Differential and Integral Equations

Asymptotic stability of solitary waves in the Benney-Luke model of water waves

Tetsu Mizumachi, Robert L. Pego, and José Raúl Quintero

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We study asymptotic stability of solitary wave solutions in the one-dimensional Benney--Luke equation, a formally valid approximation for describing two-way water-wave propagation. For this equation, as for the full water-wave problem, the classic variational method for proving orbital stability of solitary waves fails dramatically due to the fact that the second variation of the energy-momentum functional is infinitely indefinite. We establish nonlinear stability in energy norm under the spectral stability hypothesis that the linearization admits no nonzero eigenvalues of nonnegative real part. We then verify this hypothesis for waves of small energy.

Article information

Differential Integral Equations, Volume 26, Number 3/4 (2013), 253-301.

First available in Project Euclid: 5 February 2013

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 37K40: Soliton theory, asymptotic behavior of solutions 35Q35: PDEs in connection with fluid mechanics 35B35: Stability 76B15: Water waves, gravity waves; dispersion and scattering, nonlinear interaction [See also 35Q30]


Mizumachi, Tetsu; Pego, Robert L.; Quintero, José Raúl. Asymptotic stability of solitary waves in the Benney-Luke model of water waves. Differential Integral Equations 26 (2013), no. 3/4, 253--301. https://projecteuclid.org/euclid.die/1360092825

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