Stability Of Periodic Traveling Waves Research Articles

Stability of a family of travelling wave solutions in a feedforward chain of phase oscillators

Travelling waves are an important class of signal propagation phenomena in extended systems with a preferred direction of information flow. We study the generation of travelling waves in unidirectional chains of coupled oscillators communicating via a phase-dependent pulse-response interaction borrowed from mathematical neuroscience. Within the context of such systems, we develop a widely applicable, jointly numerical and analytical methodology for deducing existence and stability of periodic travelling waves. We provide careful numerical studies that support the existence of a periodic travelling wave solution as well as the asymptotic relaxation of a single oscillator to the wave when it is forced with the wave profile. Using this evidence as an assumption, we analytically prove global stability of waves in the infinite chain, with respect to initial perturbations of downstream sites. This rigorous stability result suggests that asymptotic relaxation to the travelling wave occurs even when the forcing is perturbed from the wave profile, a property of the motivating system that is supported by previous work as well as the convergence of the more sophisticated numerical algorithm that we propose in order to compute a high-precision approximation to the solution. We provide additional numerical studies that show that the wave is part of a one-parameter family, and we illustrate the structural robustness of this family with respect to changes in the coupling strength.

Slow Modulations of Periodic Waves in Hamiltonian PDEs, with Application to Capillary Fluids

Since its elaboration by Whitham, almost fifty years ago, modulation theory has been known to be closely related to the stability of periodic traveling waves. However, it is only recently that this relationship has been elucidated, and that fully nonlinear results have been obtained. These only concern dissipative systems though: reaction-diffusion systems were first considered by Doelman, Sandstede, Scheel, and Schneider [Mem. Amer. Math. Soc. 2009], and viscous systems of conservation laws have been addressed by Johnson, Noble, Rodrigues, and Zumbrun [preprint 2012]. Here, only nondissipative models are considered, and a most basic question is investigated, namely the expected link between the hyperbolicity of modulated equations and the spectral stability of periodic traveling waves to sideband perturbations. This is done first in an abstract Hamiltonian framework, which encompasses a number of dispersive models, in particular the well-known (generalized) Korteweg--de Vries equation, and the less known Euler--Korteweg system, in both Eulerian coordinates and Lagrangian coordinates. The latter is itself an abstract framework for several models arising in water waves theory, superfluidity, and quantum hydrodynamics. As regards its application to compressible capillary fluids, attention is paid here to untangle the interplay between traveling waves/modulation equations in Eulerian coordinates and those in Lagrangian coordinates. In the most general setting, it is proved that the hyperbolicity of modulated equations is indeed necessary for the spectral stability of periodic traveling waves. This extends earlier results by Serre [Comm. Partial Differential Equations 2005], Oh and Zumbrun [Arch. Ration. Mech. Anal. 2003], and Johnson, Zumbrun and Bronski [Phys. D 2010]. In addition, reduced necessary conditions are obtained in the small amplitude limit. Then numerical investigations are carried out for the modulated equations of the Euler--Korteweg system with two types of 'pressure' laws, namely the quadratic law of shallow water equations, and the nonmonotone van der Waals pressure law. Both the evolutionarity and the hyperbolicity of the modulated equations are tested, and regions of modulational instability are thus exhibited.

An index theorem for the stability of periodic travelling waves of Korteweg–de Vries type

We consider the stability of periodic travelling-wave solutions to a generalized Korteweg–de Vries (gKdV) equation and prove an index theorem relating the number of unstable and potentially unstable eigenvalues to geometric information on the classical mechanics of the travelling-wave ordinary differential equation. We illustrate this result with several examples, including the integrable KdV and modified KdV equations, the L2-critical KdV-4 equation that arises in the study of blow-up and the KdV-½ equation, which is an idealized model for plasmas.

Nonlinear stability of periodic travelling waves of Zakharov system

This paper concerns the stability of periodic travelling waves of the Zakharov system.

On the modulation equations and stability of periodic generalized Korteweg–de Vries waves via Bloch decompositions

In this paper, we consider the relation between Evans-function-based approaches to the stability of periodic travelling waves and other theories based on long-wavelength asymptotics together with Bloch wave expansions. In previous work it was shown by rigorous Evans function calculations that the formal slow modulation approximation resulting in the linearized Whitham averaged system accurately describes the spectral stability to long-wavelength perturbations. To clarify the connection between Bloch-wave-based expansions and Evans-function-based approaches, we reproduce this result without reference to the Evans function by using direct Bloch expansion methods and spectral perturbation analysis. One of the novelties of this approach is that we are able to calculate the relevant Bloch waves explicitly for arbitrary finite-amplitude solutions. Furthermore, this approach has the advantage of being applicable in the more general multi-periodic setting where no conveniently computable Evans function has yet been devised.

Stability of periodic traveling waves for complex modified Korteweg–de Vries equation

We study the existence and stability of periodic traveling-wave solutions for complex modified Korteweg–de Vries equation. We also discuss the problem of uniform continuity of the data-solution mapping.

Whitham averaged equations and modulational stability of periodic traveling waves of a hyperbolic-parabolic balance law

In this note, we report on recent ndings 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 ow 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 verication 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.

Global well-posedness and non-linear stability of periodic traveling waves for a Schrödinger-Benjamin-Ono system

The objective of this paper is two-fold: firstly, we develop a localand global (in time) well-posedness theory for a system describing the motionof two fluids with different densities under capillary-gravity waves in a deepwater flow (namely, a Schrödinger-Benjamin-Ono system) for low-regularityinitial data in both periodic and continuous cases; secondly, a family of newperiodic traveling waves for the Schrödinger-Benjamin-Ono system is given:by fixing a minimal period we obtain, via the implicit function theorem, asmooth branch of periodic solutions bifurcating a Jacobian elliptic functioncalled dnoidal, and, moreover, we prove that all these periodic traveling wavesare nonlinearly stable by perturbations with the same wavelength.

Spectral Stability of Periodic Solutions of Viscous Conservation Laws: Large Wavelength Analysis

We complete and unify the works by Oh and Zumbrun (2003a) and by the author (1994), about the spectral stability of traveling waves that are spatially periodic, in systems of n conservation laws. Our context is one-dimensional. These systems are of order larger than one, in general. For instance, they could be viscous approximations of first-order systems that are not everywhere hyperbolic. However, modelling considerations often lead to higher order terms, like capillarity in fluid dynamics; our framework remains valid in this more general setting. We make generic assumptions, saying in particular that the set of periodic traveling waves is a manifold of maximal dimension, under the restrictions given by the conserved quantities. The spectral stability of a periodic traveling wave is studied through Floquet's theory. Following Gardner (1993), we introduce an Evans function D(λ, θ), being λ the Laplace frequency and θ the phase shift. The large wavelength analysis is the description of the zero set of D around the origin. Our main result is that this zero set is described, at the leading order, by a characteristic equation This formula involves a flux F, which enters into a first-order system of conservation laws, describing the slow modulation of the periodic traveling waves. Its size N is in practice larger than n. The important consequence is that hyperbolicity of the latter system is a necessary condition for spectral stability of periodic traveling waves. Finally, we show that a similar treatment works for coupled map lattices obtained by discretizing systems of conservation laws.

Generalized instability of periodic traveling waves in anharmonic monatomic chains

We study the stability of periodic traveling waves in the quadratic-quartic (Fermi-Pasta-Ulam), the Lennard-Jones and the Morse monatomic chains. We have found that, with the only exception of zone boundary modes in the two last chains, all traveling commensurate modes are unstable. For zone boundary modes we study their stability spectra and the scaling behavior at the instability threshold. From the stability of many other commensurate modes we find that the stability eigenvalues behave continuously as a function of wavelength, indicating that stability is not recovered for incommensurate modes.

Stability of periodic travelling waves in trickle beds

In trickle bed reactors, a gas and a liquid flow together through the interstices of a random array of solid packing. A pulsing flow pattern is obtained at high gas and liquid fluxes. A macroscopic volume averaged model can be used to mathematically represent the hydrodynamics of two-phase flow in packed beds. In one dimension, the onset and characteristics of pulsing flow can be modelled as periodic travelling waves. In this paper, the stability of the one-dimensional travelling waves is computed in the context of one-dimensional and two-dimensional perturbations of the full model. The 1-D waves are imposed as solutions of a pseudo spectral discretization of the 2-D equations. Floquet theory is applied to analyze the linear stability of these time-periodic solutions. The results show marked differences between the stability of fully developed waves and the corresponding uniform flow solutions when subjected to the same perturbations. The analysis also shows that two-dimensional flow patterns are likely to evolve from the early stages of pulse growth, and not from breakup of fully-developed 1-D waves.

Bifurcation and stability of periodic traveling waves for a reaction-diffusion system

v) plane having a stable critical point at (1,O) and unstable critical points at (0,O) and (cos &, , sin &). Moreover, if we take 0 < & < rc, then flows starting with /3, < /I < rc make large excursions before decaying to (1, 0), whereas those with 0 < /? < & decay monotonically to (l,O). This “excitability” is a feature in common with the nonlinear vector field in the Hodgkin-Huxley equations, and also with the simpler FitzHugh-Nagumo equations. We begin by describing some results of numerical simulations we made of