741 research outputs found
Oscillatory and localized perturbations of periodic structures and the bifurcation of defect modes
Let denote a periodic function on the real line. The Schr\"odinger
operator, , has spectrum equal to
the union of closed real intervals separated by open spectral gaps. In this
article we study the bifurcation of discrete eigenvalues (point spectrum) into
the spectral gaps for the operator , where is
spatially localized and highly oscillatory in the sense that its Fourier
transform, is concentrated at high frequencies. Our
assumptions imply that may be pointwise large but is
small in an average sense. For the special case where
with smooth, real-valued, localized in
, and periodic or almost periodic in , the bifurcating eigenvalues are at
a distance of order from the lower edge of the spectral gap. We
obtain the leading order asymptotics of the bifurcating eigenvalues and
eigenfunctions. Underlying this bifurcation is an effective Hamiltonian
associated with the lower edge of the spectral band:
where is the Dirac distribution,
and effective-medium parameters are
explicit and independent of . The potentials we consider are a
natural model for wave propagation in a medium with localized, high-contrast
and rapid fluctuations in material parameters about a background periodic
medium.Comment: To appear in SIAM Journal on Mathematical Analysi
Nonlinear Analysis of the Eckhaus Instability: Modulated Amplitude Waves and Phase Chaos with Non-zero Average Phase Gradient
We analyze the Eckhaus instability of plane waves in the one-dimensional
complex Ginzburg-Landau equation (CGLE) and describe the nonlinear effects
arising in the Eckhaus unstable regime. Modulated amplitude waves (MAWs) are
quasi-periodic solutions of the CGLE that emerge near the Eckhaus instability
of plane waves and cease to exist due to saddle-node bifurcations (SN). These
MAWs can be characterized by their average phase gradient and by the
spatial period P of the periodic amplitude modulation. A numerical bifurcation
analysis reveals the existence and stability properties of MAWs with arbitrary
and P. MAWs are found to be stable for large enough and
intermediate values of P. For different parameter values they are unstable to
splitting and attractive interaction between subsequent extrema of the
amplitude. Defects form from perturbed plane waves for parameter values above
the SN of the corresponding MAWs. The break-down of phase chaos with average
phase gradient > 0 (``wound-up phase chaos'') is thus related to these
SNs. A lower bound for the break-down of wound-up phase chaos is given by the
necessary presence of SNs and an upper bound by the absence of the splitting
instability of MAWs.Comment: 24 pages, 14 figure
Bifurcations and stability of gap solitons in periodic potentials
We analyze the existence, stability, and internal modes of gap solitons in
nonlinear periodic systems described by the nonlinear Schrodinger equation with
a sinusoidal potential, such as photonic crystals, waveguide arrays,
optically-induced photonic lattices, and Bose-Einstein condensates loaded onto
an optical lattice. We study bifurcations of gap solitons from the band edges
of the Floquet-Bloch spectrum, and show that gap solitons can appear near all
lower or upper band edges of the spectrum, for focusing or defocusing
nonlinearity, respectively. We show that, in general, two types of gap solitons
can bifurcate from each band edge, and one of those two is always unstable. A
gap soliton corresponding to a given band edge is shown to possess a number of
internal modes that bifurcate from all band edges of the same polarity. We
demonstrate that stability of gap solitons is determined by location of the
internal modes with respect to the spectral bands of the inverted spectrum and,
when they overlap, complex eigenvalues give rise to oscillatory instabilities
of gap solitons.Comment: 18 pages, 11 figures; updated bibliograph
Complex Patterns in Extended Oscillatory Systems
Ausgedehnte dissipative Systeme können fernab vom thermodynamischen Gleichgewicht instabil gegenĂŒber Oszillationen bzw. Wellen oder raumzeitlichem Chaos werden. Die komplexe Ginzburg-Landau Gleichung (CGLE) stellt ein universelles Modell zur Beschreibung dieser raumzeitlichen Strukturen dar. Diese Arbeit ist der theoretischen Analyse komplexer Muster gewidmet. Mittels numerischer Bifurkations- und StabilitĂ€tsanalyse werden InstabilitĂ€ten einfacher Muster identifiziert und neuartige Lösungen der CGLE bestimmt. Modulierte Amplitudenwellen (MAW) und Super-Spiralwellen sind Beispiele solcher komplexer Muster. MAWs können in hydrodynamischen Experimenten und Super-Spiralwellen in der Belousov-Zhabotinsky-Reaktion beobachtet werden. Der GrenzĂŒbergang von Phasen- zu Defektchaos wird durch den Existenzbereich der MAWs erklĂ€rt. Mittels der selben numerischen Methoden wird Bursting vom Fold-Hopf-Typ in einem Modell der KalziumsignalĂŒbertragung in Zellen identifiziert
Nonlinear dynamics of waves and modulated waves in 1D thermocapillary flows. I: General presentation and periodic solutions
We present experimental results on hydrothermal traveling-waves dynamics in
long and narrow 1D channels. The onset of primary traveling-wave patterns is
briefly presented for different fluid heights and for annular or bounded
channels, i.e., within periodic or non-periodic boundary conditions. For
periodic boundary conditions, by increasing the control parameter or changing
the discrete mean-wavenumber of the waves, we produce modulated waves patterns.
These patterns range from stable periodic phase-solutions, due to supercritical
Eckhaus instability, to spatio-temporal defect-chaos involving traveling holes
and/or counter-propagating-waves competition, i.e., traveling sources and
sinks. The transition from non-linearly saturated Eckhaus modulations to
transient pattern-breaks by traveling holes and spatio-temporal defects is
documented. Our observations are presented in the framework of coupled complex
Ginzburg-Landau equations with additional fourth and fifth order terms which
account for the reflection symmetry breaking at high wave-amplitude far from
onset. The second part of this paper (nlin.PS/0208030) extends this study to
spatially non-periodic patterns observed in both annular and bounded channel.Comment: 45 pages, 21 figures (elsart.cls + AMS extensions). Accepted in
Physica D. See also companion paper "Nonlinear dynamics of waves and
modulated waves in 1D thermocapillary flows. II: Convective/absolute
transitions" (nlin.PS/0208030). A version with high resolution figures is
available on N.G. web pag
Localized Breathing Modes in Granular Crystals with Defects
We investigate nonlinear localized modes at light-mass impurities in a
one-dimensional, strongly-compressed chain of beads under Hertzian contacts.
Focusing on the case of one or two such "defects", we analyze the problem's
linear limit to identify the system eigenfrequencies and the linear defect
modes. We then examine the bifurcation of nonlinear defect modes from their
linear counterparts and study their linear stability in detail. We identify
intriguing differences between the case of impurities in contact and ones that
are not in contact. We find that the former bears similarities to the single
defect case, whereas the latter features symmetry-breaking bifurcations with
interesting static and dynamic implications
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