2,646 research outputs found
Metastability of solitary roll wave solutions of the St. Venant equations with viscosity
We study by a combination of numerical and analytical Evans function
techniques the stability of solitary wave solutions of the St. Venant equations
for viscous shallow-water flow down an incline, and related models. Our main
result is to exhibit examples of metastable solitary waves for the St. Venant
equations, with stable point spectrum indicating coherence of the wave profile
but unstable essential spectrum indicating oscillatory convective instabilities
shed in its wake. We propose a mechanism based on ``dynamic spectrum'' of the
wave profile, by which a wave train of solitary pulses can stabilize each other
by de-amplification of convective instabilities as they pass through successive
waves. We present numerical time evolution studies supporting these
conclusions, which bear also on the possibility of stable periodic solutions
close to the homoclinic. For the closely related viscous Jin-Xin model, by
contrast, for which the essential spectrum is stable, we show using the
stability index of Gardner--Zumbrun that solitary wave pulses are always
exponentially unstable, possessing point spectra with positive real part.Comment: 42 pages, 9 figure
A stability index for detonation waves in Majda's model for reacting flow
Using Evans function techniques, we develop a stability index for weak and
strong detonation waves analogous to that developed for shock waves in
[GZ,BSZ], yielding useful necessary conditions for stability. Here, we carry
out the analysis in the context of the Majda model, a simplified model for
reacting flow; the method is extended to the full Navier-Stokes equations of
reacting flow in [Ly,LyZ]. The resulting stability condition is satisfied for
all nondegenerate, i.e., spatially exponentially decaying, weak and strong
detonations of the Majda model in agreement with numerical experiments of [CMR]
and analytical results of [Sz,LY] for a related model of Majda and Rosales. We
discuss also the role in the ZND limit of degenerate, subalgebraically decaying
weak detonation and (for a modified, ``bump-type'' ignition function)
deflagration profiles, as discussed in [GS.1-2] for the full equations.Comment: 36 pages, 3 figure
On the One-dimensional Stability of Viscous Strong Detonation Waves
Building on Evans function techniques developed to study the stability of
viscous shocks, we examine the stability of viscous strong detonation wave
solutions of the reacting Navier-Stokes equations. The primary result,
following the work of Alexander, Gardner & Jones and Gardner & Zumbrun, is the
calculation of a stability index whose sign determines a necessary condition
for spectral stability. We show that for an ideal gas this index can be
evaluated in the ZND limit of vanishing dissipative effects. Moreover, when the
heat of reaction is sufficiently small, we prove that strong detonations are
spectrally stable provided the underlying shock is stable. Finally, for
completeness, the stability index calculations for the nonreacting
Navier-Stokes equations are includedComment: 66 pages, 7 figure
A robust numerical method to study oscillatory instability of gap solitary waves
The spectral problem associated with the linearization about solitary waves
of spinor systems or optical coupled mode equations supporting gap solitons is
formulated in terms of the Evans function, a complex analytic function whose
zeros correspond to eigenvalues. These problems may exhibit oscillatory
instabilities where eigenvalues detach from the edges of the continuous
spectrum, so called edge bifurcations. A numerical framework, based on a fast
robust shooting algorithm using exterior algebra is described. The complete
algorithm is robust in the sense that it does not produce spurious unstable
eigenvalues. The algorithm allows to locate exactly where the unstable discrete
eigenvalues detach from the continuous spectrum. Moreover, the algorithm allows
for stable shooting along multi-dimensional stable and unstable manifolds. The
method is illustrated by computing the stability and instability of gap
solitary waves of a coupled mode model.Comment: key words: gap solitary wave, numerical Evans function, edge
bifurcation, exterior algebra, oscillatory instability, massive Thirring
model. accepted for publication in SIAD
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