No abstract

A model equation derived by B. B. Kadomtsev & V. I. Petviashvili (1970) suggests that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal spatial direction. This prediction is rigorously confirmed for the full water-wave problem in the present paper. The theory is variational in nature. A simple but mathematically unfavourable variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle with significantly better mathematical properties using a generalisation of the Lyapunov-Schmidt reduction procedure. A nontrivial critical point of the reduced functional is detected using the direct methods of the calculus of variations.

Studied here is an initial-and boundary-value problem for the Korteweg-de Vries equationposed on a bounded interval I = {x: a x b}. This problem features non-homogeneous boundary conditions applied at x = a and x = b and is known to be well-posed in the L 2 -based Sobolev space H s (I) for any s > − 3 4 . It is shown here that this initialboundary-value problem is in fact well-posed in H s (I) for any s > −1. Moreover, the solution map that associates the solution tothe auxiliary data is not only continuous, but also analytic between the relevant function classes. The improvement on the previous theory comes about because of a more exacting appreciation of the damping that is inherent in the imposition of the boundary conditions. Published by Elsevier Inc.

Capillary-gravity waves of finite amplitude at the interface of a two-fluid flow are considered. The fluids of constant densities, extended to infinity in each direction, are inviscid and incompressible. The flow is two-dimensional and irrotational in each fluid. Assume that there is a permanent wave at the interface moving with a constant speed and the wave decays to zero at infinity with a certain rate. It is shown that the wave must decay at infinity quadratically and the solution of the exact governing equations corresponding to such a wave must satisfy two integral identities for the net displacement and the energy of the wave, while the former implies non-existence of waves of pure depression or pure elevation in the flow.

In this paper, we study the travelling gravity waves in a system of two layers of perfect fluids, the bottom one being infinitely deep, the upper one having a finite thickness h. We assume that the flow is potential and the dimensionless parameters are the ratio between densities rho = rho(2)/rho(1) and lambda = gh/c(2). We study special values of the parameters such that lambda(1 - rho) is near 1(-), where a bifurcation of a new type occurs. We formulate the problem as a spatial reversible dynamical system, where U = 0 corresponds to a uniform state (velocity c in a moving reference frame), and we consider the linearized operator around 0. We show that its spectrum contains the entire real axis (essential spectrum), with, in addition, a double eigenvalue in 0, a pair of simple imaginary eigenvalues +/-ilambda at a distance O(1) from 0, and for lambda(1 - rho) above 1, another pair of simple imaginary eigenvalues tending towards 0 as lambda(1 - rho) --> 1(+). When lambda(1 - rho)