45 research outputs found
Localised states in an extended Swift-Hohenberg equation
Recent work on the behaviour of localised states in pattern forming partial
differential equations has focused on the traditional model Swift-Hohenberg
equation which, as a result of its simplicity, has additional structure --- it
is variational in time and conservative in space. In this paper we investigate
an extended Swift-Hohenberg equation in which non-variational and
non-conservative effects play a key role. Our work concentrates on aspects of
this much more complicated problem. Firstly we carry out the normal form
analysis of the initial pattern forming instability that leads to
small-amplitude localised states. Next we examine the bifurcation structure of
the large-amplitude localised states. Finally we investigate the temporal
stability of one-peak localised states. Throughout, we compare the localised
states in the extended Swift-Hohenberg equation with the analogous solutions to
the usual Swift-Hohenberg equation
The Swift-Hohenberg equation with a nonlocal nonlinearity
It is well known that aspects of the formation of localised states in a
one-dimensional Swift--Hohenberg equation can be described by
Ginzburg--Landau-type envelope equations. This paper extends these multiple
scales analyses to cases where an additional nonlinear integral term, in the
form of a convolution, is present. The presence of a kernel function introduces
a new lengthscale into the problem, and this results in additional complexity
in both the derivation of envelope equations and in the bifurcation structure.
When the kernel is short-range, weakly nonlinear analysis results in envelope
equations of standard type but whose coefficients are modified in complicated
ways by the nonlinear nonlocal term. Nevertheless, these computations can be
formulated quite generally in terms of properties of the Fourier transform of
the kernel function. When the lengthscale associated with the kernel is longer,
our method leads naturally to the derivation of two different, novel, envelope
equations that describe aspects of the dynamics in these new regimes. The first
of these contains additional bifurcations, and unexpected loops in the
bifurcation diagram. The second of these captures the stretched-out nature of
the homoclinic snaking curves that arises due to the nonlocal term.Comment: 28 pages, 14 figures. To appear in Physica
Réverbérations d'ondes et frottement
Nous proposons qu'une origine possible de glissement saccadé et erratique de couches élastiques cisaillées l'une contre l'autre est associée à la réverbération des ondes qui rayonnent et se réfléchissent aux niveaux des interfaces en frottement solide. Dans le cadre des lois de frottement à variable interne, nous montrons que la dynamique du glissement interfacial est déterminée par un système différentiel avec délai correspondant au temps caractéristique des réverbérations fixé par l'épaisseur des couches. La stabilité du glissement stationnaire et des solutions périodiques est analysée
The meandering instability of a viscous thread
A viscous thread falling from a nozzle onto a surface exhibits the famous
rope-coiling effect, in which the thread buckles to form loops. If the surface
is replaced by a belt moving with speed , the rotational symmetry of the
buckling instability is broken and a wealth of interesting states are observed
[See S. Chiu-Webster and J. R. Lister, J. Fluid Mech., {\bf 569}, 89 (2006)].
We experimentally studied this "fluid mechanical sewing machine" in a new, more
precise apparatus. As is reduced, the steady catenary thread bifurcates
into a meandering state in which the thread displacements are only transverse
to the motion of the belt. We measured the amplitude and frequency of
the meandering close to the bifurcation. For smaller , single-frequency
meandering bifurcates to a two-frequency "figure eight" state, which contains a
significant component and parallel as well as transverse
displacements. This eventually reverts to single-frequency coiling at still
smaller . More complex, highly hysteretic states with additional frequencies
are observed for larger nozzle heights. We propose to understand this zoology
in terms of the generic amplitude equations appropriate for resonant
interactions between two oscillatory modes with frequencies and
. The form of the amplitude equations captures both the axisymmetry of
the U=0 coiling state and the symmetry-breaking effects induced by the moving
belt.Comment: 12 pages, 9 figures, revised, resubmitted to Physical Review