Exact solitary wave solutions of nonlinear wave equations

The hyperbolic function method for nonlinear wave equations is presented. In support of a computer algebra system, many exact solitary wave solutions of a class of nonlinear wave equations are obtained via the method. The method is based on the fact that the solitary wave solutions are essentially of a localized nature. Writing the solitary wave solutions of a nonlinear wave equation as the polynomials of hyperbolic functions, the nonlinear wave equation can be changed into a nonlinear system of algebraic equations. The system can be solved via Wu Elimination or Gr?bner base method. The exact solitary wave solutions of the nonlinear wave equation are obtained including many new exact solitary wave solutions.     

Wave Surface Identification Based on Stereo Vision and Wave Theory: an Initial Attempt北大核心CSCD

波高是波浪信息最基本的元素,对波高的精确测量无论是对波浪理论的研究还是数值方法的拓展,都起着指导和验证的作用。文中基于双目立体视觉原理自主搭建了波面光学测量系统,突破了传统测量设备如浪高仪等单点测量的局限性,并将波浪理论融入到数据后处理方法中,对常用的单纯依赖图像的光学测量方法进行了改进。通过在拖曳水池中对单向规则波瞬时波面的识别和重构,并将结果与浪高仪以及理论来波参数进行了对比验证,结果表明该测量系统在大范围波面的测量中误差在1%左右,最后对其在非规则的来波下进行了初步尝试。     

Transversal wave maps and transversal exponential wave maps

Transversal wave maps and wave maps are different. There are wave maps which are not transversal wave maps, and vice versa. However, if f is a wave map under certain circumstance, then f is a transversal wave map. We show that if f is a transversal exponential wave map, then the associated energy–momentum is transversally conserved. We finally obtain the relationship among transversal wave maps, transversal exponential wave maps and certain second order symmetric tensors.     

An optimal controller of an irregular wave maker

Laboratory seakeeping test under irregular wave load is essential for studying seakeeping performances of marine and offshore structures. One of the key techniques in such a sort of test is how to assure the quality of wave load. By assuring the quality of wave load, we mean that man-made irregular wave series exerting on a test model should have a given power spectrum. From a view of optimal control, the goal to assure the quality of wave load is to manage wave maker such that produced irregular wave series exerting on a test model has a given power spectrum in the optimization sense. This paper gives a control model to optimize wave maker to achieve the mentioned goal. Its application to wave maker control in a seakeeping model test of an offshore structure is demonstrated.     

Convergence of a finite difference method for combustion model problems

We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some con-     

Rogue waves in baroclinic flows

We investigate an AB system, which can be used to describe marginally unstable baroclinic wave packets in a geophysical fluid. Using the generalized Darboux transformation, we obtain higher-order rogue wave solutions and analyze rogue wave propagation and interaction. We obtain bright rogue waves with one and two peaks. For the wave packet amplitude and the mean-flow correction resulting from the self-rectification of the nonlinear wave, the positions and values of the wave crests and troughs are expressed in terms of a parameter describing the state of the basic flow, in terms of a parameter responsible for the interaction of the wave packet and the mean flow, and in terms of the group velocity. We show that the interaction of the wave packet and mean flow and also the group velocity affect the propagation and interaction of the amplitude of the wave packet and the self-rectification of the nonlinear wave.     

血液粘弹性对动脉中脉搏波传播的影响

以Womersley理论为基础,将动脉中脉搏波复波速所满足的频率方程推广到粘弹血液情形,得到了线性粘弹血管和线性粘弹血液情况下脉搏波复波速的一般关系式。应用此关系式,讨论了血液粘弹性对动脉中脉搏波波速和波的衰减的影响。结果表明,血液弹性对脉搏波传播的影响小于血管粘性的影响;对大动脉,血液弹性对脉搏波传播的影响可忽略。     

The configuration of shock wave reflection for the TSD equation

In this paper, we mainly study the nonlinear wave configuration caused by shock wave reflection for the TSD (Transonic Small Disturbance) equation and specify the existence and nonexistence of various nonlinear wave configurations. We give a condition under which the solution of the RR (Regular reflection) for the TSD equation exists. We also prove that there exists no wave configuration of shock wave reflection for the TSD equation which consists of three or four shock waves. In phase space, we prove that the TSD equation has an IR (Irregular reflection) configuration containing a centered simple wave. Furthermore, we also prove the stability of RR configuration and the wave configuration containing a centered simple wave by solving a free boundary value problem of the TSD equation.     

强非线性广义 Boussinesq 方程孤波解的波形分析及求解

上海理工大学理学院\quad 上海 200093该文建立了强非线性广义 Boussinesq 方程的耗散项、波速、渐进值与波形函数的导数之间的关系.利用适当变换和待定假设方法,作者求出了上述广义 Boussinesq 方程的扭状或钟状孤波解,还求出了以前文献中未曾提到过的余弦函数的周期波解.进一步给出了波速对波形影响的结论,即:``好'广义 Boussinesq 方程的行波当波速由小变大时,波形由钟状孤波变成余弦函数周期波解;``坏'广义 Boussinesq 方程的行波当波速由小变大时,波形由余弦函数周期波解变成钟状孤波.     

Convergence of a finite difference method for combustion model problems

We study a finite difference scheme for a combustion model problem. A projection scheme near the combustion wave, and the standard upwind finite difference scheme away from the combustion wave are applied. Convergence to weak solutions with a combustion wave is proved under the normal Courant-Friedrichs-Lewy condition. Some conditions on the ignition temperature are given to guarantee the solution containing a strong detonation wave or a weak detonation wave. Convergence to strong detonation wave solutions for the random projection method is also proved.     

Separation of internal and interaction dynamics for NLS-described wave packets with different carrier waves

We give a detailed analysis of the interaction of two NLS-described wave packets with different carrier waves for a nonlinear wave equation. By separating the internal dynamics of each wave packet from the dynamics caused by the interaction we prove that there is almost no interaction of such wave packets. We also prove the validity of a formula for the envelope shift caused by the interaction of the wave packets.     

Travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation

The travelling wave solutions of a generalized Camassa-Holm-Degasperis-Procesi equation ut-uxxt + (1 + b)umux = buxuxx + uuxxx are considered where b > 1 and m are positive integers. The qualitative analysis methods of planar autonomous systems yield its phase portraits. Its soliton wave solutions, kink or antikink wave solutions, peakon wave solutions, compacton wave solutions, periodic wave solutions and periodic cusp wave solutions are obtained. Some numerical simulations of these solutions are also give...     

Third order approximation to capillary gravity short crested waves with uniform currents

A third-order analytical solution for the capillary gravity short crested waves with uniform current (the main current direction is along the vertical wall) in front of a vertical wall is derived through a perturbation expanding technique. The validity and advantage of the new solution were proved by comparing the results of wave profiles and wave pressures with those of Huang and Jia [H. Huang, F. Jia, The patterns of surface capillary gravity short-crested waves with uniform current fields in coastal waters, Acta Mech. Sinica 22 (2006) 433–441] and Hsu [J.R.C. Hsu, Y. Tsuchiya, R. Silvester, Third-order approximation to short-crested waves, J. Fluid Mech 90 (1979) 179–196]. The important influences of currents on the wave profiles, wave frequency, the ratio of maximum crest height to the total wave height, and wave pressure are investigated for both small-scale (for example, the incident wave wavelength is 9.35 cm) and larger-scale (for example, the incident wave wavelength is 5 m) short crested wave. By numerical computation, we find wave frequency of short crested wave system is greatly affected by incident wave amplitude, incident angle, water depth, surface tension coefficient and the strength of the currents for small-scale incident wave. Furthermore, for the larger-scale short crested wave system, the higher-order solution with uniform current is particularly important for the prediction of wave profile and wave pressure for different water depth and incident angle. Computational results show that with the increase of the current speed, the crests of wave profile and wave pressure become more and more steep. In some cases, the crest value of wave pressure with strong current would be larger about six times than that of no current. Therefore, ocean engineers should consider the short crested wave-current load on marine constructs carefully.     

Coexistence of the solitary and periodic waves in convecting shallow water fluid

The existence of a solitary wave for the shallow water model in convecting circumstance was established in previous works. It is still unknown that whether there exist periodic waves. In this paper, we prove that the models possess periodic waves with a fixed range of wave speed. The amplitude and wave speed are explicitly given. Moreover, the coexistence of the solitary wave and one periodic wave is established.     

Exact traveling wave solutions and bifurcations in a nonlinear elastic rod equation

The exact parametric representations of the traveling wave solutions for a nonlinear elastic rod equation are considered. By using the method of planar dynamical systems, in different parameter regions, the phase portraits of the corresponding traveling wave system are given. Exact explicit kink wave solutions, periodic wave solutions and some unbounded wave solutions are obtained.     

Uniqueness and stability of traveling waves for periodic monostable lattice dynamical system

We study the traveling waves for a lattice dynamical system with monostable nonlinearity in periodic media. It is well known that there exists a minimal wave speed such that a traveling wave exists if and only if the wave speed is above this minimal wave speed. In this paper, we first derive a stability theorem for certain waves of non-minimal speed. Moreover, we show that wave profiles of a given speed are unique up to translations.     

Simulating wave transmission in the lee of a breakwater in spectral models due to overtopping

Spectral wave models have experienced constant development and vast improvements over the past decades. They are constantly being extended and refined in order to cover the complex wave transformation processes that take place in the coastal zone. Nevertheless, wave transmission due to overtopping has not been treated similarly yet. In this paper, a methodology to include wave generation due to overtopping in spectral wave models is presented. Incorporation of overtopping aims at better simulating the wave disturbance in the lee side of a system of offshore breakwaters and the induced hydrodynamic processes. So far, the waves generated due to wave overtopping were being neglected. The methodology consists of executing sequential simulations at small time step intervals and whenever wave overtopping occurs in a breakwater, waves are generated and transmitted in the lee side of the structure. This is achieved by modifying the boundary condition at the lee of a coastal structure to account for wave generation due to overtopping. Additionally, the transmitted spectrum source function was modified, to capture the observed transfer of energy in the higher frequencies of the spectrum due to the aforementioned overtopping process. The above methodology was implemented in the open source wave model TOMAWAC and verification with experimental measurements was carried out yielding satisfactory results. Inclusion of wave transmission due to overtopping in spectral wave models is considered to be a valuable asset, especially for the simulation of inshore hydrodynamic processes.     

The singular solutions to a nonsymmetric system of Keyfitz-Kranzer type with initial data of Riemann type

The solutions to the Riemann problem for a nonsymmetric system of Keyfitz-Kranzer type are constructed explicitly when the initial data are located in the quarter phase plane. In particular, some singular hyperbolic waves are discovered when one of the Riemann initial data is located on the boundary of the quarter phase plane, such as the delta shock wave and some composite waves in which the contact discontinuity coincides with the shock wave or the wave back of rarefaction wave. The double Riemann problem for this system with three piecewise constant states is also considered when the delta shock wave is involved. Furthermore, the global solutions to the double Riemann problem are constructed through studying the interaction between the delta shock wave and the other elementary waves by using the method of characteristics. Some interesting nonlinear phenomena are discovered during the process of constructing solutions; for example, a delta shock wave is decomposed into a delta contact discontinuity and a shock wave.     

Symbolic computation of exact solutions for a nonlinear evolution equation

In this paper, by means of the Jacobi elliptic function method, exact double periodic wave solutions and solitary wave solutions of a nonlinear evolution equation are presented. It can be shown that not only the obtained solitary wave solutions have the property of loop-shaped, cusp-shaped and hump-shaped for different values of parameters, but also different types of double periodic wave solutions are possible, namely periodic loop-shaped wave solutions, periodic hump-shaped wave solutions or periodic cusp-shaped wave solutions. Furthermore, periodic loop-shaped wave solutions will be degenerated to loop-shaped solitary wave solutions for the same values of parameters. So do cusp-shaped solutions and hump-shaped solutions. All these solutions are new and first reported here.     

Qualitative Analysis and Travelling Wave Solutions for the Generalized Pochhammer–Chree Equation with a Dissipation Term

The purpose of this paper is to reveal the influence of dissipation on travelling wave solutions of the generalized Pochhammer–Chree equation with a dissipation term, and provides travelling wave solutions for this equation. Applying the theory of planar dynamical systems, we obtain ten global phase portraits of the dynamic system corresponding to this equation under various parameter conditions. Moreover, we present the relations between the properties of travelling wave solutions and the dissipation coefficient r of this equation. We find that a bounded travelling wave solution appears as a bell profile solitary wave solution or a periodic travelling wave solution when r= 0; a bounded travelling wave solution appears as a kink profile solitary wave solution when |r| > 0 is large; a bounded travelling wave solution appears as a damped oscillatory solution when |r| > 0 is small. Further, by using undetermined coefficient method, we get all possible bell profile solitary wave solutions and approximate damped oscillatory solutions for this equation. Error estimates indicate that the approximate solutions are meaningful.