Impact of directional spreading on nonlinear KdV-soliton spectra in intermediate water

Abstract

The Korteweg–De Vries (KdV) equation is a partial differential equation used to describe the dynamics of water waves under the assumptions of shallow water, unidirectionality, weak nonlinearity and constant depth. It can be solved analytically with a suitable nonlinear Fourier transform (NFT). The NFT for the KdV equation is subsequently referred to as the KdV-NFT. The soliton part of the nonlinear Fourier spectrum provides valuable insights into the nonlinear evolution of waveforms by exposing the amplitudes and velocities of potentially hidden solitonic components. Under the KdV equation, the nonlinear spectrum evolves trivially according to simple analytic rules. This in particular reflects that solitons are conserved by the KdV equation. However, in reality, the nonlinear spectrum will change during evolution due to deviations from the KdV equation. For example, waves in the ocean are typically multi-directional. Furthermore, the water depth may range into the intermediate regime, e.g. depending on tides and peak periods. It is therefore uncertain how long the nonlinear spectrum of real-world data remains representative. In particular, it is unclear how stable the detected soliton components are during evolution. To assess the effectiveness of the KdV-NFT in representing water wave dynamics under non-ideal conditions, we generated numerical sea states with varying directional spreading in intermediate water ($kh=1.036$) using the High-Order Spectral Ocean (HOS-Ocean) model for nonlinear evolution. After applying the NFT to space series extracted from these evolving directional wave fields, we observe that the KdV-soliton spectra from the NFT are quite stable for cases with small directional spreading. We in particular observe that the largest soliton amplitude is (sometimes dramatically) more stable than the amplitude of the largest linear mode. For large directional spreading, the applicability is limited to short propagation times and distances, respectively.