Spatial dynamics and solitary hydroelastic surface waves (Q6556837)

The paper presents a comprehensive existence theory for solitary waves occurring at the interface between a thin ice sheet, modeled using the Cosserat theory of hyperelastic shells, and an ideal fluid of finite depth in irrotational motion. This study is framed within the context of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, showcasing the refinements and advancements in the theory over time. Notably, the paper introduces novel elements such as the application of a higher-order Legendre transformation to reformulate the problem as a spatial Hamiltonian system, alongside the utilization of a Riesz basis for the phase space, thus enhancing the analogy with dynamical systems.\N\NThe scientific problem addressed in the paper is the propagation of solitary waves on the surface of an ocean beneath a thin ice sheet, where the water is considered as a perfect fluid in irrotational flow, and the ice sheet is treated as an elastic shell that bends with the surface without stretching, friction, or cavitation with the fluid beneath. This study is grounded in the model developed by \textit{P. I. Plotnikov} and \textit{J. F. Toland} [Philos. Trans. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 369, No. 1947, 2942--2956 (2011; Zbl 1228.76027)], which employs the Euler equations for inviscid fluid flow in conjunction with the Cosserat theory of hyperelastic shells. The problem is described by a set of governing equations for the hydrodynamic problem in dimensionless coordinates and a moving coordinate system, with travelling waves represented by a velocity potential \(\varphi\).\N\NTo solve this problem, the authors reformulate the governing equations using a change of variable that maps the variable fluid domain to a fixed strip, transforming the equations into a more manageable form. This approach allows for a precise proof of the persistence of two homoclinic solutions as solutions to the unapproximated reduced system, corresponding to symmetric hydroelastic solitary waves. The analysis involves a formal weakly nonlinear theory, leading to the nonlinear Schrödinger equation as the leading-order reduced system. The existence of solitary wave solutions is confirmed by proving that these solutions correspond to symmetric solitary waves of elevation and depression.\N\NThe main findings of the manuscript include the rigorous formulation of the hydrodynamic problem as a spatial Hamiltonian system, the demonstration of discrete spectrum for the linear operator L, and the proof that homoclinic solutions of the reduced system correspond to solitary waves. The study employs centre-manifold reduction to detect these homoclinic solutions, confirming the predictions of the weakly nonlinear theory and establishing the existence of symmetric solitary-wave solutions. Additionally, the paper highlights the parameter regimes in which homoclinic bifurcation is detected and discusses related work in the literature, providing a comprehensive context for the presented research.\N\NIn conclusion, this research paper significantly advances the understanding of solitary hydroelastic surface waves, offering a rigorous mathematical framework and precise existence proofs for symmetric solitary-wave solutions. The novel methodological approaches and detailed spectral analysis contribute valuable insights to the field of fluid mechanics, particularly in the study of wave propagation in hydroelastic contexts. The findings have potential implications for theoretical and applied research in oceanography, coastal engineering, and related disciplines, emphasizing the relevance and importance of this work in advancing knowledge in these areas.