The Inverse KdV-Based Nonlinear Fourier Transform (KdV-NLFT): Nonlinear Superposition of Cnoidal Waves and Reconstruction of the Original Data

Abstract

Since 2008, at Leichtweiß-Institute for Hydraulic Engineering and Water Resources at TU Braunschweig a KdV-based nonlinear Fourier transform is implemented and successfully applied to numerical and hydraulic model test data of solitary wave fission behind submerged reefs [1]. The KdV-NLFT is the application of the direct and inverse scattering transform for the solution of the Korteweg-deVries equation. This approach explicitly considers both solitons and oscillatory waves (cnoidal waves) as spectral basic components for the decomposition of the original data. Furthermore, the nonlinear wave-wave interactions between the nonlinear spectral basic components are explicitly considered in the analysis. The direct KdV-NLFT decomposes the original data into cnoidal waves and provides wave heights, wave numbers or frequencies, phases and the moduli which are a measure of the nonlinearity of cnoidal waves. Details of this procedure are given in Brühl & Oumeraci [2]. The interpretation of the nonlinear spectral basic components is described in Brühl & Oumeraci [3]. The inverse KdV-NLFT which is addressed here calculates the nonlinear wave-wave interactions between cnoidal waves and provides the original data by superposition of cnoidal waves and their nonlinear interactions. The practical application of the KdV-NLFT for the analysis of long-wave propagation in shallow water is presented in Brühl & Oumeraci [4].