34 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
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
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
An instability index theory for quadratic pencils and applications
Abstract. Primarily motivated by the stability analysis of nonlinear waves in second-order in time Hamiltonian systems, in this paper we develop an instability index theory for quadratic operator pencils acting on a Hilbert space. In an extension of the known theory for linear pencils, explicit connections are made between the number of eigenvalues of a given quadratic operator pencil with positive real parts to spectral information about the individual operators comprising the coefficients of the spectral parameter in the pencil. As an application, we apply the general theory developed here to yield spectral and nonlinear stability/instability results for abstract second-order in time wave equations. More specifically, we consider the problem of the existence and stability of spatially periodic waves for the "good" Boussinesq equation. In the analysis our instability index theory provides an explicit, and somewhat surprising, connection between the stability of a given periodic traveling wave solution of the "good" Boussinesq equation and the stability of the same periodic profile, but with different wavespeed, in the nonlinear dynamics of a related generalized Korteweg-de Vries equation
Analyzing Hamiltonian spectral problems via the Krein matrix
The Krein matrix is a matrix-valued function which can be used to study Hamiltonian spectral problems. Akin to the Evans matrix, it has the property that it is singular when evaluated at an eigenvalue. Unlike the Evans matrix, it is not analytic, but is instead meromorphic. I will briefly go over its construction, and then apply it to the study of spectral stability of small periodic waves for a couple of equations.Non UBCUnreviewedAuthor affiliation: Calvin CollegeFacult