相关论文: Higher Order Modulation Equations for a Boussinesq…
We study solitary-wave and kink-wave solutions of a modified Boussinesq equation through a multiple-time reductive perturbation method. We use appropriated modified Korteweg-de Vries hierarchies to eliminate secular producing terms in each…
As a formal approximation, the nonlinear Schr\"{o}dinger (NLS) equation can be derived to describe the evolution of the envelopes of small oscillating wave packets-like solutions to the Euler-Poisson system. In this paper we rigorously…
The water wave theory traditionally assumes the fluid to be perfect, thus neglecting all effects of the viscosity. However, the explanation of several experimental data sets requires the explicit inclusion of dissipative effects. In order…
It is well known that the Korteweg-de Vries (KdV) equation and its generalizations serve as modulation equations for traveling wave solutions to generic Fermi-Pasta-Ulam-Tsingou (FPUT) lattices. Explicit approximation estimates and other…
The purpose of this paper is to extend the store of models able to support integrable defects by investigating the two-dimensional Boussinesq nonlinear wave equation. As has been previously noted in many examples, insisting that a defect…
A perturbative scheme is applied to calculate corrections to the leading, exponentially small (beyond-all-orders) amplitude of the ``trailing'' wave asymptotics of weakly localized solitons. The model considered is a Korteweg-de Vries…
Formally second-order correct, mathematical descriptions of long-crested water waves propagating mainly in one direction are derived. These equations are analogous to the first-order approximations of KdV- or BBM-type. The advantage of…
In this paper we propose a multiscale method for the acoustic wave equation in highly oscillatory media. We use a higher-order extension of the localized orthogonal decomposition method combined with a higher-order time stepping scheme and…
The modification of simplest equation method to look for exact solutions of nonlinear partial differential equations is presented. Using this method we obtain exact solutions of generalized Korteweg-de Vries equation with cubic source and…
We are interested in the description of small modulations in time and space of wave-train solutions to the complex Ginzburg-Landau equation \begin{align*} \partial_T \Psi = (1+ i \alpha) \partial_X^2 \Psi + \Psi - (1+i \beta ) \Psi…
Dispersive averaging effects are used to show that KdV equation with periodic boundary conditions possesses high frequency solutions which behave nearly linearly. Numerical simulations are presented which indicate high accuracy of this…
A generalized version of the $abcd$-Boussinesq class of systems is derived to accommodate variable bottom topography in two-dimensional space. This extension allows for the conservation of suitable energy functionals in some cases and…
In this paper, we proceed along our analysis of the Korteweg-de Vries approximation of the Gross-Pitaevskii equation initiated in a previous paper. At the long-wave limit, we establish that solutions of small amplitude to the…
The KdV equation can be derived in the shallow water limit of the Euler equations. Over the last few decades, this equation has been extended to include both higher order effects (KdV2) and an uneven river bottom. Although this equation is…
In this paper we present a modification of DJ Method [J. Math. Anal. Appl. 316 (2006), 753-763] to solve the nonlinear equations more efficiently. It is observed that the modified DJ method is faster and hence it has accelerated convergence…
We consider the initial-value problem for a system of coupled Boussinesq equations on the infinite line for localised or sufficiently rapidly decaying initial data, generating sufficiently rapidly decaying right- and left-propagating waves.…
In this paper, we investigate the well-posedness of a nonlinear dispersive model with variable coefficients that describes the evolution of surface waves propagating through a one-dimensional shallow water channel of finite length with…
The present article is the third part of a series of papers devoted to the shallow water wave modelling. In this part, we investigate the derivation of some long wave models on a deformed sphere. We propose first a suitable for our purposes…
In this paper, we consider the higher order Boussinesq (HBq) equation which models the bi-directional propagation of longitudinal waves in various continuous media. The equation contains the higher order effects of frequency dispersion. The…
In the recent paper by Kudryashov [Commun. Nonlinear Sci. Numer. Simulat., 2009, V.14, 3507-3529] seven common errors in finding exact solutions of nonlinear differential equations were listed and discussed in detail. We indicate two more…