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This paper presents an existence theory for solitary waves at the interface between a thin ice sheet (modelled using the Cosserat theory of hyperelastic shells) and an ideal fluid (of finite depth and in irrotational motion). The theory takes the form of a review of the Kirchgässner reduction to a finite-dimensional Hamiltonian system, highlighting the refinements in the theory over the years and presenting some novel aspects including the use of a higher-order Legendre transformation to formulate the problem as a spatial Hamiltonian system, and a Riesz basis for the phase space to complete the analogy with a dynamical system. The reduced system is to leading order given by the focussing cubic nonlinear Schrödinger equation, agreeing with the result of formal weakly nonlinear theory (which is included for completeness). We give a precise proof of the persistence of two of its homoclinic solutions as solutions to the unapproximated reduced system which correspond to symmetric hydroeleastic solitary waves.

The existence of a line solitary–wave solution to the water–wave problem with strong surface–tension effects was predicted on the basis of a model equation in the celebrated 1895 paper by D. J. Korteweg and G. de Vries and rigorously confirmed a century later by C. J. Amick and K. Kirchgässner in 1989. A model equation derived by B. B. Kadomtsev and V. I. Petviashvili in 1970 suggests that the Korteweg–de Vries line solitary wave belongs to a family of periodically modulated solitary waveswhich have a solitary–wave profile in the direction of motion and are periodic in the transverse direction. This prediction is rigorously confirmed for the full water–wave problem in the present paper. It is shown that the Korteweg–de Vries solitary wave undergoes a dimension–breaking bifurcationthat generates a family of periodically modulated solitary waves. The term dimension–breaking phenomenondescribes the spontaneous emergence of a spatially inhomogeneous solution of a partial differential equation from a solution which is homogeneous in one or more spatial dimensions.

Abstract:: We consider modulating pulse solutions for a nonlinear wave equation on the infinite line. Such a solution consists of a permanent pulse-like envelope steadily advancing in the laboratory frame and modulating an underlying wave-train. The problem is formulated as an infinite-dimensional dynamical system with one stable, one unstable and infinitely many neutral directions. Using a partial normal form and invariant-manifold theory we establish the existence of modulating pulse solutions which decay to small-amplitude disturbances at large distances.

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ω at the free surface. It is shown that there is a countably infinite set of frequencies {ω0, ω1, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t^1/2 as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary-value problems in non-smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.

This article studies the evolutionary problem for linear gravity waves on the surface of water in a uniform, symmetric channel which is excited by an antisymmetric pressure force of frequency ωat the free surface. It is shown that there is a countably infinite set of frequencies {ω0, ω1, …} which give rise to resonance phenomena: the amplitude of the wave motion grows like t1/2as t→∞ in a sense which is precisely specified. Under pressure forcing at any other frequency the solution obeys the principle of limiting amplitude. These results are obtained by combining methods developed for problems in acoustic waveguides with regularity theory for elliptic boundary‐value problems in non‐smooth domains. Copyright © 1999 John Wiley & Sons, Ltd.

This article establishes the existence of a trapped-mode solution to a linearized water-wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls; the forces of gravity and surface tension are operative. A trapped mode corresponds to an eigenvalue of the composition of an inverse differential operator and a Neumann–Dirichlet operator for an elliptic boundary-value problem in the fluid domain. The existence of such an eigenvalue is established by extending previous results dealing with the case when surface tension is absent. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons, Ltd.

This short article discusses the spectrum of the Neumann Laplacian in the infinite domain Ω⊂ℝⁿ , n 2 created by inserting a compact obstacle P into the uniform cylinder Ω0 =(−∞, ∞)×Ω'. The main result is the existence of at least one embedded eigenvalue when P is an (n −2)-dimensional surface whose unit normal is parallel to Ω' at each point of P . The special case when P is symmetric about {0}×Ω' is also treated. It is shown that there is at least one symmetric eigenvector and, when P is sufficiently long, at least one antisymmetric eigenvector. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

This short article discusses the spectrum of the Neumann Laplacian in the infinite domain Ω⊂ℝn, n⩾2 created by inserting a compact obstacle Pinto the uniform cylinder Ω0=(−∞, ∞)×Ω′. The main result is the existence of at least one embedded eigenvalue when Pis an (n−2)‐dimensional surface whose unit normal is parallel to Ω′ at each point of P. The special case when Pis symmetric about {0}×Ω′ is also treated. It is shown that there is at least one symmetric eigenvector and, when Pis sufficiently long, at least one antisymmetric eigenvector. © 1998 B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

This article establishes the existence of a trapped‐mode solution to a linearized water‐wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls; the forces of gravity and surface tension are operative. A trapped mode corresponds to an eigenvalue of the composition of an inverse differential operator and a Neumann–Dirichlet operator for an elliptic boundary‐value problem in the fluid domain. The existence of such an eigenvalue is established by extending previous results dealing with the case when surface tension is absent. © 1998 B.G. Teubner Stuttgart–John Wiley & Sons, Ltd.

This article establishes the existence of trapped-mode solutions of a linearized water-wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls. A trapped mode corresponds to an eigenvalue of a non-local Neumann–Dirichlet operator for an elliptic boundary-value problem in the fluid domain. The existence of such an eigenvalue is established by generalizing previous results concerning spectral theory for differential operators to this non-local operator. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

This article establishes the existence of trapped‐mode solutions of a linearized water‐wave problem. The fluid occupies a symmetric horizontal channel that is uniform everywhere apart from a confined region which either contains a thin vertical plate spanning the depth of the channel or has indentations in the channel walls. A trapped mode corresponds to an eigenvalue of a non‐local Neumann–Dirichlet operator for an elliptic boundary‐value problem in the fluid domain. The existence of such an eigenvalue is established by generalizing previous results concerning spectral theory for differential operators to this non‐local operator. © 1997 by B. G. Teubner Stuttgart–John Wiley & Sons Ltd.

It has long been recognized that linear theories cannot capture even the most basic features of waves observed on water surfaces: solitary waves being a particularly famous example where linear equations will not suffice. Hydrodynamic waves are inherently nonlinear. However the basic Euler equations which govern the behaviour of inviscid incompressible fluid flows are notoriously difficult mathematically and are certainly not well understood even at the beginning of the 21st century. When a constant–pressure nonlinear free–boundary condition is added to represent a water wave they become more awkward still, and more nonlinear.