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...

In this paper, we use the approximation of shallow water waves (Margaritondo 2005 Eur. J. Phys. 26 401) to understand the behavior of a tsunami in a variable depth. We deduce the shallow water wave equation and the continuity equation that must be satisfied when a wave encounters a discontinuity in the sea depth. A short explanation about how the tsunami hit the west coast of India is given bas...

Long's equation describes stationary flows to all orders of nonlinearity and dispersion. Dissipation is neglected. In this paper, Long's equation is used to attempt to model the propagation of a solibore -a train of internal waves in shallow water at the deepening phase of the internal tide. 1. The Solibore Phenomenon The internal tide in shallow water often has a sawtooth shape rather than a s...

We consider the linearization of the three-dimensional water waves equation with surface tension about a flat interface. Using oscillatory integral methods, we prove that solutions of this equation demonstrate dispersive decay at the somewhat surprising rate of t−5/6. This rate is due to competition between surface tension and gravitation at O(1) wave numbers and is connected to the fact that, ...

Submitted: Nov 12, 2013; Accepted: Dec 18, 2013; Published: Dec 22, 2013 Abstract: In this article, we have employed an enhanced (G′/G)-expansion method to find the exact solutions first and then the solitary wave solutions of the nonlinear generalized shallow water wave equation. Here we have derived solitons, singular solitons and periodic wave solutions through the enhanced (G′/G)-expansion ...

The Benney-Luke equation (BL) is a model for the evolution of three-dimensional weakly nonlinear, long water waves of small amplitude. In this paper we propose a nearly conservative scheme for the numerical resolution of (BL). Moreover, it is known ([PQ99] and [Pau03]) that (BL) is linked to the Kadomtsev-Petviashvili equation for almost one-dimensional waves propagating in one direction. We st...

The nature of transverse instabilities of dark solitons for the (2+1)-dimensional defocusing nonlinear Schrödinger/Gross–Pitaevskĭi (NLS/GP) equation is considered. Special attention is given to the small (shallow) amplitude regime, which limits to the Kadomtsev–Petviashvili (KP) equation. We study analytically and numerically the eigenvalues of the linearized NLS/GP equation. The dispersion re...

In this paper, we investigate a new approach for the numerical solution of the two-dimensional depth-integrated shallow water equations, based on coupling discontinuous and continuous Galerkin methods. In this approach, we couple a discontinuous Galerkin method applied to the primitive continuity equation, coupled to a continuous Galerkin method applied to the so-called ‘‘wave continuity equati...