Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform

Details Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform Internal solitary waves in the ocean: Analysis using the periodic, inverse scattering transform

2007-08-25

arxiv.org/abs/0708.3421v2

Mathematics and Computers in Simulation 80 (1), 2009, pp. 192-201

Physics

Fluid Dynamics

10 pages, 4 figures (6 images); v2: corrected a few minor mistakes and typos, version accepted for publication in Math. Comput

Scientific paper

10.1016/j.matcom.2009.06.005

The periodic, inverse scattering transform (PIST) is a powerful analytical tool in the theory of integrable, nonlinear evolution equations. Osborne pioneered the use of the PIST in the analysis of data form inherently nonlinear physical processes. In particular, Osborne's so-called nonlinear Fourier analysis has been successfully used in the study of waves whose dynamics are (to a good approximation) governed by the Korteweg--de Vries equation. In this paper, the mathematical details and a new application of the PIST are discussed. The numerical aspects of and difficulties in obtaining the nonlinear Fourier (i.e., PIST) spectrum of a physical data set are also addressed. In particular, an improved bracketing of the "spectral eigenvalues" (i.e., the +/-1 crossings of the Floquet discriminant) and a new root-finding algorithm for computing the latter are proposed. Finally, it is shown how the PIST can be used to gain insightful information about the phenomenon of soliton-induced acoustic resonances, by computing the nonlinear Fourier spectrum of a data set from a simulation of internal solitary wave generation and propagation in the Yellow Sea.

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