81 research outputs found
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
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
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
- …