Waves in an elastic tube filled with a heterogeneous fluid of variable viscosity

By treating the artery as a prestressed thin elastic tube and the blood as an incompressible heterogeneous fluid with variable viscosity, we studied the propagation of weakly non-linear waves in such a composite medium through the use of reductive perturbation method. By assuming a variable density and a variable viscosity for blood in the radial direction we obtained the perturbed Korteweg-deVries equation as the evolution equation when the viscosity is of order of ε^3/2. We observed that the perturbed character is the combined result of the viscosity and the heterogeneity of the blood. A progressive wave type of solution is presented for the evolution equation and the result is discussed. The numerical results indicate that for a certain value of the density parameter sigma, the wave equation loses its dispersive character and the evolution equation degenerates. It is further shown that, for the perturbed KdV equation both the amplitude and the wave speed decay in the time parameter τ.     

Generation and evolution of oblique solitary waves in supercritical flows

It is considered that a thin strut sits in a supercritical shallow water flow sheet over a homogeneous or very mildly varying topography. This stationary 3-D problem can be reduced from a Boussinesq-type equation into a KdV equation with a forcing term due to uneven topography, in which the transverse coordinate Y plays a same role as the time in original KdV equation. As the first example a multi-soliton wave pattern is shown by means of N-soliton solution. The second example deals with the generation of solitary wave-train by a wedge-shaped strut on an even bottom. Whitham's average method is applied to show that the shock wave jump at the wedge vertex develops to a cnoidal wave train and eventually to a solitary wavetrain. The third example is the evolution of a single oblique soliton over a periodically varying topography. The adiabatic perturbation result due to Karpman & Maslov (1978) is applied. Two coupled ordinary differential equations with periodic disturbance are obtained for the soliton amplitude and phase. Numerical solutions of these equations show chaotic patterns of this perturbed soliton.     

Korteweg-de vries-type equations in the theory of surface waves in electrically conducting liquids

Equations are obtained which describe the propagation of long waves of small, but finite amplitude in an ideal weakly conducting liquid and on the basis of these equations the influence of MHD interaction effects on the characteristics of the solitary waves is investigated. The wave equations are derived under less rigorous constraints on the external magnetic field and the MHD interaction parameter than in [1–3]. It is shown that the evolution of the free surface is described by the KdV-Burgers or KdV equations with a dissipative perturbation, and that the propagation velocity of the solitary waves depends on the strength of the external magnetic field.Translated from Izvestiya Akademii Nauk SSSR, Mekhanika Zhidkosti i Gaza, No. 6, pp. 177–180, November–December, 1989.     

Modulation theory for the steady forced KdV–Burgers equation and the construction of periodic solutions

We present a multiple-scale perturbation technique for deriving asymptotic solutions to the steady Korteweg–de Vries (KdV) equation, perturbed by external sinusoidal forcing and Burger’s damping term, which models the near resonant forcing of shallow water in a container. The first order solution in the perturbation hierarchy is the modulated cnoidal wave equation. Using the second order equation in the hierarchy, a system of differential equations is found describing the slowly varying properties of the cnoidal wave. We analyse the fixed point solutions of this system, which correspond to periodic solutions to the perturbed KdV equation. These solutions are then compared to the experimental results of Chester and Bones (1968).     

Forced dissipative Boussinesq equation for solitary waves excited by unstable topography

In the paper, the effects of topographic forcing and dissipation on solitary Rossby waves are studied. Special attention is given to solitary Rossby waves excited by unstable topography. Based on the perturbation analysis, it is shown that the nonlinear evolution equation for the wave amplitude satisfies a forced dissipative Boussinesq equation. By using the modified Jacobi elliptic function expansion method and the pseudo-spectral method, the solutions of homogeneous and inhomogeneous dissipative Boussinesq equation are obtained, respectively. With the help of these solutions, the evolutional character of Rossby waves under the influence of dissipation and unstable topography is discussed.     

Analysis of lumps,single-stripe,breather-wave,and two-wave solutions to the generalized perturbed-KdV equation by means of Hirota’s bilinear method

In this paper, we implement the Hirota’s bilinear method to extract diverse wave profiles to the generalized perturbed-KdV equation when the test function approaches are taken into consideration. Several novel solutions such as lump-soliton, lump-periodic, single-stripe soliton, breather waves, and two-wave solutions are obtained to the proposed model. We conduct some graphical analysis including 2D and 3D plots to show the physical structures of the recovery solutions. On the other hand, this work contains a correction of previous published results for a special case of the perturbed KdV. Moreover, we investigate the significance of the nonlinearity, perturbation, and dispersion parameters being acting on the propagation of the perturbed KdV. Finally, our obtained solutions are verified by inserting them into the governing equation.

     

A kind of new algebraic Rossby solitary waves generated by periodic external source

In the article, by employing multiple-scale, perturbation method, a new model is derived to describe the algebraic Rossby solitary waves generated by periodic external source in stratified fluid. The local conservation laws and analytic solutions of the model are obtained, and the breakup properties are discussed. By numeric simulation, some problems on the generation and evolution of the algebraic solitary waves under the influence of periodic external source are theoretically investigated. The results show that besides the solitary waves, an additional harmonic wave appears in the region of the external source forcing. Furthermore, the periodic variation of the external source forcing can prevent solitary waves from breaking. Meanwhile, the detuning parameter has an important effect on the breakup of the algebraic Rossby solitary waves.     

Internal solitary waves moving over the low slopes of topographies

Internal solitary waves moving over uneven bottoms are analyzed based on the reductive perturbation method, in which the amplitude, slope and horizontal lengthscale of a topography on the bottom are of the orders of , ^5/2 and ^−3/2, respectively, where the small parameter is also a measure of the wave amplitude. A free surface condition is adopted at the top of the fluid layer. That condition contains two parameters, δ and Δ, the first of which concerns the discontinuity of the basic density between the outer layer and the inner one; the second concerns the discontinuity of the mean density between them. An amplitude equation for the disturbance of order decomposes into a Korteweg-de Vries (KdV) equation and a system of algebraic equations for a stationary disturbance around a topography on the bottom. Solitary waves moving over a localized hill are studied in a simple case where both the basic flow speed and the Brunt-Vaisalla frequency are constant over the fluid layer. For this case, the expression for the amplitude of the stationary disturbance contains singular points with respect to basic flow speed. These singularities correspond to the resonant conditions modified by the free surface condition. The advancing speeds of solitary waves are changed by the influence of bottom topography, in a case where the long internal waves propagate in the direction opposite to the basic flow, but their waveforms remain almost unchanged.     

Wave number shocks for the leading tail of Korteweg-de Vries solitary waves in slowly varying media

The leading tail for slowly varying solitary waves for the perturbed Korteweg-de Vries (KdV) equation is analyzed. The path of the core of the solitary wave is obtained and shown to provide a moving boundary for the leading tail. The leading tail is predicted to be triple valued within a penumbral caustic (envelope of characteristics) caused by the initial acceleration of the core. A rescaling in the neighborhood of the singularity shows that the solution there satisfies the diffusion equation. The solution involves an incomplete Airy-type exponential integral, where critical points (significant for Laplace's asymptotic method) satisfy the structure of the penumbral caustic. A wave number shock develops, which separates two different solitary wave tails, one due to the moving core and the other due to the initial condition. The shock velocity is that predicted from conservation of waves.     

Comparisons of CIP,compact and CIP‐CSL2 schemes for reproducing internal solitary waves

Six different models were evaluated for reproducing internal solitary waves which occur and propagate in a stratified flow field with a sharp interface. Three stages were used to compute internal solitary waves in a stratified field: (1) first‐phase computation of momentum equations, (2) second‐phase computation of momentum equations, which corresponds to computing the Poisson's equation, and (3) density computation. The six models discussed in this paper consisted of combinations of four different schemes, a three‐point combined compact difference scheme (CCD), a normal central difference scheme (CDS), a cubic‐polynomial interpolation (CIP), and an exactly conservative semi‐Lagrangian scheme (CIP‐CSL2). The residual cutting method was used to solve the Poisson's equation. Three tests were used to confirm the validity of the computations using KdV theory; i.e. the incremental wave speed and amplitude of internal solitary waves, the maximum horizontal velocity and amplitude, and the wave form. In terms of the shape of an internal solitary wave, using CIP for momentum equations was found to provide better performance than CCD. These results suggest one of the most appropriate scheme for reproducing internal solitary waves may be one in which CIP is used for momentum equations and CCD to solve the Poisson's equation. Copyright © 2005 John Wiley & Sons, Ltd.     

Rossby solitary waves excited by the unstable topography in weak shear flow

A new forced KdV equation including topography is derived and the numerical solutions are given. The topographic variable should be related with the temporal and spatial function, which is called unstable topography. The physical features of the solitary waves about the mass and energy are discussed by theoretical analysis. In further studies, the pseudo-spectral numerical methods are used to discuss the evolution of solitary wave generated by the topography when meridional wave number \(m=1\); in a similar way, we analyze the solitary wave when meridional wave number \(m=2\). At last, we make the comparison for the characteristics of waves between \(m=1\) and \(m=2\), the wave of meridional number \(m=1\) plays a leading role.     

Solitons and other nonlinear waves of the Boussinesq equation

This paper studies the dynamics of shallow water waves that are governed by the Boussinesq equations. A few perturbation terms are taken into account. The ansatz method is used to carry out the perturbed Boussinesq equation. Later on, the mapping method is used to extract a few more analytical solutions. Additionally, the Weierstrass elliptic function method is also used to obtain solitary waves and singular soliton solutions. Finally, the Lie symmetry approach is used to extract a few more additional solutions.     

Head-on collision between two mKdV solitary waves in a two-layer fluid system

In this paper, based on the equations presented in [2], the head-on collision between two solitary waves described by the modified KdV equation (the mKdV equation, for short) is investigated by using the reductive perturbation method combined with the PLK method. These waves propagate at the interface of a two-fluid system, in which the density ratio of the two fluids equals the square of the depth ratio of the fluids. The second order perturbation solution is obtained. It is found that in the case of disregarding the nonuniform phase shift, the solitary waves preserve their original profiles after collision, which agrees with Fornberg and Whitham's numerical result of overtaking collision161 whereas after considering the nonuniform phase shift, the wave profiles may deform after collision.     

分层流体中内孤立波在台阶上的反射和透射

基于匹配渐近展开和格林函数的方法，研究了两层流体系统中内孤立波在台阶地形上透射、 反射及其分裂的演化特征. 通过保角变换和求解奇异Fredholm积分方程，获得了反映地形 效应对Boussinesq方程影响的约化边界条件，藉此建立了KdV演化方程的``初值'问题， 根据散射反演理论获得了反射波和透射波的解析表达式. 分析结果表明：上下流体层的厚度 比、密度比以及台阶高度对于反射和透射波振幅及其分裂具有显著的影响. 尤其当上层流体 厚度小于下层厚度时，由于存在临界点，在其附近反射波的幅值随台阶高度的演化由单调增 变为单调减，透射波的幅值由单调减变为单调增；上台阶的反射波与入射波反相，其最大幅 值可达到入射波的数倍；此外，下台阶反射波也可发展为单支孤立波，它区别于单层流体中 反射波仅为衰减的振荡波列.     

On permanent nonlinear waves in a rotating fluid

The governing equation for long nonlinear gravity waves in a rotating fluid changes with the value of the Coriolis parameter f. (1) When f is large, i.e. in the strong rotation case, in an infinite ocean, there are only Sverdrup waves; in a semi-infinite ocean or in a channel, there are either solitary Kelvin waves, for which the governing equation is a KdV equation, or Poincaré waves, which can be obtained by superposition of two Sverdrup waves. (2) When f is small, i.e. in the weak rotation case, in an infinite ocean there are solitary or cnoidal waves governed by the Ostrovskiy equation, and we provide an explicit solution for both solitary and cnoidal Ostrovskiy progressive waves; and in a semi-infinite ocean or a channel, there are Sverdrup waves, which are governed either by Ostrovskiy equations or by the Grimshaw-Melville equation. (3) When f is very small, i.e. in the very weak rotation case, in an infinite ocean, or in a channel, there are solitary waves with a horizontal crest, but with a velocity component or a pressure gradient, which are governed by KdV equations as in the non-rotating case. Physically, that means that the most determining factor is the ratio of the Rossby radius of deformation over a characteristic length of the wave.     

Instability and long-time evolution of cnoidal wave solutions for a Benney–Luke equation

We investigate numerically the stability of periodic traveling wave solutions (cnoidal waves) for a generalized Benney–Luke equation. By using a high-accurate Fourier spectral method, we find different kinds of evolution depending on the period of the perturbation. A cnoidal wave solution with period T is orbitally stable with regard to perturbations having the same period T, within certain range of wave velocities. This is a fact proved recently by Angulo and Quintero [Existence and orbital stability of cnoidal waves for a 1D boussinesq equation, International Journal of Mathematics and Mathematical Sciences (2007), in press, doi:10.1155/2007/52020] and our numerical experiments are consistent with their theory. In the present work we show numerically that cnoidal waves with period T become unstable when perturbed by small amplitude disturbances whose period is an integer multiple of T. Particularly, if the period of the perturbation is 2T, the evolution of the deviation of the solution from the orbit of the cnoidal wave is found to be approximately a time-periodic function. In other cases, the numerical experiments indicate a non-periodic behavior.     

被均匀流缓慢调制的有限水深毛细重力波

非线性的存在会产生高次谐波,这些谐波又反作用于原来的低次谐波,使波幅发生缓慢变化,从而产生缓慢调制现象．这里从考虑均匀流作用下的毛细重力水波基本方程出发,在不可压缩、无旋、无黏条件假设下,使用多重尺度分析方法推导出了在均匀流影响下有限深水毛细重力波振幅所满足的非线性Schr?dinger方程(NLSE)．分析了NLSE解的调制不稳定性．给出了毛细重力波调制不稳定的条件和钟型孤立波产生的条件．分析了无量纲最大不稳定增长率随无量纲水深和表面张力的变化趋势．同时给出了无量纲不稳定增长率随无量纲微扰动波数变化的曲线,呈现出了先增大后减小的趋势．最后指出均匀顺流减小了无量纲不稳定增长率及最大增长率,逆流增大了它们．由表面张力作用产生的毛细波及重力与表面张力共同作用产生的毛细重力波,与流的相互作用可以改变海表粗糙度和海洋上层流场结构,进而影响海气界面动量、热量及水汽的交换．了解海表这些短波动力机制,对卫星遥感的精确测量、海气相互作用的研究及海气耦合模式的改进等有重要意义．      

Symmetry and Uniqueness of the Solitary-Wave Solution for the Ostrovsky Equation

We consider herein the Ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water, or the capillary waves in a plasma with the effects of rotation. Using the modified sliding method, we prove that the solitary wave moving to the left to the Ostrovsky equation is symmetric about the origin and unique up to translations. We also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed, weak rotation, and dispersive parameter.     

Solitary waves for a nonlinear dispersive long wave equation

All the possible traveling wave solutions of Whitham-Broer-Kaup （WBK） equation are investigated in the present paper. By employing phase plane analysis, transition boundaries are derived to divide the parameter space into several regions associated with different types of phase portraits corresponding to different forms of wave solutions. All the exact expressions of bounded wave solutions are obtained as well as their existence conditions. The mechanism of bifurcation between different waves with varying Hamiltonian value has been revealed. It is pointed out that as the periods of two coexisted periodic waves tend to infinity, they may evolve to two solitary waves. Furthermore, when their trajectories pass through the common saddle point, the two solitary waves may merge into a periodic wave, and its amplitude is nearly equal to the sum of the amplitudes of the two solitary wave solutions.