This paper presents an extension of the variational meshless method to the calculation of the dispersion diagram of metallic waveguides including inhomogeneous dielectric regions. The method is based ...on the combination of the variational formulation of a 2-D boundary problem and of the meshless method using radial basis functions. The problem requires a vector representation of the field, and it leads to a well-conditioned, real, and symmetric eigenproblem, where the matrices depend on the propagation constant <inline-formula> <tex-math notation="LaTeX">\beta </tex-math></inline-formula>. By solving the eigenproblem for several values of <inline-formula> <tex-math notation="LaTeX">\beta </tex-math></inline-formula>, a spurious-free dispersion diagram of the guiding structure is obtained. Several structures are studied to demonstrate the accuracy and reliability of the proposed technique. The simulation results are compared with the analytical ones, when available, and with those given by a commercial FEM code, always showing a very good agreement with a smaller number of unknowns.
We investigate the dynamics of Gaussian and Super-Gaussian optical beams in weakly nonlocal nonlinear media with cubic quintic nonlinearities. Through the variational method, a set of internal ...parameters of beam propagation are constructed and their properties, illustrated from numerical simulations. We discover that Gaussian and Super-Gaussian optical beams generally perform stable propagation with different dynamics depending on whether beams order, quintic nonlinearity and/or nonlocality are taken into account. The evolution of the light beams is periodic due to the competition between nonlinearities and diffraction. The phase front curvature increases with the beams order when the width decreases. The numerical study of the interaction between a Gaussian and Super-Gaussian optical beams, shows that with or without the quintic nonlinearity and/or nonlocality, various behaviours such as repulsion, attraction and soft shock waves are observed. Moreover, new phenomena are exhibited in the case of the Super-Gaussian, according to their shape.
•Gaussian and Super-Gaussian beams in weakly nonlinear nonlocal media is investigated.•Reduction of the breathing period and the phase front curvature are observed.•The phase front curvature increase globally with the beam order.•Super-Gaussain interactions with the medium can generate attractive or repulsive force.•Quintic nonlinearity and nonlocality lead to the stabization of the beams propagation.
This paper is devoted to the study of existence and multiplicity of weak solutions to a Hamiltonian integro-differential system. The main tool used is the theory of min–max based on Mountain-Pass ...theorem. Hamiltonian integro-differential considered system is of Fredholm type and the imposed Dirichlet boundary conditions are occurred at the integral bounds. Furthermore, we demonstrate some cases in which the weak solutions are equivalent with classical solutions
We present a new variational method for multi-view stereovision and non-rigid three-dimensional motion estimation from multiple video sequences. Our method minimizes the prediction error of the shape ...and motion estimates. Both problems then translate into a generic image registration task. The latter is entrusted to a global measure of image similarity, chosen depending on imaging conditions and scene properties. Rather than integrating a matching measure computed independently at each surface point, our approach computes a global image-based matching score between the input images and the predicted images. The matching process fully handles projective distortion and partial occlusions. Neighborhood as well as global intensity information can be exploited to improve the robustness to appearance changes due to non-Lambertian materials and illumination changes, without any approximation of shape, motion or visibility. Moreover, our approach results in a simpler, more flexible, and more efficient implementation than in existing methods. The computation time on large datasets does not exceed thirty minutes on a standard workstation. Finally, our method is compliant with a hardware implementation with graphics processor units. Our stereovision algorithm yields very good results on a variety of datasets including specularities and translucency. We have successfully tested our motion estimation algorithm on a very challenging multi-view video sequence of a non-rigid scene.PUBLICATION ABSTRACT
In this paper, we investigate a nonlocal and nonlinear elliptic problem,(P){−a(∫Ω|∇u|2dx)Δu=λu+up in Ω,u=0 on ∂Ω, where N≤3, Ω⊂RN is a bounded domain with smooth boundary ∂Ω, a is a nondegenerate ...continuous function, p>1 and λ∈R. We show several effects of the nonlocal coefficient a on the structure of the solution set of (P). We first introduce a scaling observation and describe the solution set by using that of an associated semilinear problem. This allows us to get unbounded continua of solutions (λ,u) of (P). A rich variety of new bifurcation and multiplicity results are observed. We also prove that the nonlocal coefficient can induce up to uncountably many solutions in a convenient way. Lastly, we give some remarks from the variational point of view.
This article deals with the existence of the following quasilinear degenerate singular elliptic equation:
(
P
λ
)
-
div
(
w
(
x
)
|
∇
u
|
p
-
2
∇
u
)
=
g
λ
(
u
)
,
u
>
0
in
Ω
,
u
=
0
on
∂
Ω
,
where
Ω
...⊂
R
n
is a smooth bounded domain,
n
≥
3
,
λ
>
0
,
p
>
1
, and
w
is a Muckenhoupt weight. Using variational techniques, for
g
λ
(
u
)
=
λ
f
(
u
)
u
-
q
and certain assumptions on
f
, we show existence of a solution to
(
P
λ
)
for each
λ
>
0
. Moreover, when
g
λ
(
u
)
=
λ
u
-
q
+
u
r
, we establish existence of at least two solutions to
(
P
λ
)
in a suitable range of the parameter
λ
. Here, we assume
q
∈
(
0
,
1
)
and
r
∈
(
p
-
1
,
p
s
∗
-
1
)
.
We give three conditions on initial data for the blowing up of the corresponding solutions to some system of Klein-Gordon equations on the three dimensional Euclidean space. We first use Levine's ...concavity argument to show that the negativeness of energy leads to the blowing up of local solutions in finite time. For the data of positive energy, we give a sufficient condition so that the corresponding solution blows up in finite time. This condition embodies datum with arbitrarily large energy. At last we use Payne-Sattinger's potential well argument to classify the datum with energy not so large (to be exact, below the ground states) into two parts: one part consists of datum leading to blowing-up solutions in finite time, while the other part consists of datum that leads to the global solutions.
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