Hilbert transform view of water-wave theory (Q6541830)

The paper provides an innovative perspective on the mathematical foundations of water surface waves. The study re-examines traditional approaches, particularly the multiple-scale analysis, through the lens of the Hilbert transform, aiming to refine our understanding of non-local and dispersive effects in water-wave dynamics.\N\NThe scientific problem addressed in the manuscript pertains to the inadequacies of existing methodologies for describing weakly nonlinear wave phenomena, especially in planar and cylindrical geometries. The standard multiple-scale approach, while effective in specific contexts, is shown to have limitations in addressing the intricate interplay of nonlinear and dispersive forces. By leveraging the Hilbert transform, the study seeks to derive new models that overcome these limitations, offering both a reinterpretation of classical equations, such as Zakharov's equation, and novel formulations, including a weakly nonlinear envelope equation for cylindrical wave dynamics.\N\NTo tackle this problem, the author employs the Hilbert transform to simplify the mathematical treatment of water waves by focusing on dynamics at the free surface rather than the entire fluid domain. This approach reduces the algebraic complexity inherent in traditional methods and highlights key features of wave phenomena that were previously obscured. For planar waves, the author derives a general integro-differential equation, termed the ''Hilbert wave equation,'' which encapsulates the dynamics of small-amplitude solutions without the narrow wavepacket assumption. Similarly, for cylindrical waves, the analysis identifies the unique challenges posed by radial symmetry, particularly the coupling of first- and second-order effects, which leads to a modified nonlinear Schrödinger equation with an inverse-square potential. The study also explores the extension of these ideas to finite-depth scenarios, revealing the transition from non-local dispersive dynamics in deep water to weakly dispersive, local dynamics in shallow water.\N\NThe findings of the study are substantial. In addition to providing a more compact derivation of classical results, the research elucidates the sources of certain limitations in the traditional approaches, such as the indeterminacy encountered in cylindrical geometries. The work introduces a generalised framework that seamlessly integrates dispersion and non-locality, avoiding issues like triad resonances that complicate other models. The derivation of the concentric nonlinear Schrödinger equation (cNLS) represents a significant contribution, as it accounts for the radial spreading of waves and bridges the gap between Cartesian and cylindrical formulations. The study also underscores the utility of the Hilbert transform in achieving deeper insights into wave turbulence and extending the analysis to broader geometrical settings.\N\NThe significance of this research lies in its potential to advance the theoretical understanding of water-wave mechanics while offering practical tools for the analysis of free-surface flows in diverse settings. The novel models and methodologies proposed are not only mathematically rigorous but also adaptable to complex physical scenarios, paving the way for further exploration in both theoretical and applied fluid dynamics. This work exemplifies how foundational mathematical insights can lead to a re-evaluation and enhancement of classical theories, with implications for fields ranging from oceanography to wave turbulence.