Nonlinear evolution and higher harmonics in extreme water waves based on higher order Peregrine solutions of the nonlinear Schrödinger equation
Phys. Rev. Fluids 10, 064802 – Published 23 June, 2025
DOI: https://doi.org/10.1103/948t-8dgh
Abstract
Extreme waves have posed a significant threat to human safety and the integrity of marine structures at sea or near coastal regions. As a possible explanation for the formation of extreme waves, modulation instability has been studied extensively in recent years. Peregrine solutions of the nonlinear Schrödinger equation (NLSE) describe the nonlinear evolution of slowly varying wave envelopes. Limited studies have been found in the literature for Peregrine solutions at higher orders as mechanisms of the formation of large-amplitude waves. Additionally, the characteristics of higher harmonics generated during the evolution of Peregrine solutions remain unexplored. This study employs a high-precision numerical model based on a high-order spectral method to simulate the nonlinear evolution of the first-, the second-, and the third-order Peregrine solutions. Experiments were carried out in a wave flume to implement the high order solutions, which demonstrates the physical understanding of the solutions in the application in water waves and validates the model. A phase manipulation method is utilized to decompose the higher harmonics of the wave elevations. The results examine the evolution of wave profile, skewness, and kurtosis for the first three orders of Peregrine solutions. Higher harmonics of the wave elevations during modulation and demodulation are extracted and analyzed for the first time. The one-dimensional and two-dimensional Fourier transforms are used to analyze the amplitude spectrum and the joint wavenumber-frequency spectrum, respectively, during the evolution of the wave trains. Through spectral analysis, the nonlinear energy transfer characteristics of Peregrine solutions are identified.
