u)$ is a general nonlinear term which may also depend on the eigenvalue $\lambda$. We give a variational characterization of the…
This paper presents a method for computing eigenvalues and eigenvectors for some types of nonlinear eigenvalue problems. The main idea is to approximate the functions involved in the eigenvalue problem by rational functions and then apply a…
We define and study the properties of Darboux-type transformations between Sturm--Liouville problems with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter (including the Dirichlet boundary…
An optimal control approach based on multiple parameter genetic algorithms is applied to the design of plasmonic nanoconstructs with pre-determined optical properties and functionalities. We first develop nanoscale metallic lenses that…
Quasi-normal-eigenvalue optimization is studied under constraints $b_1(x) \le B(x) \le b_2 (x)$ on structure functions $B$ of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when…
We consider a two-spectra inverse problem for the one-dimensional Schr\"{o}dinger equation with boundary conditions containing rational Herglotz--Nevanlinna functions of the eigenvalue parameter and provide a complete solution of this…
In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image-processing, graph partition and classification, nonlinear physics and more. The canonical eigenproblem we…
We investigate nonlinear one- and two-dimensional photonic crystals by applying a finite element-iterative method.Numerical results show the essential influence of nonlinear elements embedded into a quarter-wave stack and the sharp photonic…
We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying…
We use our vector Maxwell's nonlinear eigenmode solver to study the stationary solutions in 2D cross-section waveguides with Kerr nonlinear cores. This solver is based on the fixed power algorithm within the finite element method. First,…
We study the spectral problems associated with the finite-difference operators $H_N = 2 \cosh(p) + V_N(x)$, where $V_N(x)$ is an arbitrary polynomial potential of degree $N$. These systems can be regarded as a solvable deformation of the…