68,308 research outputs found
Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints
The Einstein constraint equations have been the subject of study for more
than fifty years. The introduction of the conformal method in the 1970's as a
parameterization of initial data for the Einstein equations led to increased
interest in the development of a complete solution theory for the constraints,
with the theory for constant mean curvature (CMC) spatial slices and closed
manifolds completely developed by 1995. The first general non-CMC existence
result was establish by Holst et al. in 2008, with extensions to rough data by
Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC
theory remains mostly open; moreover, recent work of Maxwell on specific
symmetry models sheds light on fundamental non-uniqueness problems with the
conformal method as a parameterization in non-CMC settings. In parallel with
these mathematical developments, computational physicists have uncovered
surprising behavior in numerical solutions to the extended conformal thin
sandwich formulation of the Einstein constraints. In particular, numerical
evidence suggests the existence of multiple solutions with a quadratic fold,
and a recent analysis of a simplified model supports this conclusion. In this
article, we examine this apparent bifurcation phenomena in a methodical way,
using modern techniques in bifurcation theory and in numerical homotopy
methods. We first review the evidence for the presence of bifurcation in the
Hamiltonian constraint in the time-symmetric case. We give a brief introduction
to the mathematical framework for analyzing bifurcation phenomena, and then
develop the main ideas behind the construction of numerical homotopy, or
path-following, methods in the analysis of bifurcation phenomena. We then apply
the continuation software package AUTO to this problem, and verify the presence
of the fold with homotopy-based numerical methods.Comment: 13 pages, 4 figures. Final revision for publication, added material
on physical implication
On the existence of oscillating solutions in non-monotone Mean-Field Games
For non-monotone single and two-populations time-dependent Mean-Field Game
systems we obtain the existence of an infinite number of branches of
non-trivial solutions. These non-trivial solutions are in particular shown to
exhibit an oscillatory behaviour when they are close to the trivial (constant)
one. The existence of such branches is derived using local and global
bifurcation methods, that rely on the analysis of eigenfunction expansions of
solutions to the associated linearized problem. Numerical analysis is performed
on two different models to observe the oscillatory behaviour of solutions
predicted by bifurcation theory, and to study further properties of branches
far away from bifurcation points.Comment: 24 pages, 10 figure
Composite "zigzag" structures in the 1D complex Ginzburg-Landau equation
We study the dynamics of the one-dimensional complex Ginzburg Landau equation
(CGLE) in the regime where holes and defects organize themselves into composite
superstructures which we call zigzags. Extensive numerical simulations of the
CGLE reveal a wide range of dynamical zigzag behavior which we summarize in a
`phase diagram'. We have performed a numerical linear stability and bifurcation
analysis of regular zigzag structures which reveals that traveling zigzags
bifurcate from stationary zigzags via a pitchfork bifurcation. This bifurcation
changes from supercritical (forward) to subcritical (backward) as a function of
the CGLE coefficients, and we show the relevance of this for the `phase
diagram'. Our findings indicate that in the zigzag parameter regime of the
CGLE, the transition between defect-rich and defect-poor states is governed by
bifurcations of the zigzag structures.Comment: 20 pages, 11 figure
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