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December 2004 Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator
William Artiles, André Nachbin
Methods Appl. Anal. 11(4): 475-492 (December 2004).


New nonlinear evolution equations are derived that generalize the system by Matsuno [16] and a terrain-following Boussinesq system by Nachbin [23]. The regime considers finite-amplitude surface gravity waves on a two-dimensional incompressible and inviscid fluid of, highly variable, finite depth. The asymptotic simplification of the nonlinear potential theory equations is performed through a perturbation anaylsis of the Dirichlet-to-Neumann operator on a highly corrugated strip. This is achieved through the use of a curvilinear coordinate system. Rather than doing a long wave expansion for the velocity potential, a Fourier-type operator is expanded in a wave steepness parameter. The novelty is that the topography can vary on a broad range of scales. It can also have a complex profile including that of a multiply-valued function. The resulting evolution equations are variable coefficient Boussinesq-type equations. These equations represent a fully dispersive system in the sense that the original (hyperbolic tangent) dispersion relation is not truncated. The formulation is done over a periodically extended domain so that, as an application, it produces efficient Fourier (FFT) solvers. A preliminary communication of this work has been published in the Physical Review Letters [1].


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William Artiles. André Nachbin. "Asymptotic nonlinear wave modeling through the Dirichlet-to-Neumann operator." Methods Appl. Anal. 11 (4) 475 - 492, December 2004.


Published: December 2004
First available in Project Euclid: 13 April 2006

zbMATH: 1177.76041
MathSciNet: MR2195366

Primary: 76B15
Secondary: 35B40, 35R35

Rights: Copyright © 2004 International Press of Boston


Vol.11 • No. 4 • December 2004
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