First order systems of odes with nonlinear nonlocal boundary conditions

In this article, we prove an existence of solutions for a non-local boundary value problem with nonlinearity in a nonlocal condition. Our method is based upon the Mawhin's coincidence theory

Surface-plasmon resonances of arbitrarily shaped nanometallic structures in the small-screening-length limit

According to the hydrodynamic Drude model, surface-plasmon resonances of metallic nanostructures blueshift owing to the nonlocal response of the metal's electron gas. The screening length characterising the nonlocal effect is often small relative to the overall dimensions of the metallic structure, which enables us to derive a coarse-grained nonlocal description using matched asymptotic expansions; a perturbation theory for the blueshifts of arbitrary shaped nanometallic structures is then developed. The effect of nonlocality is not always a perturbation and we present a detailed analysis of the "bonding" modes of a dimer of nearly touching nanowires where the leading-order eigenfrequencies and eigenmode distributions are shown to be a renormalisation of those predicted assuming a local metal permittivity

A nested hybridizable discontinuous Galerkin method for computing second-harmonic generation in three-dimensional metallic nanostructures

In this paper, we develop a nested hybridizable discontinuous Galerkin (HDG) method to numerically solve the Maxwell's equations coupled with the hydrodynamic model for the conduction-band electrons in metals. By means of a static condensation to eliminate the degrees of freedom of the approximate solution defined in the elements, the HDG method yields a linear system in terms of the degrees of freedom of the approximate trace defined on the element boundaries. Furthermore, we propose to reorder these degrees of freedom so that the linear system accommodates a second static condensation to eliminate a large portion of the degrees of freedom of the approximate trace, thereby yielding a much smaller linear system. For the particular metallic structures considered in this paper, the resulting linear system obtained by means of nested static condensations is a block tridiagonal system, which can be solved efficiently. We apply the nested HDG method to compute the second harmonic generation (SHG) on a triangular coaxial periodic nanogap structure. This nonlinear optics phenomenon features rapid field variations and extreme boundary-layer structures that span multiple length scales. Numerical results show that the ability to identify structures which exhibit resonances at $\omega$ and $2\omega$ is paramount to excite the second harmonic response.Comment: 31 pages, 7 figure

Inhomogeneous broadening of tunneling conductance in double quantum wells

The lineshape of the tunneling conductance in double quantum wells with a large-scale roughness of heterointerfaces is investigated. Large-scale variations of coupled energy levels and scattering due to the short-range potential are taken into account. The interplay between the inhomogeneous broadening, induced by the non-screened part of large-scale potential, and the homogeneous broadening due to the scattering by short-range potentials is considered. It is shown that the large inhomogeneous broadening can be strongly modified by nonlocal effects involved in the proposed mechanism of inhomogeneity. Related change of lineshape of the resonant tunneling conductance between Gaussian and Lorentzian peaks is described. The theoretical results agree quite well with experimental data.Comment: 11 pages, 5 figure

Nonexistence of positive solutions of nonlinear boundary value problems

We discuss the nonexistence of positive solutions for nonlinear boundary value problems. In particular, we discuss necessary restrictions on parameters in nonlocal problems in order that (strictly) positive solutions exist. We consider cases that can be written in an equivalent integral equation form which covers a wide range of problems. In contrast to previous work, we do not use concavity arguments, instead we use positivity properties of an associated linear operator which uses ideas related to the $u_0$-positive operators of Krasnosel'skii

Optimal intervals for uniqueness of solutions for nonlocal boundary value problems

For the nth order differential equation (see PDF for equation) we obtain optimal bounds on the length of intervals on which solutions are unique for certain nonlocal three point boundary value problems. These bounds are obtained through an application of the Pontryagin Maximum Principle from the theory of optimal control