804 research outputs found
Whitham Averaged Equations and Modulational Stability of Periodic Traveling Waves of a Hyperbolic-Parabolic Balance Law
In this note, we report on recent findings concerning the spectral and
nonlinear stability of periodic traveling wave solutions of
hyperbolic-parabolic systems of balance laws, as applied to the St. Venant
equations of shallow water flow down an incline. We begin by introducing a
natural set of spectral stability assumptions, motivated by considerations from
the Whitham averaged equations, and outline the recent proof yielding nonlinear
stability under these conditions. We then turn to an analytical and numerical
investigation of the verification of these spectral stability assumptions.
While spectral instability is shown analytically to hold in both the Hopf and
homoclinic limits, our numerical studies indicates spectrally stable periodic
solutions of intermediate period. A mechanism for this moderate-amplitude
stabilization is proposed in terms of numerically observed "metastability" of
the the limiting homoclinic orbits.Comment: 27 pages, 5 figures. Minor changes throughou
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
Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto-Sivashinsky equation
In this paper we consider the spectral and nonlinear stability of periodic
traveling wave solutions of a generalized Kuramoto-Sivashinsky equation. In
particular, we resolve the long-standing question of nonlinear modulational
stability by demonstrating that spectrally stable waves are nonlinearly stable
when subject to small localized (integrable) perturbations. Our analysis is
based upon detailed estimates of the linearized solution operator, which are
complicated by the fact that the (necessarily essential) spectrum of the
associated linearization intersects the imaginary axis at the origin. We carry
out a numerical Evans function study of the spectral problem and find bands of
spectrally stable periodic traveling waves, in close agreement with previous
numerical studies of Frisch-She-Thual, Bar-Nepomnyashchy,
Chang-Demekhin-Kopelevich, and others carried out by other techniques. We also
compare predictions of the associated Whitham modulation equations, which
formally describe the dynamics of weak large scale perturbations of a periodic
wave train, with numerical time evolution studies, demonstrating their
effectiveness at a practical level. For the reader's convenience, we include in
an appendix the corresponding treatment of the Swift-Hohenberg equation, a
nonconservative counterpart of the generalized Kuramoto-Sivashinsky equation
for which the nonlinear stability analysis is considerably simpler, together
with numerical Evans function analyses extending spectral stability analyses of
Mielke and Schneider.Comment: 78 pages, 11 figure
A New Method to Predict Vessel Capsizing in a Realistic Seaway
A recently developed approach, in the area of nonlinear oscillations, is used to analyze the single degree of freedom equation of motion of a oating unit (such as a ship) about a critical axis (such as roll). This method makes use of a closed form analytic solution, exact upto the rst order, and takes into account the the complete unperturbed (no damping or forcing) dynamics. Using this method very-large-amplitude nonlinear vessel motion in a random seaway can be analysed with techniques similar to those used to analyse nonlinear vessel motions in a regular (periodic) or random seaway. The practical result being that dynamic capsizing studies can be undertaken considering the shortterm irregularity of the design seaway. The capsize risk associated with operation in a given sea state can be evaluated during the design stage or when an operating area change is being considered. Moreover, this technique can also be used to guide physical model tests or computer simulation studies to focus on critical vessel and environmental conditions which may result in dangerously large motion amplitudes. Extensive comparitive results are included to demonstrate the practical usefulness of this approach. The results are in the form of solution orbits which lie in the stable or unstable manifolds and are then projected onto the phase plane
Stability of Viscous St. Venant Roll-Waves: From Onset to the Infinite-Froude Number Limit
International audienceWe study the spectral stability of roll-wave solutions of the viscous St. Venant equationsmodeling inclined shallow-water flow, both at onset in the small-Froude number or âweakly unstableâlimit F â 2+ and for general values of the Froude number F , including the limit F â +â. In the former,F â 2+ , limit, the shallow water equations are formally approximated by a Korteweg de Vries/Kuramoto-Sivashinsky (KdV-KS) equation that is a singular perturbation of the standard Korteweg de Vries (KdV)equation modeling horizontal shallow water flow. Our main analytical result is to rigorously validate thisformal limit, showing that stability as F â 2+ is equivalent to stability of the corresponding KdV-KSwaves in the KdV limit. Together with recent results obtained for KdV-KS by JohnsonâNobleâRodriguesâZumbrun and Barker, this gives not only the first rigorous verification of stability for any single viscous St.Venant roll wave, but a complete classification of stability in the weakly unstable limit. In the remainderof the paper, we investigate numerically and analytically the evolution of the stability diagram as Froudenumber increases to infinity. Notably, we find transition at around F = 2.3 from weakly unstable todifferent, large-F behavior, with stability determined by simple power law relations. The latter stabilitycriteria are potentially useful in hydraulic engineering applications, for which typically 2.5 †F †6.0
- âŠ