TY - JOUR

T1 - Modulational instability in the Whitham equation with surface tension and vorticity

AU - Hur, Vera Mikyoung

AU - Johnson, Mathew A.

N1 - Funding Information:
VMH is supported by the National Science Foundation grant CAREER DMS-1352597 , an Alfred P. Sloan research fellowship, and a Beckman fellowship of the Center for Advanced Study at the University of Illinois at Urbana–Champaign . MAJ is supported by the National Science Foundation under grant DMS-1211183 . The authors thank the anonymous referees for their careful reading of the manuscript and many helpful comments and references.
Publisher Copyright:
© 2015 Elsevier Ltd.

PY - 2015/12/1

Y1 - 2015/12/1

N2 - We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability of gravity waves. In the case of weak surface tension, we find intervals of stable and unstable wave numbers, whose boundaries are associated with the extremum of the group velocity, the resonance between the first and second harmonics, the resonance between long and short waves, and a resonance between dispersion and the nonlinearity. For each constant vorticity, we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value, and stable otherwise. Moreover it can be made stable for a sufficiently large vorticity. The results agree with those based upon numerical computations or formal multiple-scale expansions to the physical problem.

AB - We study modulational stability and instability in the Whitham equation, combining the dispersion relation of water waves and a nonlinearity of the shallow water equations, and modified to permit the effects of surface tension and constant vorticity. When the surface tension coefficient is large, we show that a periodic traveling wave of sufficiently small amplitude is unstable to long wavelength perturbations if the wave number is greater than a critical value, and stable otherwise, similarly to the Benjamin-Feir instability of gravity waves. In the case of weak surface tension, we find intervals of stable and unstable wave numbers, whose boundaries are associated with the extremum of the group velocity, the resonance between the first and second harmonics, the resonance between long and short waves, and a resonance between dispersion and the nonlinearity. For each constant vorticity, we show that a periodic traveling wave of sufficiently small amplitude is unstable if the wave number is greater than a critical value, and stable otherwise. Moreover it can be made stable for a sufficiently large vorticity. The results agree with those based upon numerical computations or formal multiple-scale expansions to the physical problem.

KW - Constant vorticity

KW - Modulational instability

KW - Surface tension

KW - Water waves

KW - Whitham equation

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U2 - 10.1016/j.na.2015.08.019

DO - 10.1016/j.na.2015.08.019

M3 - Article

AN - SCOPUS:84941890978

VL - 129

SP - 104

EP - 118

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

M1 - 10635

ER -