1,092 research outputs found
Linear elliptic system with nonlinear boundary conditions without Landesman-Lazer conditions
The boundary value problem is examined for the system of elliptic equations
of from where is positive
semidefinite matrix on and It is assumed that
is a bounded function which may vanish
at infinity. The proofs are based on Leray-Schauder degree methods
Interaction of modulated pulses in the nonlinear Schroedinger equation with periodic potential
We consider a cubic nonlinear Schroedinger equation with periodic potential.
In a semiclassical scaling the nonlinear interaction of modulated pulses
concentrated in one or several Bloch bands is studied. The notion of closed
mode systems is introduced which allows for the rigorous derivation of a finite
system of amplitude equations describing the macroscopic dynamics of these
pulses
Resonances and superlattice pattern stabilization in two-frequency forced Faraday waves
We investigate the role weakly damped modes play in the selection of Faraday
wave patterns forced with rationally-related frequency components m*omega and
n*omega. We use symmetry considerations to argue for the special importance of
the weakly damped modes oscillating with twice the frequency of the critical
mode, and those oscillating primarily with the "difference frequency"
|n-m|*omega and the "sum frequency" (n+m)*omega. We then perform a weakly
nonlinear analysis using equations of Zhang and Vinals (1997, J. Fluid Mech.
336) which apply to small-amplitude waves on weakly inviscid, semi-infinite
fluid layers. For weak damping and forcing and one-dimensional waves, we
perform a perturbation expansion through fourth order which yields analytical
expressions for onset parameters and the cubic bifurcation coefficient that
determines wave amplitude as a function of forcing near onset. For stronger
damping and forcing we numerically compute these same parameters, as well as
the cubic cross-coupling coefficient for competing waves travelling at an angle
theta relative to each other. The resonance effects predicted by symmetry are
borne out in the perturbation results for one spatial dimension, and are
supported by the numerical results in two dimensions. The difference frequency
resonance plays a key role in stabilizing superlattice patterns of the SL-I
type observed by Kudrolli, Pier and Gollub (1998, Physica D 123).Comment: 41 pages, 13 figures; corrected figure 1b and minor typos in tex
Two and three-dimensional oscillons in nonlinear Faraday resonance
We study 2D and 3D localised oscillating patterns in a simple model system
exhibiting nonlinear Faraday resonance. The corresponding amplitude equation is
shown to have exact soliton solutions which are found to be always unstable in
3D. On the contrary, the 2D solitons are shown to be stable in a certain
parameter range; hence the damping and parametric driving are capable of
suppressing the nonlinear blowup and dispersive decay of solitons in two
dimensions. The negative feedback loop occurs via the enslaving of the
soliton's phase, coupled to the driver, to its amplitude and width.Comment: 4 pages; 1 figur
On the tilting of protostellar disks by resonant tidal effects
We consider the dynamics of a protostellar disk surrounding a star in a
circular-orbit binary system. Our aim is to determine whether, if the disk is
initially tilted with respect to the plane of the binary orbit, the inclination
of the system will increase or decrease with time. The problem is formulated in
the binary frame in which the tidal potential of the companion star is static.
We consider a steady, flat disk that is aligned with the binary plane and
investigate its linear stability with respect to tilting or warping
perturbations. The dynamics is controlled by the competing effects of the m=0
and m=2 azimuthal Fourier components of the tidal potential. In the presence of
dissipation, the m=0 component causes alignment of the system, while the m=2
component has the opposite tendency. We find that disks that are sufficiently
large, in particular those that extend to their tidal truncation radii, are
generally stable and will therefore tend to alignment with the binary plane on
a time-scale comparable to that found in previous studies. However, the effect
of the m=2 component is enhanced in the vicinity of resonances where the outer
radius of the disk is such that the natural frequency of a global bending mode
of the disk is equal to twice the binary orbital frequency. Under such
circumstances, the disk can be unstable to tilting and acquire a warped shape,
even in the absence of dissipation. The outer radius corresponding to the
primary resonance is always smaller than the tidal truncation radius. For disks
smaller than the primary resonance, the m=2 component may be able to cause a
very slow growth of inclination through the effect of a near resonance that
occurs close to the disk center. We discuss these results in the light of
recent observations of protostellar disks in binary systems.Comment: 21 pages, 7 figures, to be published in the Astrophysical Journa
Fractional-Period Excitations in Continuum Periodic Systems
We investigate the generation of fractional-period states in continuum
periodic systems. As an example, we consider a Bose-Einstein condensate
confined in an optical-lattice potential. We show that when the potential is
turned on non-adiabatically, the system explores a number of transient states
whose periodicity is a fraction of that of the lattice. We illustrate the
origin of fractional-period states analytically by treating them as resonant
states of a parametrically forced Duffing oscillator and discuss their
transient nature and potential observability.Comment: 10 pages, 6 figures (some with multiple parts); revised version:
minor clarifications of a couple points, to appear in Physical Review
Bright and Gap Solitons in Membrane-Type Acoustic Metamaterials
We study analytically and numerically envelope solitons (bright and gap
solitons) in a one-dimensional, nonlinear acoustic metamaterial, composed of an
air-filled waveguide periodically loaded by clamped elastic plates. Based on
the transmission line approach, we derive a nonlinear dynamical lattice model
which, in the continuum approximation, leads to a nonlinear, dispersive and
dissipative wave equation. Applying the multiple scales perturbation method, we
derive an effective lossy nonlinear Schr\"odinger equation and obtain
analytical expressions for bright and gap solitons. We also perform direct
numerical simulations to study the dissipation-induced dynamics of the bright
and gap solitons. Numerical and analytical results, relying on the analytical
approximations and perturbation theory for solions, are found to be in good
agreement
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