20,392 research outputs found
Orbital stability in the cubic defocusing NLS equation: I. Cnoidal periodic waves
Periodic waves of the one-dimensional cubic defocusing NLS equation are
considered. Using tools from integrability theory, these waves have been shown
in [Bottman, Deconinck, and Nivala, 2011] to be linearly stable and the
Floquet-Bloch spectrum of the linearized operator has been explicitly computed.
We combine here the first four conserved quantities of the NLS equation to give
a direct proof that cnoidal periodic waves are orbitally stable with respect to
subharmonic perturbations, with period equal to an integer multiple of the
period of the wave. Our result is not restricted to the periodic waves of small
amplitudes.Comment: 28 pages, 3 figures. Main result strengthened by removing a smallness
condition. Limiting case of the black soliton now postponed to a companion
pape
A delay differential model of ENSO variability: Parametric instability and the distribution of extremes
We consider a delay differential equation (DDE) model for El-Nino Southern
Oscillation (ENSO) variability. The model combines two key mechanisms that
participate in ENSO dynamics: delayed negative feedback and seasonal forcing.
We perform stability analyses of the model in the three-dimensional space of
its physically relevant parameters. Our results illustrate the role of these
three parameters: strength of seasonal forcing , atmosphere-ocean coupling
, and propagation period of oceanic waves across the Tropical
Pacific. Two regimes of variability, stable and unstable, are separated by a
sharp neutral curve in the plane at constant . The detailed
structure of the neutral curve becomes very irregular and possibly fractal,
while individual trajectories within the unstable region become highly complex
and possibly chaotic, as the atmosphere-ocean coupling increases. In
the unstable regime, spontaneous transitions occur in the mean ``temperature''
({\it i.e.}, thermocline depth), period, and extreme annual values, for purely
periodic, seasonal forcing. The model reproduces the Devil's bleachers
characterizing other ENSO models, such as nonlinear, coupled systems of partial
differential equations; some of the features of this behavior have been
documented in general circulation models, as well as in observations. We
expect, therefore, similar behavior in much more detailed and realistic models,
where it is harder to describe its causes as completely.Comment: 22 pages, 9 figure
Periodic solutions to a -Laplacian neutral Duffing equation with variable parameter
We study a type of -Laplacian neutral Duffing functional differential equation with variable parameter to establish new results on the existence of -periodic solutions. The proof is based on a famous continuation theorem for coincidence degree theory. Our research enriches the contents of neutral equations and generalizes known results. An example is given to illustrate the effectiveness of our results
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