173 research outputs found
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
Real-valued, time-periodic localized weak solutions for a semilinear wave equation with periodic potentials
We consider the semilinear wave equation for three different classes (P1), (P2), (P3) of periodic potentials
. (P1) consists of periodically extended delta-distributions, (P2) of
periodic step potentials and (P3) contains certain periodic potentials V,q\in
H^r_{\per}(\R) for . Among other assumptions we suppose that
for some and . In each class we can find
suitable potentials that give rise to a critical exponent such that
for both in the "+" and the "-" case we can use variational
methods to prove existence of time-periodic real-valued solutions that are
localized in the space direction. The potentials are constructed explicitely in
class (P1) and (P2) and are found by a recent result from inverse spectral
theory in class (P3). The critical exponent depends on the regularity
of . Our result builds upon a Fourier expansion of the solution and a
detailed analysis of the spectrum of the wave operator. In fact, it turns out
that by a careful choice of the potentials and the spatial and temporal
periods, the spectrum of the wave operator
(considered on suitable space of time-periodic functions) is bounded away from
. This allows to find weak solutions as critical points of a functional on a
suitable Hilbert space and to apply tools for strongly indefinite variational
problems
Existence of cylindrically symmetric ground states to a nonlinear curl-curl equation with non-constant coefficients
We consider the nonlinear curl-curl problem in related to the nonlinear Maxwell
equations with Kerr-type nonlinear material laws. We prove the existence of a
symmetric ground-state type solution for a bounded, cylindrically symmetric
coefficient and subcritical cylindrically symmetric nonlinearity . The
new existence result extends the class of problems for which ground-state type
solutions are known. It is based on compactness properties of symmetric
functions due to Lions, new rearrangement type inequalities from Brock and the
recent extension of the Nehari-manifold technique by Szulkin and Weth.Comment: 13 page
Localized Modes of the Linear Periodic Schr\"{o}dinger Operator with a Nonlocal Perturbation
We consider the existence of localized modes corresponding to eigenvalues of
the periodic Schr\"{o}dinger operator with an interface.
The interface is modeled by a jump either in the value or the derivative of
and, in general, does not correspond to a localized perturbation of the
perfectly periodic operator. The periodic potentials on each side of the
interface can, moreover, be different. As we show, eigenvalues can only occur
in spectral gaps. We pose the eigenvalue problem as a gluing problem for
the fundamental solutions (Bloch functions) of the second order ODEs on each
side of the interface. The problem is thus reduced to finding matchings of the
ratio functions , where are
those Bloch functions that decay on the respective half-lines. These ratio
functions are analyzed with the help of the Pr\"{u}fer transformation. The
limit values of at band edges depend on the ordering of Dirichlet and
Neumann eigenvalues at gap edges. We show that the ordering can be determined
in the first two gaps via variational analysis for potentials satisfying
certain monotonicity conditions. Numerical computations of interface
eigenvalues are presented to corroborate the analysis.Comment: 1. finiteness of the number of additive interface eigenvalues proved
in a remark below Corollary 3.6.; 2. small modifications and typo correction
"Boundary blowup" type sub-solutions to semilinear elliptic equations with Hardy potential
Semilinear elliptic equations which give rise to solutions blowing up at the
boundary are perturbed by a Hardy potential. The size of this potential effects
the existence of a certain type of solutions (large solutions): if the
potential is too small, then no large solution exists. The presence of the
Hardy potential requires a new definition of large solutions, following the
pattern of the associated linear problem. Nonexistence and existence results
for different types of solutions will be given. Our considerations are based on
a Phragmen-Lindelof type theorem which enables us to classify the solutions and
sub-solutions according to their behavior near the boundary. Nonexistence
follows from this principle together with the Keller-Osserman upper bound. The
existence proofs rely on sub- and super-solution techniques and on estimates
for the Hardy constant derived in Marcus, Mizel and Pinchover.Comment: 23 pages, 3 figure
Characterization of balls by Riesz-Potentials
Using elementary differential inequality methods we prove an improvement
of a theorem of Kantorovich concerning solutions of nonlinear equations in
Banach spaces
Sharp parameter ranges in the uniform anti-maximum principle forsecond-order ordinary differential operators
We consider the equation (pu')'-qu+λwu = f in (0,1) subject to homogenous boundary conditions at x = 0 and x = 1, e.g., u'(0) = u'(1) = 0. Let λ1 be the first eigenvalue of the corresponding Sturm-Liouville problem. If f ≤ 0 but ≢ 0 then it is known that there exists δ > 0 (independent on f) such that for λ ∈ (λ1, λ1 + δ] any solution u must be negative. This so-called uniform anti-maximum principle (UAMP) goes back to Clément, Peletier [4]. In this paper we establish the sharp values of δ for which (UAMP) holds. The same phenomenon, including sharp values of δ, can be shown for the radially symmetric p-Laplacian on balls and annuli in ℝn provided 1 ≤ n < p. The results are illustrated by explicitly computed example
Existence of solutions to nonlinear, subcritical higher-order elliptic Dirichlet problems
We consider the -th order elliptic boundary value problem on
a bounded smooth domain with Dirichlet boundary conditions
on . The operator is a uniformly elliptic linear operator
of order whose principle part is of the form . We assume that
is superlinear at the origin and satisfies , , where are positive functions and is subcritical. By
combining degree theory with new and recently established a priori estimates,
we prove the existence of a nontrivial solution
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