Direct and inverse nonlinear Fourier transform based on the Korteweg-deVries equation (KdV-NLFT) - A spectral analysis of nonlinear surface waves in shallow water

Abstract

In the conventional analysis methods such as fast Fourier (FFT), wavelet (WT) and Hilbert-Huang transform (FFT) the spectral decomposition of the original data does not consider pos-sible effects of the relative water depth on the shape, stability or nonlinearity of the deter-mined spectral components. In the KdV-NFLT cnoidal waves are used as basis for the spec-tral decomposition of the original data. This allows the consideration of the water depth as a governing parameter for the analysis and the original surface data are decomposed adaptively into those nonlinear physical oscillatory waves and solitary waves that really occur in shallow water. Furthermore, the real nonlinear wave-wave interactions between the nonlinear cnoidal waves are considered and calculated explicitly in the KdV-NLFT. The main topics of the thesis are: (i) The numerical implementation and verification of the implemented inverse and direct scattering transform (IST and DST) of the Korteweg-deVries equation as a generalized Fourier transform (KdV-NLFT). (ii) The practical application of the implemented KdV-NLFT to selected shallow-water problems such as the reliable identifica-tion of solitons from signals measured over and behind submerged reefs (soliton fission) and the analysis of the nonlinear propagation of long-period waves in shallow water. (iii) A com-parative analysis using KdV-NLFT and conventional analysis methods such as the linear fast Fourier transform (FFT) in the frequency domain and the Hilbert-Huang transform (HHT) in the time-frequency domain. (iv) Then, based on the results of these comparative analyses rec-ommendations will be given for the practical application of the nonlinear KdV-NLFT and the conventional methods FFT and HHT for the spectral analysis of nonlinear shallow-water time and space series. Finally, the results of the thesis clearly show that the KdV-NLFT provides a decisive insight into the underlying nonlinear processes of the analysed shallow-water wave problems that cannot be obtained by application of the conventional analysis methods.