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Publication 12-CNA-002

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

Tetsu Mizumachi
Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan

Robert L. Pego
Department of Mathematical Sciences, Carnegie Mellon University,
Pittsburgh, PA 15213

José Raúl Quintero
Departamento  Matemáticas, Universidad del Valle, Colombia

Abstract: 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 non-zero eigenvalues of non-negative real part. We also verify this hypothesis for waves of small energy.

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