The theory of internal waves between two bodies of immiscible fluid is important both for its interest to ocean engineering and as a source of numerous interesting mathematical model equations that exhibit nonlinearity and dispersion. In this paper we derive a Hamiltonian formulation of the problem of a dynamic free interface (with rigid lid upper boundary conditions), and of a free surface and a free interface, this latter situation occurring more commonly in experiment and in nature.From the formulation, we develop a Hamiltonian perturbation theory for the long-wave limits, and we carry out a systematic analysis of the principal longwave scaling regimes. This analysis provides a uniform treatment of the classical works of Peters and Stoker [28], Benjamin [3,4], Ono [26], and many others. Our considerations include the Boussinesq and Korteweg-de Vries (KdV) regimes over finite-depth fluids, the Benjamin-Ono regimes in the situation in which one fluid layer is infinitely deep, and the intermediate long-wave regimes. In addition, we describe a novel class of scaling regimes of the problem, in which the amplitude of the interface disturbance is of the same order as the mean fluid depth, and the characteristic small parameter corresponds to the slope of the interface.Our principal results are that we highlight the discrepancies between the case of rigid lid and of free surface upper boundary conditions, which in some circumstances can be significant. Motivated by the recent results of Choi and Camassa [6,7], we also derive novel systems of nonlinear dispersive long-wave equations in the large-amplitude, small-slope regime. Our formulation of the dynamical free-surface, free-interface problem is shown to be very effective for perturbation calculations; in addition, it holds promise as a basis for numerical simulations.

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This article concerns the pairwise nonlinear interaction of solitary waves in the free surface of a body of water lying over a horizontal bottom. Unlike solitary waves in many completely integrable model systems, solitary waves for the full Euler equations do not collide elastically; after interactions there is a nonzero residual wave which trails the post-collision solitary waves. In this report on new numerical and experimental studies of such solitary wave interactions, we verify that this is the case, both in head-on collisions (the counter-propagating case) and overtaking collisions (the co-propagating case), quantifying the degree to which interactions are inelastic. In the situation in which two identical solitary waves undergo a head-on collision, we compare the asymptotic predictions of Su and Mirie [6] and Byatt-Smith [23], the wavetank experiments of Maxworthy [5], and the numerical results of Cooker, Weidman and Bale [4] with independent numerical simulations, in which we quantify the phase change, the run-up, and the form of the residual wave and its Fourier signature in both small and large amplitude interactions. This updates the prior numerical observations of inelastic interactions in Fenton and Rienecker [3]. In the case of two non-identical solitary waves, our precision wavetank experiments are compared with numerical simulations, again observing the run up, phase lag, and the generation of a residual from the interaction. Considering overtaking solitary wave interactions, we compare our experimental observations, numerical simulations, and the asymptotic predictions of Zou and Su [14], and again we quantify the inelastic residual after collisions in the simulations. Geometrically, our numerical simulations of overtaking interactions fit into the three categories of KdV two-soliton solutions defined in Lax [16], with however a modification in the parameter regime. In all cases we have considered, collisions are seen to be inelastic, although the degree to which interactions depart from elastic is very small. Finally, we give several theoretical results: (1) a relationship between the change in amplitude of solitary waves due to a pairwise collision and the energy carried away from the interaction by the residual component, and (2) a rigorous estimate of the size of the residual component of pairwise solitary wave collisions. This estimate is consistent with the analytic results of Schneider and Wayne [20], Wright [22] and Bona, Colin and Lannes [21]. However in the light of our numerical data, both (1) and (2) indicate a need to re-evaluate the asymptotic results in [6, 14].

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou [RP]. We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being V.E. Zakharov's Hamiltonian [Z] for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length scale as, or longer than, the order of the wavelength of the surface waves.In the two dimensional case of waves in a channel, we give an alternate derivation of the effective KdV equation that is obtained in [RP]. In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three dimensional long wave equations in a Boussinesq scaling regime, and again in certain cases an effective KP system in the appropriate unidirectional limit.The computations for these results are performed in the framework of an asymptotic analysis of multiple scale operators. In the present case this involves the DirichletNeumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

This paper is concerned with the two-dimensional problem of nonlinear gravity waves travelling at the interface between a thin ice sheet and an ideal fluid of infinite depth. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff’s hypothesis, which yields a conservative and nonlinear expression for the bending force. A Hamiltonian formulation for this hydroelastic problem is proposed in terms of quantities evaluated at the fluid–ice interface. For small-amplitude waves, a nonlinear Schrödinger equation is derived and its analysis shows that no solitary wavepackets exist in this case. For larger amplitudes, both forced and free steady waves are computed by direct numerical simulations using a boundary-integral method. In the unforced case, solitary waves of depression as well as of elevation are found, including overhanging waves with a bubble-shaped profile for wave speeds $c$ much lower than the minimum phase speed ${c}_{\mathit{min}} $. It is also shown that the energy of depression solitary waves has a minimum at a wave speed ${c}_{m} $ slightly less than ${c}_{\mathit{min}} $, which suggests that such waves are stable for $c\lt {c}_{m} $ and unstable for $c\gt {c}_{m} $. This observation is verified by time-dependent computations using a high-order spectral method. These computations also indicate that solitary waves of elevation are likely to be unstable.