The scientific problem addressed in this paper is the analysis of Wilton ripple resonances, particularly focusing on higher-order resonances that involve wavenumbers beyond the commonly studied triad resonance (\(N = 2\)). Traditional studies predominantly consider resonances between two harmonics (\(k = 1\) and \(k = 2\)), but this work extends the exploration to scenarios where the resonance occurs between \(k = 1\) and \(N\) harmonics for \(N\) greater than \(2\). The primary aim is to derive explicit formulas for the coefficients in the asymptotic series expansions associated with these higher-order resonances and to understand the behaviour and convergence of these solutions in weakly nonlinear models.
The methods employed involve asymptotic analysis of weakly nonlinear dispersive partial differential equations. The authors utilise the example of the Kawahara equation to illustrate the construction of the asymptotic solutions. The technique begins by expanding the solution and the wave speed correction in terms of a small amplitude parameter, \(\varepsilon\). The method involves deriving recursive formulas for the coefficients at each order of the asymptotic expansion, taking into account the specific characteristics of the linear operator associated with each model. This approach is generalized for different values of \(N\), providing a comprehensive framework to analyse higher-order resonances.
The main findings of the manuscript reveal that for higher-order resonances (\(N > 3\)), the amplitude of the resonant harmonics appears at an order of \(\varepsilon^{N-2}\), which is significantly smaller than the main wave amplitude. This indicates that higher-order resonances are less pronounced and harder to detect compared to lower-order resonances. Additionally, the study highlights the importance of the spectral gap and the behaviour of the linear operator at high frequencies in determining the convergence of the asymptotic series. It was observed that certain linear operators could lead to a small radius of convergence due to a small recurring divisor in the recursive formulas.
In conclusion, this research significantly advances the understanding of Wilton ripples in the context of higher-order resonances. The results provide critical insights into the asymptotic behaviour of solutions to weakly nonlinear dispersive partial differential equations, and the derived formulas and methodologies can be utilised in future studies to further explore the existence and stability of these resonances. The findings have important implications for the study of nonlinear wave interactions and contribute to the broader field of mathematical physics and fluid dynamics.