276 research outputs found
The Krein Matrix: General Theory and Concrete Applications in Atomic Bose-Einstein Condensates
When finding the nonzero eigenvalues for Hamiltonian eigenvalue problems it
is especially important to locate not only the unstable eigenvalues (i.e.,
those with positive real part), but also those which are purely imaginary but
have negative Krein signature. These latter eigenvalues have the property that
they can become unstable upon collision with other purely imaginary
eigenvalues, i.e., they are a necessary building block in the mechanism leading
to the so-called Hamiltonian-Hopf bifurcation. In this paper we review a
general theory for constructing a meromorphic matrix-valued function, the
so-called Krein matrix, which has the property of not only locating the
unstable eigenvalues, but also those with negative Krein signature. These
eigenvalues are realized as zeros of the determinant. The resulting finite
dimensional problem obtained by setting the determinant of the Krein matrix to
zero presents a valuable simplification. In this paper the usefulness of the
technique is illustrated through prototypical examples of spectral analysis of
states that have arisen in recent experimental and theoretical studies of
atomic Bose-Einstein condensates. In particular, we consider one-dimensional
settings (the cigar trap) possessing real-valued multi-dark-soliton solutions,
and two-dimensional settings (the pancake trap) admitting complex multi-vortex
stationary waveforms.Comment: 26 pages, 16 figures (revised version on April 18 2013
Existence and stability of hole solutions to complex Ginzburg-Landau equations
We consider the existence and stability of the hole, or dark soliton,
solution to a Ginzburg-Landau perturbation of the defocusing nonlinear
Schroedinger equation (NLS), and to the nearly real complex Ginzburg-Landau
equation (CGL). By using dynamical systems techniques, it is shown that the
dark soliton can persist as either a regular perturbation or a singular
perturbation of that which exists for the NLS. When considering the stability
of the soliton, a major difficulty which must be overcome is that eigenvalues
may bifurcate out of the continuous spectrum, i.e., an edge bifurcation may
occur. Since the continuous spectrum for the NLS covers the imaginary axis, and
since for the CGL it touches the origin, such a bifurcation may lead to an
unstable wave. An additional important consideration is that an edge
bifurcation can happen even if there are no eigenvalues embedded in the
continuous spectrum. Building on and refining ideas first presented in Kapitula
and Sandstede (Physica D, 1998) and Kapitula (SIAM J. Math. Anal., 1999), we
show that when the wave persists as a regular perturbation, at most three
eigenvalues will bifurcate out of the continuous spectrum. Furthermore, we
precisely track these bifurcating eigenvalues, and thus are able to give
conditions for which the perturbed wave will be stable. For the NLS the results
are an improvement and refinement of previous work, while the results for the
CGL are new. The techniques presented are very general and are therefore
applicable to a much larger class of problems than those considered here.Comment: 41 pages, 4 figures, submitte
Stability analysis for pitchfork bifurcations of solitary waves in generalized nonlinear Schroedinger equations
Linear stability of both sign-definite (positive) and sign-indefinite
solitary waves near pitchfork bifurcations is analyzed for the generalized
nonlinear Schroedinger equations with arbitrary forms of nonlinearity and
external potentials in arbitrary spatial dimensions. Bifurcations of
linear-stability eigenvalues associated with pitchfork bifurcations are
analytically calculated. It is shown that the smooth solution branch switches
stability at the bifurcation point. In addition, the two bifurcated solution
branches and the smooth branch have the opposite (same) stability when their
power slopes have the same (opposite) sign. One unusual feature on the
stability of these pitchfork bifurcations is that the smooth and bifurcated
solution branches can be both stable or both unstable, which contrasts such
bifurcations in finite-dimensional dynamical systems where the smooth and
bifurcated branches generally have opposite stability. For the special case of
positive solitary waves, stronger and more explicit stability results are also
obtained. It is shown that for positive solitary waves, their linear stability
near a bifurcation point can be read off directly from their power diagram.
Lastly, various numerical examples are presented, and the numerical results
confirm the analytical predictions both qualitatively and quantitatively.Comment: 28 pages, 6 figure
Stability criterion for bright solitary waves of the perturbed cubic-quintic Schroedinger equation
The stability of the bright solitary wave solution to the perturbed
cubic-quintic Schroedinger equation is considered. It is shown that in a
certain region of parameter space these solutions are unstable, with the
instability being manifested as a small positive eigenvalue. Furthermore, it is
shown that in the complimentary region of parameter space there are no small
unstable eigenvalues. The proof involves a novel calculation of the Evans
function, which is of interest in its own right. As a consequence of the
eigenvalue calculation, it is additionally shown that N-bump bright solitary
waves bifurcate from the primary wave.Comment: 28 pages, LaTeX, submitted; author info at
http://www.math.unm.edu/~kapitul
Spectral stability of nonlinear waves in KdV-type evolution equations
This paper concerns spectral stability of nonlinear waves in KdV-type
evolution equations. The relevant eigenvalue problem is defined by the
composition of an unbounded self-adjoint operator with a finite number of
negative eigenvalues and an unbounded non-invertible symplectic operator
. The instability index theorem is proven under a generic
assumption on the self-adjoint operator both in the case of solitary waves and
periodic waves. This result is reviewed in the context of other recent results
on spectral stability of nonlinear waves in KdV-type evolution equations.Comment: 15 pages, no figure
Assessing Women\u27s Understanding of Mensuration: Knowledge Gaps and Educational Needs
The menstrual cycle is more complex than what seems to be taught in the education system. School systems focus on the basics of the menstrual cycle, like for example what to do when a young female gets her menstrual cycle for the first time and telling the young female that she can get pregnant but then do not go into detail on the timeframe when pregnancy can occur. It consists of more information than what is taught in the public school system and should be better taught to young females to prevent the unpreventable. The main objective of the study was to evaluate how much women know or think they know about the menstrual cycle, and where in their education they learned the information. In order to generate responses, a survey was created and sent out to women that asked multiple questions. The questions were regarding the menstrual cycle and were specific to the knowledge that they have obtained or have not obtained in their lifetime.https://digitalcommons.misericordia.edu/research_posters2023/1008/thumbnail.jp
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