157 research outputs found
Dynamics of stripe patterns in type-I superconductors subject to a rotating field
The evolution of stripe patterns in type-I superconductors subject to a
rotating in-plane magnetic field is investigated magneto-optically. The
experimental results reveal a very rich and interesting behavior of the
patterns. For small rotation angles, a small parallel displacement of the main
part of the stripes and a co-rotation of their very ends is observed. For
larger angles, small sideward protrusions develop, which then generate a zigzag
instability, ultimately leading to a breaking of stripes into smaller segments.
The short segments then start to co-rotate with the applied field although they
lag behind by approximately . Very interestingly, if the rotation is
continued, also reconnection of segments into longer stripes takes place. These
observations demonstrate the importance of pinning in type-I superconductors.Comment: To appear in Phys. Rev.
The Nikolaevskiy equation with dispersion
The Nikolaevskiy equation was originally proposed as a model for seismic
waves and is also a model for a wide variety of systems incorporating a
neutral, Goldstone mode, including electroconvection and reaction-diffusion
systems. It is known to exhibit chaotic dynamics at the onset of pattern
formation, at least when the dispersive terms in the equation are suppressed,
as is commonly the practice in previous analyses. In this paper, the effects of
reinstating the dispersive terms are examined. It is shown that such terms can
stabilise some of the spatially periodic traveling waves; this allows us to
study the loss of stability and transition to chaos of the waves. The secondary
stability diagram (Busse balloon) for the traveling waves can be remarkably
complicated.Comment: 24 pages; accepted for publication in Phys. Rev.
Long-Time Asymptotics for the Korteweg-de Vries Equation via Nonlinear Steepest Descent
We apply the method of nonlinear steepest descent to compute the long-time
asymptotics of the Korteweg-de Vries equation for decaying initial data in the
soliton and similarity region. This paper can be viewed as an expository
introduction to this method.Comment: 31 page
Reformulating the Schrodinger equation as a Shabat-Zakharov system
We reformulate the second-order Schrodinger equation as a set of two coupled
first order differential equations, a so-called "Shabat-Zakharov system",
(sometimes called a "Zakharov-Shabat" system). There is considerable
flexibility in this approach, and we emphasise the utility of introducing an
"auxiliary condition" or "gauge condition" that is used to cut down the degrees
of freedom. Using this formalism, we derive the explicit (but formal) general
solution to the Schrodinger equation. The general solution depends on three
arbitrarily chosen functions, and a path-ordered exponential matrix. If one
considers path ordering to be an "elementary" process, then this represents
complete quadrature, albeit formal, of the second-order linear ODE.Comment: 18 pages, plain LaTe
Negaton and Positon Solutions of the KDV Equation
We give a systematic classification and a detailed discussion of the
structure, motion and scattering of the recently discovered negaton and positon
solutions of the Korteweg-de Vries equation. There are two distinct types of
negaton solutions which we label and , where is the
order of the Wronskian used in the derivation. For negatons, the number of
singularities and zeros is finite and they show very interesting time
dependence. The general motion is in the positive direction, except for
certain negatons which exhibit one oscillation around the origin. In contrast,
there is just one type of positon solution, which we label . For
positons, one gets a finite number of singularities for odd, but an
infinite number for even values of . The general motion of positons is in
the negative direction with periodic oscillations. Negatons and positons
retain their identities in a scattering process and their phase shifts are
discussed. We obtain a simple explanation of all phase shifts by generalizing
the notions of ``mass" and ``center of mass" to singular solutions. Finally, it
is shown that negaton and positon solutions of the KdV equation can be used to
obtain corresponding new solutions of the modified KdV equation.Comment: 20 pages plus 12 figures(available from authors on request),Latex
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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
The Kundu--Eckhaus equation and its discretizations
In this article we show that the complex Burgers and the Kundu--Eckhaus
equations are related by a Miura transformation. We use this relation to
discretize the Kundu--Eckhaus equation.Comment: 10 page
Parametric Forcing of Waves with Non-Monotonic Dispersion Relation: Domain Structures in Ferrofluids?
Surface waves on ferrofluids exposed to a dc-magnetic field exhibit a
non-monotonic dispersion relation. The effect of a parametric driving on such
waves is studied within suitable coupled Ginzburg-Landau equations. Due to the
non-monotonicity the neutral curve for the excitation of standing waves can
have up to three minima. The stability of the waves with respect to long-wave
perturbations is determined a phase-diffusion equation. It shows that the
band of stable wave numbers can split up into two or three sub-bands. The
resulting competition between the wave numbers corresponding to the respective
sub-bands leads quite naturally to patterns consisting of multiple domains of
standing waves which differ in their wave number. The coarsening dynamics of
such domain structures is addressed.Comment: 23 pages, 6 postscript figures, composed using RevTeX. Submitted to
PR
Mixed-Mode Oscillations in a Stochastic, Piecewise-Linear System
We analyze a piecewise-linear FitzHugh-Nagumo model. The system exhibits a
canard near which both small amplitude and large amplitude periodic orbits
exist. The addition of small noise induces mixed-mode oscillations (MMOs) in
the vicinity of the canard point. We determine the effect of each model
parameter on the stochastically driven MMOs. In particular we show that any
parameter variation (such as a modification of the piecewise-linear function in
the model) that leaves the ratio of noise amplitude to time-scale separation
unchanged typically has little effect on the width of the interval of the
primary bifurcation parameter over which MMOs occur. In that sense, the MMOs
are robust. Furthermore we show that the piecewise-linear model exhibits MMOs
more readily than the classical FitzHugh-Nagumo model for which a cubic
polynomial is the only nonlinearity. By studying a piecewise-linear model we
are able to explain results using analytical expressions and compare these with
numerical investigations.Comment: 25 pages, 10 figure
On the Cauchy Problem for the Korteweg-de Vries Equation with Steplike Finite-Gap Initial Data I. Schwartz-Type Perturbations
We solve the Cauchy problem for the Korteweg-de Vries equation with initial
conditions which are steplike Schwartz-type perturbations of finite-gap
potentials under the assumption that the respective spectral bands either
coincide or are disjoint.Comment: 29 page
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