Related papers: Ordering of two small parameters in the shallow wa…
Based on an extension of the Foldy--Wouthuysen method to two-body equations, the problem of expansion of equal-time relativistic equations for two Dirac particles in powers of $1/c$ to higher orders is considered. For the case of two…
A new operator equation for periodic gravity waves on water of finite depth is derived and investigated; it is equivalent to Babenko's equation considered in \cite{KD}. Both operators in the proposed equation are nonlinear and depend on the…
In the seminal work of Benjamin,\cite{Ben} in the late 70's, he has derived the ubiquitous Benjamin model, which is a reduced model in the theory of water waves. Notably, it contains two parameters in its dispersion part and under some…
We prove the local well-posedness for a nonlinear equation modeling the evolution of the free surface for waves of moderate amplitude in the shallow water regime.
The classical approach to linking lattice dynamics properties to continuum equations of motion, the "method of long waves," is extended to include higher order terms. The additional terms account for non-local and non-linear effects. In the…
We consider the Isobe-Kakinuma model for water waves in both cases of the flat and the variable bottoms. The Isobe-Kakinuma model is a system of Euler-Lagrange equations for an approximate Lagrangian which is derived from Luke's Lagrangian…
Wave shape (e.g. wave skewness and asymmetry) impacts sediment transport, remote sensing and ship safety. Previous work showed that wind affects wave shape in intermediate and deep water. Here, we investigate the effect of wind on wave…
In the present paper, we consider the linear wave equation with the scale-invariant damping and mass. It is known that the global behavior of the solution depends on the size of the coefficients in front of the damping and mass at initial…
We study the evolution of nonlinear surface gravity water-wave packets developing from modulational instability over an uneven bottom. A nonlinear Schr\"odinger equation (NLSE) with coefficients varying in space along propagation is used as…
This paper provides an alternative methodology for analysis of three-wave interactions under the exact dispersion relation associated with gravity waves in fluid of intermediate depth. A Korteweg-de Vries type of equation with exact…
In this paper, we derive asymptotic models for the propagation of two and three-dimensional gravity waves at the free surface and the interface between two layers of immiscible fluids of different densities, over an uneven bottom. We assume…
In this paper we are concerned with a initial boundary-value problem for a coupled system of two KdV equations, posed on the positive half line, under the effect of a localized damping term. The model arises when modeling the propagation of…
We justify rigorously an Isobe-Kakinuma model for water waves as a higher order shallow water approximation in the case of a flat bottom. It is known that the full water wave equations are approximated by the shallow water equations with an…
In this paper we study symmetry reductions and exact solutions of the shallow water wave (SWW) equation $$u_{xxxt} + \alpha u_x u_{xt} + \beta u_t u_{xx} - u_{xt} - u_{xx} = 0,\eqno(1)$$ where $\alpha$ and $\beta$ are arbitrary, nonzero,…
The problem of interest in this article are waves on a layer of finite depth governed by the Euler equations in the presence of gravity, surface tension, and vertical electric fields. Perturbation theory is used to identify canonical…
We study the transverse dynamics of two-dimensional traveling periodic waves for the gravity--capillary water-wave problem. The governing equations are the Euler equations for the irrotational flow of an inviscid fluid layer with free…
The Newton, Gauss--Newton and Levenberg--Marquardt methods all use the first derivative of a vector function (the Jacobian) to minimise its sum of squares. When the Jacobian matrix is ill-conditioned, the function varies much faster in some…
We are concerned with hyperbolic systems of order-one linear PDEs originated on non-characteristic manifolds. We put forward a simple but effective method of transforming such initial conditions to standard initial conditions (i.e. when the…
Hammack & Segur (1978) conducted a series of surface water-wave experiments in which the evolution of long waves of depression was measured and studied. This present work compares time series from these experiments with predictions from…
We study the long time evolution of internal waves in two dimensional subcritical channels with flat horizontal ends. We show the leading profiles of solutions are the outgoing solutions to the stationary equations. This is done by showing…