Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

Claassen, Kyle M.; Johnson, Mathew A.

doi:10.48550/arxiv.1710.09950

Cited by 5 publications

(18 citation statements)

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“…Since such equations are often used to describe water waves in the long-wave regime where the forces of gravity and surface tension are both important [13,19], our study gives insight into the mechanism for instability in the context of more complicated equations describing water waves. It is known that equations admitting bidirectional waves can exhibit high-frequency instabilities [6,4] and in this work, we show that resonance provides another mechanism, even in the context of one-directional wave propagation.…”

Section: Introductionsupporting

confidence: 57%

Stability of Periodic Traveling Wave Solutions to the Kawahara Equation

Trichtchenko¹,

Deconinck²,

Kollár³

2018

SIAM J. Appl. Dyn. Syst.

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We analyse the stability of periodic, travelling-wave solutions to the Kawahara equation and some of its generalizations. We determine the parameter regime for which these solutions can exhibit resonance. By examining perturbations of small-amplitude solutions, we show that generalised resonance is a mechanism for high-frequency instabilities. We derive a quadratic equation which fully determines the stability region for these solutions. Focussing on perturbations of the small-amplitude solutions, we obtain asymptotic results for how their instabilities develop and grow. Numerical computation is used to confirm these asymptotic results and illustrate regimes where our asymptotic analysis does not apply.

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Section: Introductionsupporting

confidence: 57%

Stability of Periodic Traveling Wave Solutions to the Kawahara Equation

Trichtchenko¹,

Deconinck²,

Kollár³

2018

SIAM J. Appl. Dyn. Syst.

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“…Indeed, (1.4) is known to be locally well-posed in standard Sobolev spaces only provided that the Cauchy data has strictly positive surface elevation. The work [9] provides numerical evidence that this surface elevation restriction is sharp. This evolutionary perspective motivates the search also for periodic traveling wave solutions of (1.4) with strictly positive wave height.…”

Section: Numerical Observationsmentioning

confidence: 89%

“…We consider the stability of waves in these and more general full-dispersion models outside of the small-amplitude regime as an important open problem. We note, however, the recent work [9] where the global global bifurcation and spectral stability of large amplitude waves of (2.1), and other related bidirecitonal full-dispersion water wave models, have been numerically investigated. The interested reader is referred to this paper for a number of numerical observations concerning the stability of large amplitude waves that is so far unproven.…”

Section: Numerical Observationsmentioning

confidence: 96%

“…We thereby hope to motivate the analytical theory by pointing out some key features of solutions along the global bifurcation branch that will be central to our later analysis, as well as some that are conjectures that have not yet been proved analytically. See also [9], where the authors numerically investigate the global bifurcation and dynamic stability of periodic traveling wave solutions of various bidirectional, full-dispersion water wave models.…”

Section: Numerical Observationsmentioning

confidence: 99%

“…The system (1.4) can be derived as an ad-hoc bidirectionalization of the Whitham equation or, as in [1] and [26], via a formal expansion of the Dirichlet-Neumann operator appearing in the free-surface water-wave equations. Although the analytical existence theory for this equation is conditional [16,23], it displays nice qualitative properties for solutions with strictly positive surface elevation: it significantly outperforms the KdV equation in experimental settings [8,34], and the steady periodic waves of supercritical wave speed are stable in the appropriate regimes [9]. A proof of conditionally stable solitary waves in the spirit of [13] is in preparation, too [27].…”

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confidence: 99%

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Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

Ehrnström

Johnson

Claassen

2018

Arch Rational Mech Anal

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We consider the existence of periodic traveling waves in a bidirectional Whitham equation, combining the full two-way dispersion relation from the incompressible Euler equations with a canonical shallow water nonlinearity. Of particular interest is the existence of a highest, cusped, traveling wave solution, which we obtain as a limiting case at the end of the main bifurcation branch of 2π-periodic traveling wave solutions continuing from the zero state. Unlike the unidirectional Whitham equation, containing only one branch of the full Euler dispersion relation, where such a highest wave behaves like |x| 1/2 near its crest, the cusped waves obtained here behave like |x log |x||. Although the linear operator involved in this equation can be easily represented in terms of an integral operator, it maps continuous functions out of the Hölder and Lipschitz scales of function spaces by introducing logarithmic singularities. Since the nonlinearity is also of higher order than that of the unidirectional Whitham equation, several parts of our proofs and results deviate from those of the corresponding unidirectional equation, with the analysis of the logarithmic singularity being the most subtle component. This paper is part of a longer research programme for understanding the interplay between nonlinearities and dispersion in the formation of large-amplitude waves and their singularities.

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A comparative study of bi-directional Whitham systems

Dinvay

Dutykh

Kalisch

2019

Applied Numerical Mathematics

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In 1967, Whitham proposed a simplified surface water-wave model which combined the full linear dispersion relation of the full Euler equations with a weakly linear approximation. The equation he postulated which is now called the Whitham equation has recently been extended to a system of equations allowing for bi-directional propagation of surface waves. A number of different two-way systems have been put forward, and even though they are similar from a modeling point of view, these systems have very different mathematical properties.In the current work, we review some of the existing fully dispersive systems, such as found in [1,3,8,17,23,24]. We use state-of-the-art numerical tools to try to understand existence and stability of solutions to the initial-value problem associated to these systems. We also put forward a new system which is Hamiltonian and semi-linear. The new system is shown to perform well both with regard to approximating the full Euler system, and with regard to well posedness properties.

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Numerical Bifurcation and Spectral Stability of Wavetrains in Bidirectional Whitham Models

Cited by 5 publications

References 0 publications

Stability of Periodic Traveling Wave Solutions to the Kawahara Equation

Stability of Periodic Traveling Wave Solutions to the Kawahara Equation

Existence of a Highest Wave in a Fully Dispersive Two-Way Shallow Water Model

A comparative study of bi-directional Whitham systems

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