1,438 research outputs found
Amplitude equations near pattern forming instabilities for strongly driven ferromagnets
A transversally driven isotropic ferromagnet being under the influence of a
static external and an uniaxial internal anisotropy field is studied. We
consider the dissipative Landau-Lifshitz equation as the fundamental equation
of motion and treat it in ~dimensions. The stability of the spatially
homogeneous magnetizations against inhomogeneous perturbations is analyzed.
Subsequently the dynamics above threshold is described via amplitude equations
and the dependence of their coefficients on the physical parameters of the
system is determined explicitly. We find soft- and hard-mode instabilities,
transitions between sub- and supercritical behaviour, various bifurcations of
higher codimension, and present a series of explicit bifurcation diagrams. The
analysis of the codimension-2 point where the soft- and hard-mode instabilities
coincide leads to a system of two coupled Ginzburg-Landau equations.Comment: LATeX, 25 pages, submitted to Z.Phys.B figures available via
[email protected] in /pub/publications/frank/zpb_95
(postscript, plain or gziped
Pulses and Snakes in Ginzburg--Landau Equation
Using a variational formulation for partial differential equations (PDEs)
combined with numerical simulations on ordinary differential equations (ODEs),
we find two categories (pulses and snakes) of dissipative solitons, and analyze
the dependence of both their shape and stability on the physical parameters of
the cubic-quintic Ginzburg-Landau equation (CGLE). In contrast to the regular
solitary waves investigated in numerous integrable and non-integrable systems
over the last three decades, these dissipative solitons are not stationary in
time. Rather, they are spatially confined pulse-type structures whose envelopes
exhibit complicated temporal dynamics. Numerical simulations reveal very
interesting bifurcations sequences as the parameters of the CGLE are varied.
Our predictions on the variation of the soliton amplitude, width, position,
speed and phase of the solutions using the variational formulation agree with
simulation results.Comment: 30 pages, 14 figure
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
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