Pascal Noble

Nonlinear modulational stability of periodic traveling-wave solutions of the generalized Kuramoto–Sivashinsky equation

Abstract In this paper we consider the spectral and nonlinear stabilities 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.

Nonlinear Stability of Viscous Roll Waves

Extending results of Oh and Zumbrun and of Johnson and Zumbrun for parabolic conservation laws, we show that spectral stability implies nonlinear stability for spatially periodic viscous roll wave solutions of the one-dimensional St. Venant equations for shallow water flow down an inclined ramp. The main new issues to be overcome are incomplete parabolicity and the nonconservative form of the equations, which lead to undifferentiated quadratic source terms that cannot be handled using the estimates of the conservative case. The first is resolved by treating the equations in the more favorable Lagrangian coordinates, for which one can obtain large-amplitude nonlinear damping estimates similar to those carried out by Mascia and Zumbrun in the related shock wave case, assuming only symmetrizability of the hyperbolic part. The second is resolved by the observation that, similarly as in the relaxation and detonation cases, sources occurring in nonconservative components experience decay that is greater than ex...

Nonlocalized modulation of periodic reaction diffusion waves: The Whitham equation

In a companion paper, we established nonlinear stability with detailed diffusive rates of decay of spectrally stable periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized (L1) perturbation. Here, we determine time-asymptotic behavior under such perturbations, showing that solutions consist of a leading order of a modulation whose parameter evolution is governed by an associated Whitham averaged equation.

Spectral stability of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation in the Korteweg-de Vries limit

We study the spectral stability of a family of periodic wave trains of the Korteweg-de Vries/Kuramoto-Sivashinsky equation @tv +v@xv +@ 3v + @ 2 xv +@ 4 xv = 0; > 0, in the Korteweg-de Vries limit ! 0, a canonical limit describing small-amplitude weakly unstable thin lm ow. More precisely, we carry out a rigorous singular perturbation analysis reducing the problem of spectral stability in this limit to the validation of a set of three conditions, each of which have been numerically analyzed in previous studies and shown to hold simultaneously on a non-empty set of parameter space. The main technical diculty

Nonlocalized Modulation of Periodic Reaction Diffusion Waves: Nonlinear Stability

Extending results of Johnson and Zumbrun showing stability under localized (L1) perturbations, we show that spectral stability implies nonlinear modulational stability of periodic traveling-wave solutions of reaction diffusion systems under small perturbations consisting of a nonlocalized modulation plus a localized perturbation. The main new ingredient is a detailed analysis of linear behavior under modulational data

Slow modulations of periodic waves in Hamiltonian PDEs, with application to capillary fluids

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Thin power-law film flow down an inclined plane: consistent shallow-water models and stability under large-scale perturbations

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