Various classes of steady and unsteady dark solitary waves (DSWs) are known to exist in modulation equations for water waves in finite depth. However, there is a class of steady DSWS of the full water-wave problem which are missed by the classical modulation equations such as the Hasimoto-Ono, Benney-Roskes, and Davey-Stewartson. These steady DSWs, recently discovered by Bridges and Donaldson, ...

We aim to improve the techniques to predict tsunami wave heights along the coast. The modeling of tsunamis with the shallow water equations has been very successful, but often shortcomings arise, for example because wave dispersion is neglected. To bypass the latter shortcoming, we use the (linearized) variational Boussinesq model derived by Klopman et al. [12]. Another shortcoming is that comp...

Abstract. The main purpose of this article is to present an approximate solution for the one dimensional wave equation subject to an integral conservation condition in terms of second kind Chebyshev polynomials. The operational matrices of integration and derivation are introduced and utilized to reduce the wave equation and the conditions into the matrix equations which correspond to a system ...

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the ...

The rupture of a 3D stationary free liquid film under the competing effects of surface tension and van der Waals forces is studied as a linearized stability problem in a purely irrotational analysis utilizing the dissipation method. The results of the foregoing analysis are compared with a 2D long-wave approximation that has given rise to an extensive literature on the rupture problem. The irro...

In this paper we study, from a numerical point of view, some aspects of stability of solitary-wave solutions of the Bona–Smith systems of equations. These systems are a family of Boussinesq-type equations and were originally proposed for modelling the two-way propagation of one-dimensional long waves of small amplitude in an open channel of water of constant depth. We study numerically the beha...

In this paper, we shall study traveling wave solutions for a set of onedimensional nonlinear, nonlocal, evolutionary partial differential equations. This class of equations originally arose at quadratic order in the asymptotic expansion for shallow water waves [4,10]. The famous Korteweg–de Vries equation – which is nonlinear, but local – arises uniquely at linear order in this shallow water wa...

We derive transport equations for the propagation of water wave action in the presence of a static, spatially random surface drift. Using the Wigner distribution W(x,k, t) to represent the envelope of the wave amplitude at position x contained in waves with wavevector k, we describe surface wave transport over static flows consisting of two length scales; one varying smoothly on the wavelength ...