Convergence of periodic wavetrains in the limit of large wavelength

The Korteweg-de Vries equation was originally derived as a model for unidirectional propagation of water waves. This equation possesses a special class of traveling-wave solutions corresponding to surface solitary waves. It also has permanent-wave solutions which are periodic in space, the so-called cnoidal waves. A classical observation of Korteweg and de Vries was that the solitary wave is obtained as a certain limit of cnoidal wavetrains.This result is extended here, in the context of the Korteweg-de Vries equation. It is demonstrated that a general class of solutions of the Korteweg-de Vries equation is obtained as limiting forms of periodic solutions, as the period becomes large.     

Well-Posedness for Multicomponent Schrödinger–gKdV Systems and Stability of Solitary Waves with Prescribed Mass

In this paper we prove the well-posedness issues of the associated initial value problem, the existence of nontrivial solutions with prescribed \(L^2\)-norm, and the stability of associated solitary waves for two classes of coupled nonlinear dispersive equations. The first problem here describes the nonlinear interaction between two Schrödinger type short waves and a generalized Korteweg-de Vries type long wave and the second problem describes the nonlinear interaction of two generalized Korteweg-de Vries type long waves with a common Schrödinger type short wave. The results here extend many of the previously obtained results for two-component coupled Schrödinger–Korteweg-de Vries systems.     

Soliton Solutions for the Inverse Korteweg-De Vries Equation

We construct soliton solutions of the inverse Korteweg-de Vries equation by developing the tanh-function method and symbolic-computation techniques.     

Application of the Riemann–Hilbert method to the vector modified Korteweg-de Vries equation

Nonlinear Dynamics - Under investigation in this paper is the inverse scattering transform of the vector modified Korteweg-de Vries (vmKdV) equation, which can be reduced to several integrable...     

Evolution of initial discontinuity for the defocusing complex modified KdV equation

Nonlinear Dynamics - The complete classification of solutions to the defocusing complex modified Korteweg-de Vries (cmKdV) equation with the step-like initial condition is given by Whitham theory....     

A note on nonlinear dispersion relations

In this note the nonlinear dispersion relations for the Korteweg-de Vries equation, the Benjamin-Bona-Mahony equation and the time regularized long wave equation are presented and discussed.     

Damping of large-amplitude solitary waves

We consider the damping of large-amplitude solitary waves in the framework of the extended Korteweg-de Vries equation (that is, the usual Korteweg-de Vries equation supplemented with a cubic nonlinear term) modified by the inclusion of a small damping term. The damping of a solitary wave is studied for several different forms of friction, using both the analytical adiabatic asymptotic theory and numerical simulations. When the coefficient of the cubic nonlinear term has the opposite sign to the coefficient of the linear dispersive term, the extended Kortweg-de Vries equation can support large-amplitude “thick” solitary waves. Under the influence of friction, these “thick” solitary waves decay and may produce one or more secondary solitary waves in this process. On the other hand, when the coefficient of the cubic nonlinear term has the same sign as the coefficient of the linear dispersive term, but the opposite sign to the coefficient of the quadratic nonlinear term, the action of friction may cause a solitary wave to decay into a wave packet.     

An extended traffic flow model on a gradient highway with the consideration of the relative velocity

In this paper, an extended traffic flow model on a single-lane gradient highway is proposed with the consideration of the relative velocity. The stability condition is obtained by the use of linear stability analysis. It is shown that the stability of traffic flow on the gradient varies with the slope and the coefficient of the relative velocity: when the slope is constant, the stable regions increase with the increase of the coefficient of the relative velocity; when the coefficient of the relative velocity is constant, the stable regions increase with the decrease of the slope in downhill and increase with the increase of the slope in uphill. The Burgers, Korteweg-de Vries, and modified Korteweg-de Vries equations are derived to describe the triangular shock waves, soliton waves, and kink-antikink waves in the stable, metastable, and unstable region, respectively. The numerical simulation shows a good agreement with the analytical result, which shows that the traffic congestion can be suppressed by introducing the relative velocity.     

三阶非线性发展方程的线性化有理逼近方法

研究了三阶非线性发展方程的初边值问题的解。采用基于Sinc函数的微分求积法发展了线性化有理逼近方法。通常的配点法不适用于上述三阶问题的求解。本文把提出的方法用于求解KdV方程,取得了良好的效果。     

Decompositions of Backlund transformations for the Korteweg-de Vries equation

In this paper, decomposition relations such as B=SB₁S, connecting scale transformations and Backlund transformations for the Korteweg-de Vries K-dV, modified K-dV, higher-order K-dV and cylindrical equations, are obtained.     

On the invariants of nonlinear differential equations of third order

We present a method for finding invariants of nonlinear differential equations of third order. Some propositions are given, which allow to conjecture the existence (or nonexistence) of invariants of a given nonlinear equation. As applications, invariants of the reduced Korteweg-de Vries and Nagumo equations, besides others, are studied.     

Multiple scale fourier transformation: An application to nonlinear dispersive waves

A Fourier transformation involving multiple scales is applied to describe the far-field asymptotic behaviour of nonlinear dispersive waves. It is shown that a nonlinear asymptotic perturbation can be carried out in terms of simple calculations with respect to Dirac delta functions involving a multiple scale wave number and frequency space. Fourier transformed versions of the nonlinear Schrödinger and Korteweg-de Vries equations are derived explicitly.     

Discrete models leading to a weakly dissipative korteweg-de vries equation

It is shown how a simple one-dimensional non-linear lattice model in which damping is present leads to a continuum approximation which is a generalization of the Korteweg-de Vries equation. The effect of the damping in the discrete model is to introduce a damping term in the continuum approximation which depends only on a power of the undifferentiated dependent variable. Numerical results are presented which show that such damping is extremely weak and that dispersive effects predominate.     

Approximate equations for long water waves

In the first part of this paper the Hamiltonian theory of water waves is used to obtain some equations in local coordinates. These equations are approximations of the Boussinesq type. They are stable with respect to short wave perturbations, e.g. rounding off errors in digital computing. In the second part the relation of Boussinesq equations to Korteweg-de Vries and Benjamin-Bona-Mahony equations is investigated.     

Two-mode fifth-order KdV equations: necessary conditions for multiple-soliton solutions to exist

In this work we establish two wave modes for the integrable fifth-order Korteweg-de Vries (TfKdV) equations. We determine necessary conditions of the nonlinearity and dispersion parameters of the equation for multiple-soliton solutions to exist. We apply the simplified Hirota method to derive multiple-soliton solutions under these conditions. We also examine the dispersion relations and the phase shifts of the developed models.     

Alternating segment explicit-implicit scheme for nonlinear third-order KdV equation

A group of asymmetric difference schemes to approach the Korteweg-de Vries(KdV)equation is given here.According to such schemes,the full explicit difference scheme and the fun implicit one,an alternating segment explicit-implicit difference scheme for solving the KdV equation is constructed.The scheme is linear unconditionally stable by the analysis of linearization procedure,and is used directly on the parallel computer. The numerical experiments show that the method has high accuracy.     

Nonlinear waves on the free surface of an electrically conducting liquid

Gravity waves of small but finite amplitude on the free surface of an electrically conducting liquid are examined. For the waves whose propagation is described by the Korteweg-de Vries equation (in the absence of a magnetic field), equations are derived. In addition to nonlinearity and dispersion, these equations include the influence of applied magnetic fields. As an example, the effect of magnetic damping on a solitary wave is presented.     

Transient waves produced by flow past a bump

Inviscid, incompressible, irrotational transient free surface flow past a bump is considered. Asymptotic methods are used to show that for Froude numbers sufficiently close to 1, the transient flow begins by satisfying the classical linear theory. The free surface then grows with time to become nonlinear then satisfying the Korteweg-de Vries (KdV) equation with a forcing term. When solved numerically, this equation shows that solitary waves are generated at the bump and run upstream.     

On long waves in shallow water with shear flow

Various evolution equations for long interfacial waves with shear flow are derived with the assumptions that: (1) the wavelength of the interfacial waves is much larger than the total depth of the fluids, and (2) that the spanwise dependence is much weaker than the streamwise dependence. A particular case of interest occurs when the interfacial waves are at or near direct resonance. In this case the Boussinesq equation replaces the Korteweg-de Vries equation. Various special solutions in two dimensions are discussed.     

Stability of quasilongitudinally propagating solitons in a plasma with hall dispersion

By analogy with the generalization obtained in [16] for the Korteweg-de Vries equation, the derivative nonlinear Schrödinger equation is extended to the weakly non-one-dimensional case. On the basis of the equation obtained the stability of solitons propagating at small angles to the undisturbed magnetic field relative to non-one-dimensional perturbations is investigated.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 2, pp. 159–165, March–April, 1987.