150 research outputs found
The limiting absorption principle for periodic differential operators and applications to nonlinear Helmholtz equations
We prove an -version of the limiting absoprtion principle for a class of
periodic elliptic differential operators of second order. The result is applied
to the construction of nontrivial solutions of nonlinear Helmholtz equations
with periodic coefficient functions
A priori bounds and global bifurcation results for frequency combs modeled by the Lugiato-Lefever equation
In nonlinear optics -periodic solutions
of the stationary Lugiato-Lefever equation serve as a model for frequency combs, which are optical
signals consisting of a superposition of modes with equally spaced frequencies.
We prove that nontrivial frequency combs can only be observed for special
ranges of values of the forcing and detuning parameters and , as it
has been previously documented in experiments and numerical simulations. E.g.,
if the detuning parameter is too large then nontrivial frequency combs
do not exist, cf. Theorem 2. Additionally, we show that for large ranges of
parameter values nontrivial frequency combs may be found on continua which
bifurcate from curves of trivial frequency combs. Our results rely on the proof
of a priori bounds for the stationary Lugiato-Lefever equation as well as a
detailed rigorous bifurcation analysis based on the bifurcation theorems of
Crandall-Rabinowitz and Rabinowitz. We use the software packages AUTO and
MATLAB to illustrate our results by numerical computations of bifurcation
diagrams and of selected solutions
Bifurcations of nontrivial solutions of a cubic Helmholtz system
This paper presents local and global bifurcation results for radially
symmetric solutions of the cubic Helmholtz system \begin{equation*}
\begin{cases} -\Delta u - \mu u = \left( u^2 + b \: v^2 \right) u &\text{ on }
\mathbb{R}^3, \\ -\Delta v - \nu v = \left( v^2 + b \: u^2 \right) v &\text{ on
} \mathbb{R}^3. \end{cases} \end{equation*} It is shown that every point along
any given branch of radial semitrivial solutions or diagonal
solutions (for ) is a bifurcation point. Our
analysis is based on a detailed investigation of the oscillatory behavior of
solutions at infinity that are shown to decay like as
.Comment: 31 page
Dual Variational Methods for a nonlinear Helmholtz system
This paper considers a pair of coupled nonlinear Helmholtz equations
\begin{align*}
-\Delta u - \mu u = a(x) \left( |u|^\frac{p}{2} + b(x) |v|^\frac{p}{2}
\right)|u|^{\frac{p}{2} - 2}u, \end{align*} \begin{align*}
-\Delta v - \nu v = a(x) \left( |v|^\frac{p}{2} + b(x) |u|^\frac{p}{2}
\right)|v|^{\frac{p}{2} - 2}v \end{align*} on where
. The existence of nontrivial strong solutions
in is established using dual variational methods.
The focus lies on necessary and sufficient conditions on the parameters
deciding whether or not both components of such solutions are nontrivial.Comment: Published version. Contains minor revisions: Quote added,
explanations on p.12 concerning F_{\mu\nu} = \infty, correction of exponent
on p.1
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