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Abstract We derive the Whitham modulation equations for the Zakharov–Kuznetsov equation via a multiple scales expansion and averaging two conservation laws over one oscillation period of its periodic traveling wave solutions. We then use the Whitham modulation equations to study the transverse stability of the periodic traveling wave solutions. We find that all periodic solutions traveling along the first spatial coordinate are linearly unstable with respect to purely transversal perturbations, and we obtain an explicit expression for the growth rate of perturbations in the long wave limit. We validate these predictions by linearizing the equation around its periodic solutions and solving the resulting eigenvalue problem numerically. We also calculate the growth rate of the solitary waves analytically. The predictions of Whitham modulation theory are in excellent agreement with both of these approaches. Finally, we generalize the stability analysis to periodic waves traveling in arbitrary directions and to perturbations that are not purely transversal, and we determine the resulting domains of stability and instability.
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Stability of Periodic, Traveling-Wave Solutions to theCapillary Whitham Equation
Recently, the Whitham and capillary Whitham equations were shown to accurately modelthe evolution of surface waves on shallow water. In order to gain a deeper understanding of theseequations, we compute periodic, traveling-wave solutions for both and study their stability. Wepresent plots of a representative sampling of solutions for a range of wavelengths, wave speeds, waveheights, and surface tension values. Finally, we discuss the role these parameters play in the stabilityof these solutions.
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- Award ID(s):
- 1716120
- PAR ID:
- 10094041
- Date Published:
- Journal Name:
- Fluids
- Volume:
- 4
- Issue:
- 1
- ISSN:
- 2311-5521
- Page Range / eLocation ID:
- 58
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract The Whitham equation was proposed as a model for surface water waves that combines the quadratic flux nonlinearityof the Korteweg–de Vries equation and the full linear dispersion relationof unidirectional gravity water waves in suitably scaled variables. This paper proposes and analyzes a generalization of Whitham's model to unidirectional nonlinear wave equations consisting of a general nonlinear flux functionand a general linear dispersion relation. Assuming the existence of periodic traveling wave solutions to this generalized Whitham equation, their slow modulations are studied in the context of Whitham modulation theory. A multiple scales calculation yields the modulation equations, a system of three conservation laws that describe the slow evolution of the periodic traveling wave's wavenumber, amplitude, and mean. In the weakly nonlinear limit, explicit, simple criteria in terms of generalandestablishing the strict hyperbolicity and genuine nonlinearity of the modulation equations are determined. This result is interpreted as a generalized Lighthill–Whitham criterion for modulational instability.more » « less
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/fluids4010058
