A study of solitary waves by He's semi-inverse variational principle

The dynamics of solitary waves in the presence of perturbation terms is studied in this paper with the aid of the semi-inverse variational principle. In this paper, shallow water waves as well as internal gravity waves in a density-stratified ocean are considered. These are respectively modeled by the Korteweg–de Vries equation as well as the compound Korteweg–de Vries equation. An analytical solution of the solitary wave is found in each case.     

Time behaviour of the error when simulating finite-band periodic waves. The case of the KdV equation

This paper is devoted to study the error growth of numerical time integrators for N-phase or N-band quasi-periodic (in time) solutions of the periodic Korteweg–de Vries equation. It is shown that the preservation, through numerical time integration, of conserved quantities of the periodic problem of the equation, may be an element to take into account in the selection of the numerical method. We explain why the inclusion of these properties of conservation provides a better error propagation. In particular, we emphasize how the preservation of invariants makes influence in the simulation of some physical parameters of the waves.     

Solitary dynamo waves

Long dynamo waves are a characteristic feature of interface dynamo models with spatially localized α and Ω effects. The evolution of such waves is described by the modified Korteweg–de Vries equation. Solutions to this equation take the form of solitary waves, breathers, and snoidal and cnoidal waves, and represent nonlinear waves of magnetic activity that migrate towards the equator, as observed on the Sun. Averaging techniques extend the theory to longer times and relate the amplitude of these waves to the dynamo number.     

Radially ingoing and outgoing dispersive and dissipative structures in electron‐ion plasmas in the presence of electron trapping

In this article, non‐linear propagation of ingoing and outgoing electrostatic waves on the ion time scale in an unmagnetized, non‐relativistic electron‐ion (ei) plasma in the presence of warm ions, ion kinematic viscosity, and trapped Maxwellian electrons was examined in a non‐planar geometry. In the weak non‐linearity limit, modified soliton and shock equations were derived with the inclusion of electron trapping in cylindrical and spherical geometries. The finite difference method was used to solve all these equations in the non‐planar geometries using the planar versions of these equations as an initial input. The results were compared with their counterparts with quadratic non‐linearity and the main differences were expounded. It was shown that the spatio‐temporal scales over which the shocks form for the non‐planar trapped Burgers equation are much shorter by comparison with the shocks admitted by the non‐planar trapped Korteweg de Vries Burgers equation. It was also found that unlike their non‐linear shock counterparts, the solitary structures admitted by the non‐planar trapped Korteweg de Vries equation exhibit a phase shift.     

Periodic and chaotic behaviour in a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations

The dynamical behaviour of a reduced form of the perturbed generalized Korteweg–de Vries and Kadomtsev–Petviashvili equations (extension of the Korteweg–de Vries equation to two space variables) are studied in this paper. Harmonic solutions of non-resonance and primary resonance are obtained using the perturbation method. Chaotic motion under harmonic excitations is studied using the Melnikov method.A wide range of solutions for the reduced perturbed generalized Korteweg–de Vries equations, in which non-linear phenomena appearing within transition from regular harmonic response (periodic solutions) to chaotic motion, are obtained using the time integration Runge–Kutta method. When chaos is found, it is detected by examining the phase plane, the Poincaré map, the sensitivity solution of the solution to initial conditions, and by calculating the largest Lyapunov exponent.     

Magnetosonic shock waves in dense astrophysical electron–positron–ion plasmas

Multifluid quantum magnetohydrodynamic model (QMHD) is used to investigate small but finite amplitude magnetosonic shock waves in dense) electron–positron–ion (e–p–i) plasmas. The Korteweg–de Vries–Burgers (KdVB) equation is derived by using reductive perturbation method. It is noticed that variations in the positron density modify the profile of magnetosonic shocks in dense e–p–i plasmas significantly. The numerical results are also presented by taking into account the dense plasma parameters from published literature of astrophysical conditions, in compact stars.     

Nonlinear structures for extended Korteweg–de Vries equation in multicomponent plasma

Using the fluid hydrodynamic equations of positive and negative ions, as well as q-nonextensive electron density distribution, an extended Korteweg–de Vries (EKdV) equation describing a small but finite amplitude dust ion-acoustic waves (DIAWs) is derived. Extended homogeneous balance method is used to obtain a new class of solutions of the EKdV equation. The effects of different physical parameters on the propagating nonlinear structures and their relevance to particle acceleration in space plasma are reported.     

采用混合模型数值模拟从深海到浅海内波的传播

在南海东沙岛附近, 从MODIS遥感图像发现内波传播是从深海经陆架坡再到浅海, 由于深海和浅海环境条件的差异以及传播模型的适用条件不同, 因此 不能采用同一模型模拟内波的传播, 需用两种模型来分别模拟内波在深海和浅海中的传播. 采用差分法, 首先用非线性薛定谔方程模拟了深海内波的传播, 然后用EKdV方程模拟了内波在浅海中的继续传播. 模拟结果与实际的MODIS遥感内波图像相符合, 并与应用单一模型模拟结果相比, 混合模型模拟该海区的内波传播更接近遥感实测, 表明了混合模型的合理性.     

Analysis of the stability and density waves for traffic flow

In this paper, the optimal velocity model of traffic is extended to take into account the relative velocity. The stability and density waves for traffic flow are investigated analytically with the perturbation method. The stability criterion is derived by the linear stability analysis. It is shown that the triangular shock wave, soliton wave and kink wave appear respectively in our model for density waves in the three regions: stable, metastable and unstable regions. These correspond to the solutions of the Burgers equation, Korteweg－de Vries equation and modified Korteweg－de Vries equation. The analytical results are confirmed to be in good agreement with those of numerical simulation. All the results indicate that the interaction of a car with relative velocity can affect the stability of the traffic flow and raise critical density.     

Wave Breaking in a Class of Nonlocal Dispersive Wave Equations

Abstract

The Korteweg de Vries (KdV) equation is well known as an approximation model for small amplitude and long waves in di!erent physical contexts, but wave breaking phenomena related to short wavelengths are not captured in. In this work we consider a class of nonlocal dispersive wave equations which also incorporate physics of short wavelength scales. The model is identified by a renormalization of an infinite dispersive di!erential operator, followed by further specifications in terms of conservation laws associated with the underlying equation. Several well-known models are thus rediscovered. Wave breaking criteria are obtained for several models including the Burgers-Poisson system, the Camassa-Holm type equation and an Euler-Poisson system. The wave breaking criteria for these models are shown to depend only on the negativity of the initial velocity slope relative to other global quantities.     

Nonlinear propagation of ultra-low-frequency electromagnetic modes in a magnetized dusty plasma

A theoretical investigation has been made of nonlinear propagation of ultra-low-frequency electromagnetic waves in a magnetized two fluid (negatively charged dust and positively charged ion fluids) dusty plasma. These are modified Alfvén waves for small value of and are modified magnetosonic waves for large , where is the angle between the directions of the external magnetic field and the wave propagation. A nonlinear evolution equation for the wave magnetic field, which is known as Korteweg de Vries (K-dV) equation and which admits a stationary solitary wave solution, is derived by the reductive perturbation method. The effects of external magnetic field and dust characteristics on the amplitude and the width of these solitary structures are examined. The implications of these results to some space and astrophysical plasma systems, especially to planetary ring-systems, are briefly mentioned. Received 8 July 1999 and Received in final form 11 October 1999     

Some super systems with local bi-Hamiltonian operators

For a class of super skew-symmetric operators with eight parameters, two kinds of super Hamiltonian operators are identified by the method of functional multi-vectors. Appropriate decompositions of these operators lead to compatible super Hamiltonian pairs which in turn produce nonlinear super bi-Hamiltonian systems from the trivial x-translation flow. As examples, besides the super Korteweg–de Vries equations and other known ones, new super generalizations are obtained for the Riemann equation, the Hunter–Saxton equation and the Camassa–Holm equation, all of them admit two compatible local super Hamiltonian operators.     

Nonlinear Waves in an Inhomogeneous Fluid Filled Elastic Tube

In a thin-walled, homogeneous, straight, long, circular, and incompressible fluid filled elastic tube, small but finite long wavelength nonlinear waves can be describe by a KdV (Korteweg de Vries) equation, while the carrier wave modulations are described by a nonlinear Schrödinger equation (NLSE). However if the elastic tube is slowly inhomogeneous, then it is found, in this paper, that the carrier wave modulations are described by an NLSE-like equation. There are soliton-like solutions for them, but the stability and instability regions for this soliton-like waves will change, depending on what kind of inhomogeneity the tube has.     

Ion-acoustic shock waves in multi-electron temperature collisional plasma

The nonlinear propagation of ion-acoustic waves in a collision-dominated double electron temperature plasma is considered. Accounting for the ion viscosity and the ion heat conductivity, it is shown by means of two-warm fluid equations that the nonlinear evolution of the ion-acoustic waves is governed by the Korteweg—de Vries—Burgers equation. Stationary shock solution of the KdV—Burgers equation is presented.     

Dissipative ion‐acoustic waves in collisional electron‐positron‐ion plasmas with Kappa distribution

In this work, linear and non‐linear structures of ion‐acoustic waves (IAWs) are investigated in a collisional plasma consisting of warm ions, superthermal electrons, and positrons. A dissipative effect is assumed due to ion‐neutral collisions. The linear properties of IAWs are investigated. It is shown that the dynamics of the IAWs is governed by the damped Korteweg‐de Vries (K‐dV) equation. It is seen that the ion‐neutral collisions modify the basic features of ion‐acoustic solitary waves significantly. Also, the effect of the plasma parameters on the dissipative IAWs is discussed in detail.     

BERNOULLI NUMBERS AND SOLITONS — REVISITED

In the present paper we propose a new proof of the Grosset–Veselov formula connecting one-soliton solution of the Korteweg–de Vries equation to the Bernoulli numbers. The approach involves Eulerian numbers and Riccati's differential equation.     

Helical solitons in vector modified Korteweg–de Vries equations

We study existence of helical solitons in the vector modified Korteweg–de Vries (mKdV) equations, one of which is integrable, whereas another one is non-integrable. The latter one describes nonlinear waves in various physical systems, including plasma and chains of particles connected by elastic springs. By using the dynamical system methods such as the blow-up near singular points and the construction of invariant manifolds, we construct helical solitons by the efficient shooting method. The helical solitons arise as the result of co-dimension one bifurcation and exist along a curve in the velocity-frequency parameter plane. Examples of helical solitons are constructed numerically for the non-integrable equation and compared with exact solutions in the integrable vector mKdV equation. The stability of helical solitons with respect to small perturbations is confirmed by direct numerical simulations.     

New exact solutions of Burgers’ type equations with conformable derivative

In this paper, the new exact solutions for some nonlinear partial differential equations are obtained within the newly established conformable derivative. We use the first integral method to establish the exact solutions for time-fractional Burgers’ equation, modified Burgers’ equation, and Burgers–Korteweg–de Vries equation. We report that this method is efficient and it can be successfully used to obtain new analytical solutions of nonlinear FDEs.     

Some features of one-soliton solutions of the Korteweg - de Vries equation

Using the method of inverse scattering problem [1, 2], we study solutions of the Korteweg - de Vries equation under initial conditions in the form of two nonsoliton pulses with not very large amplitudes. It is shown that if the distance between these pulses is not large, then they evolve to one soliton and an oscillating nonlinear tail for t → ∞. As the distance between the pulses or the pulse amplitudes increase, two solitons and an oscillating nonlinear tail are formed. Similar behavior is observed for solutions of the nonlinear Schrödinger equation. The only difference is that three, but not two, solitons are formed if the distance between two initial inphase pulses increases. The results of analytical consideration are illustrated by the numerical solution of the Korteweg - de Vries equation.     

Positons of the modified Korteweg de Vries equation

The concept of positons, i.e. certain multiparametric solutions of the Korteweg de Vries equation with new properties, is extended to the modified Korteweg de Vries equation. It is shown that the essential features of positons carry over to this case; the collision of positons, the solitary-wave-positon interaction and simple generalizations are discussed in detail. Suggestions for future research and possible applications of the present work are sketched.