581 research outputs found
Resonantly driven wobbling kinks
The amplitude of oscillations of the freely wobbling kink in the
theory decays due to the emission of second-harmonic radiation. We study the
compensation of these radiation losses (as well as additional dissipative
losses) by the resonant driving of the kink. We consider both direct and
parametric driving at a range of resonance frequencies. In each case, we derive
the amplitude equations which describe the evolution of the amplitude of the
wobbling and the kink's velocity. These equations predict multistability and
hysteretic transitions in the wobbling amplitude for each driving frequency --
the conclusion verified by numerical simulations of the full partial
differential equation. We show that the strongest parametric resonance occurs
when the driving frequency equals the natural wobbling frequency and not double
that value. For direct driving, the strongest resonance is at half the natural
frequency, but there is also a weaker resonance when the driving frequency
equals the natural wobbling frequency itself. We show that this resonance is
accompanied by translational motion of the kink.Comment: 19 pages in a double-column format; 8 figure
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
Solvability and asymptotic properties for an elliptic geophysical fluid flows model in a planar exterior domain
In this paper, we study the solvability and asymptotic properties of a recently derived gyre model of nonlinear elliptic Schrödinger equation arising from the geophysical fluid flows. The existence theorems and the asymptotic properties for radial positive solutions are established due to space theory and analytical techniques, some special cases and specific examples are also given to describe the applicability of model in gyres of geophysical fluid flows
Existence of solutions for fractional differential equations with three-point boundary conditions at resonance in
In this paper, by applying the coincidence degree theory which was first introduced by Mawhin, we obtain an existence result for the fractional three-point boundary value problems in , where the dimension of the kernel of fractional differential operator with the boundary conditions can take any value in . This is our novelty. Several examples are presented to illustrate the result
Quasi-Exact Solvability and the direct approach to invariant subspaces
We propose a more direct approach to constructing differential operators that
preserve polynomial subspaces than the one based on considering elements of the
enveloping algebra of sl(2). This approach is used here to construct new
exactly solvable and quasi-exactly solvable quantum Hamiltonians on the line
which are not Lie-algebraic. It is also applied to generate potentials with
multiple algebraic sectors. We discuss two illustrative examples of these two
applications: an interesting generalization of the Lam\'e potential which
posses four algebraic sectors, and a quasi-exactly solvable deformation of the
Morse potential which is not Lie-algebraic.Comment: 17 pages, 3 figure
Libration driven multipolar instabilities
We consider rotating flows in non-axisymmetric enclosures that are driven by
libration, i.e. by a small periodic modulation of the rotation rate. Thanks to
its simplicity, this model is relevant to various contexts, from industrial
containers (with small oscillations of the rotation rate) to fluid layers of
terrestial planets (with length-of-day variations). Assuming a multipolar
-fold boundary deformation, we first obtain the two-dimensional basic flow.
We then perform a short-wavelength local stability analysis of the basic flow,
showing that an instability may occur in three dimensions. We christen it the
Libration Driven Multipolar Instability (LDMI). The growth rates of the LDMI
are computed by a Floquet analysis in a systematic way, and compared to
analytical expressions obtained by perturbation methods. We then focus on the
simplest geometry allowing the LDMI, a librating deformed cylinder. To take
into account viscous and confinement effects, we perform a global stability
analysis, which shows that the LDMI results from a parametric resonance of
inertial modes. Performing numerical simulations of this librating cylinder, we
confirm that the basic flow is indeed established and report the first
numerical evidence of the LDMI. Numerical results, in excellent agreement with
the stability results, are used to explore the non-linear regime of the
instability (amplitude and viscous dissipation of the driven flow). We finally
provide an example of LDMI in a deformed spherical container to show that the
instability mechanism is generic. Our results show that the previously studied
libration driven elliptical instability simply corresponds to the particular
case of a wider class of instabilities. Summarizing, this work shows that
any oscillating non-axisymmetric container in rotation may excite intermittent,
space-filling LDMI flows, and this instability should thus be easy to observe
experimentally
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