Nature-based Solutions (NbS) use natural processes to address social, economic, and environmental challenges, including climate change, and this study explores their application for coastal protection. Before the actual implementation of NbS structures, co-creation with stakeholders to identify and assess potential NbS solutions for the selected local site is essential. However, existing frameworks for selecting NbS measures in coastal protection remain limited in handling the complex interactions among multiple indicators and the inherent uncertainty in expert evaluations. To address this gap, we develop an integrated assessment framework specifically for NbS in coastal protection. We first identified 18 distinct coastal protection measures through literature reviews. The proposed framework combines the Fuzzy Delphi Method (FDM) and entropy-based intuitionistic fuzzy TOPSIS (IF-TOPSIS) to facilitate expert-driven multi-criteria decision analysis (MCDA), both of which use fuzzy theory to address uncertainty and ambiguity in expert judgments. The 18 consensus indicators from an initial pool of 63 evaluation indicators are obtained in FDM analysis. Subsequently, IF-TOPSIS is applied to rank the 18 measures against these evaluation indicators based on membership, non-membership, and hesitancy degrees. The proposed framework is demonstrated through a case study at the Golden Coast, Tainan, Taiwan to illustrate its practical applicability.

Soliton gases are large ensembles of random solitons with distinct characteristics arising from integrable system dynamics. They have been widely studied in theory and experiments, and were observed in natural lagoons. However, it remains an open question whether they occur naturally in the open ocean. Nonlinear ocean states containing solitons have been observed in the literature, but the dominance of solitons over other wave components required for a soliton gas has not been demonstrated. Our study provides the first field evidence of soliton gas sea states in the deep ocean, measured in Taiwan waters. The soliton energy ratio derived from the nonlinear Fourier transform (NFT) is used as a key parameter to quantify how close sea states are to soliton gases. We identify eleven measurements with extremely high soliton energy ratios. They are characterized by short-period waves with relatively small wave heights, accompanied by extreme steepness and Benjamin–Feir Index (BFI) values. These states are exceptionally rare, representing only 0.054% of our dataset. Since directional interference can artificially increase the estimated soliton energy ratio obtained from measured time series, we further apply a probabilistic directional filtering method to remove the directional interference. Three wave records from the Eluanbi station are found to retain high soliton energy ratios after the directional interference has been removed, confirming that they are indeed soliton gases.

The separation of wind sea and swell is crucial for advancing wave dynamics research, improving wave forecasting, and optimizing the design of coastal and offshore structures. In this study, we highlight the limitations of the widely used wave age method for separating two-dimensional wind sea and swell. Specifically, under strong wind conditions, waves require extended durations to reach full development—an aspect not accounted for by the wave age method, which assumes fully developed seas and thus tends to overestimate wind sea. Furthermore, changes in wind direction and wave refraction in shallow waters can lead to misclassification. To overcome these issues, we propose a novel algorithm for directional spectral separation, grounded in wind wave growth theory and incorporating wave refraction effects. The proposed method improves separation accuracy and delivers more consistent results across a range of wave conditions.

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.

When a large number of solitons dominates the dynamics of a system, scientists describe this collective behaviour of solitons as a soliton gas. Soliton gases are currently the subject of intense practical and theoretical investigations. The existence of soliton gases has been confirmed in experiments, but is not clear what kind of sea states might lead to soliton gases. Therefore, in order to determine the wave parameters for sea states that lead to soliton gases, large numbers of surface wave elevations are generated by the well-known JOSNWAP model in this paper. Here, we only discuss soliton gases in deep water governed by the nonlinear Schrödinger (NLS) equation. The nonlinear Fourier transform (NFT) with vanishing boundary conditions is applied to the simulated ocean surface waves. The resulting nonlinear Fourier spectrum is used to calculate the energy of radiation waves and solitons. We investigate which JONSWAP parameters result in sea states that can be characterized as soliton gases, and find that a large Phillip’s parameter α, a large peak enhancement parameter γ and a short peak period T_P are important factors for soliton gas conditions. The results allow researchers to estimate how likely soliton gases are in deep waters. Furthermore, we find that the appearance of rogue waves is slightly increased in highly nonlinear sea states with soliton gas-like conditions.

Rogue waves are extreme waves in the ocean that appear from nowhere and disappear without a trace. They are usually modelled by the nonlinear Schrödinger equation (NLS), which describes nonlinear phenomena such as modulational instability and solitons on finite backgrounds. In this study, the periodic nonlinear Fourier transform (NFT) for the NLS equation is applied to simulate ocean surface waves in deep water. The temporal and spatial structures of surface waves are obtained by evolving JONSWAP time series using the NLS equation. Several parameters extracted from the NFT spectra of the initial time series are investigated as predictors for the maximum wave height during evolution. We investigate several parameters from the literature, and find that with suitably optimized coefficients, a NFT-based parameter based on the largest unstable mode has a good correlation with the overall maximum wave amplitude. This new spectral criterion can contribute to rogue wave forecasting under extreme sea states.