Bifurcations of travelling wave solutions for a two-component camassa-holm equation

By using the method of dynamical systems to the two-component generalization of the Camassa-Holm equation, the existence of solitary wave solutions, kink and anti-kink wave solutions, and uncountably infinite many breaking wave solutions, smooth and non-smooth periodic wave solutions is obtained. Under different parametric conditions, various sufficient conditions to guarantee the existence of the above solutions are given. Some exact explicit parametric representations of travelling wave solutions are listed.     

一类二次KdV类型水波方程的行波解

应用平面动力系统理论研究了一类非线性KdV方程的行波解的动力学行为．在参数空间的不同区域内,给出了系统存在孤立波解,周期波解,扭子和反扭子波解的充分条件,并计算出所有可能的精确行波解的参数表示．     

一类非线性波方程的光滑与非光滑行波解

讨论了一类偏微分方程的行波解。该方程的行波方程对应于一个平面三次多项式系统,因而可将行波解的研究化为对平面系统所定义的相轨线的拓扑分类研究。应用平面动力系统理论在三参数空间内作定性分析,首先获得三次多项式系统的完整拓扑分类,再将相平面分析的结果返回到非线性波解u(ξ) 。考虑到解关于变量ξ=x-ct在“奇线”近旁的不连续性,可得到各种光滑与非光滑行波的存在条件。     

Bifurcation and nonsmooth dynamics of solitary waves in the generalized long-short wave equations

Generalized wave equations, which model the resonant interaction between the long wave and the short wave, are considered. To understand the underlying complex dynamics, the bifurcations and nonsmooth behaviors of solitary waves for this system are investigated by qualitative techniques in dynamical systems. These complex behaviors may serve as mechanisms for fascinating physical phenomena such as solitons, chaos and turbulence.     

多级透水板消波的一个特解

本文对无限长常水深平底渠道中一小振幅入射波经由多个间隔相等、透水性能一致的细孔透水板的反射和透射进行了研究,得到了相邻两板间距l为入射波半波长的倍数时的一个特解.结果表明,当无量纲的孔隙影响参数G₀等于透水板个数的一半时消波效果最佳,入射波能量的50%能被消掉.此时反射波与透射波的振幅相等.     

广义Drinfeld-Sokolov方程的行波解的分支

用Ansatz方法和动力系统理论研究了广义Drinfeld-Sokolov方程的行波解．在给定的两组参数条件下,得到了广义Drinfeld-Sokolov方程更多的孤立波解,扭子和反扭子波解及周期波解,并给出这些行波解精确的参数表示．     

BIFURCATIONS OF TRAVELLING WAVE SOLUTIONS TO A COUPLED NONLINEAR WAVE SYSTEM

Employ theory of bifurcations of dynamical systems to a system of coupled nonlin-ear equations, the existence of solitary wave solutions, kink wave solutions, anti-kink wave solutions and periodic wave solutions is obtained. Under different parametric conditions, various suffcient conditions to guarantee the existence of the above so-lutions are given. Some exact explicit parametric representations of travelling wave solutions are derived.     

Traveling wave solutions for Generalized Drinfeld–Sokolov equations

In this paper, solitary waves and periodic waves for Generalized Drinfeld–Sokolov equations are studied, by using the theory of dynamical systems. Bifurcation parameter sets are shown. Under given parameter conditions, explicit formulas of solitary wave, kink (anti-kink) wave and periodic wave solutions are obtained.     

Seismic wave imaging in visco-acoustic media

Realistic representation of the earth may be achieved by combining the mechanical properties of elastic solids and viscous liquids. That is to say, the amplitude will be attenuated with different frequency and the phase will be changed in the seismic data acquisition. In the seismic data processing, this effect must be compensated. In this paper, we put forward a visco-acoustic wave propagator which is of better calculating stability and     

Research on traveling wave solutions for a class of (3+1)-dimensional nonlinear equation

Nonlinear wave phenomena are of great importance in the nature, and have became for a long time a challenging research topic for both pure and applied mathematicians. In this paper the solitary wave, kink (anti-kink) wave and periodic wave solutions for a class of (3+1)-dimensional nonlinear equation were obtained by some effective methods from the dynamical systems theory.     

管道内激波的不稳定性

本文将无限大激波阵面的激波不稳定性理论[1]推广到矩形截面管道内的激波不稳定性问题.首先,给出这个问题的数学提法,包括扰动方程与三类边界条件.其次,给出扰动方程的普遍解.上游和下游的普遍解分别含有5个待定常数.再次,在一类边界条件和一个假定下,证明了激波前扰动为0,激波后两个声扰动之一为0.边界条件是,X→±∞处扰动物理量为0.假定只讨论激波不稳定性问题,从而可先设ω=iγ,γ是不稳定性增长率,为正实数.另一类边界条件是管壁上法向速度扰动为0,它使波数只能取一组离散值.最后,用扰动激波上的5个守恒方程这一边界条件来决定激波后4个待定常数和扰动激波振幅这个未知量时,导出了色散关系.结果表明,正实数γ确是存在.不稳定激波有两种模式,一种模式为γ=-W·k(W<0)它代表激波的绝对不稳定性,是新得到的模式.另一种模式与过去工作中给出的[2,3]大体相同.本文则进一步给出了这种模式的激波不稳定性增长率,并指出j2((?V/?P)_H=1+2M为最不稳定点(即无量纲化的不稳定性增长率Г=∞).如果不假定ω是纯虚数,而是复数,其虚部为正实数Im(ω)≥0.本文也严格证明了其不稳定性判据仍有两种模式,ω仍为纯虚数.     

有限水深中的极限非线性波

本文用参数优化方法.计算出有限水深中的极限Stokes波.计算的精度高,效率也高.从而进一步证明了用优化手段求stokes波动解里的待定系数是一个合理的、行之有效的方法.     

Strong solutions to stochastic wave equations with values in Riemannian manifolds

Let M be a d-dimensional compact Riemannian manifold. We prove existence of a unique global strong solution of the stochastic wave equation , where Y is a C¹-smooth transformation and W is a spatially homogeneous Wiener process on whose spectral measure has finite moments up to order 2.     

Nonlinear behavior of electromagnetic waves on surface of antiferromagnet

The properties of the nonlinear TM waves on the interface between a dielectric and an antiferromagnet are studied. The relationship between the field components of TM wave is discussed in detail, and the dispersion characteristics as well as the position of the peak field are exposed. The theoretical analysis shows that for the nonlinear TM waves there exist passband(s) and stopband(s) which can be switched into each other by varying the power. It is revealed that, in the case of , the nonlinear TM waves on the interface are backward surface waves with the group and phase velocities opposite. Project supported by the National Natural Science Foundation of China (Grant No. 69477020).     

Bifurcation of traveling wave solutions of the $K(m, n)$ equation with generalized evolution term

In this paper, by using bifurcation theory and methods of plane dynamic system, we investigate the bifurcations of the traveling wave system corresponding to the $K(m, n)$ equation with generalized evolution term. Under different parameter conditions, some exact explicit parametric representations of traveling wave solution are obtained.     

一个曲面波方程的精确解

Abstract. Some exact travelling wave solutions and rational travelling wave solutions of a sur-face wave equation in a convecting fluid are given in this paper.     

Nonlinear Superposition of Delta Waves in Multi-dimensional Space

We study the behavior of the solution u^ε to the semilinear wave equation with initial data aξ + u_i(i = 1, 2) in multidimensional space, where u_i is a classical function and aξ is smooth and converges to a distribution a_i as ε → 0. In some circumstances one can prove the convergence of u^ε, and our results express a striking superposition principle. The singular part of the solution propagates linearly. The classical part shows the nonlinear effects. And, the limit of the nonlinear solution u^ε, delta wave, as the data become more singular is the sum of the two parts.     

The bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation

By using methods from dynamical systems theory, this paper researches the bifurcation and exact travelling wave solutions for the modified Benjamin-Bona-Mahoney (mBBM) equation. Implicit exact parametric representations of all travelling wave solutions as well as some explicit analytic solutions are given. Specially, breaking wave solutions are obtained, which KdV equation does not include.     

Wave front sets of the Riemann function of elastic interface problems

We obtain an inner and an outer estimates of wave front sets and analytic wave front sets of the Riemann function of elastic interface problems by using the localization method due to Wakabayashi. In our problem the outer estimate of wave front sets and analytic wave front sets of the Riemann function coincides with the inner estimate of those. The strong point of our results is to catch the lateral wave as well as the incident, the reflected, and the refracted waves. Copyright © 2004 John Wiley & Sons, Ltd.     

Bifurcation of travelling wave solutions in a nonlinear variant of the RLW equation

By using the method of planar dynamical systems to a nonlinear variant of the regularized long-wave equation (RLW equation in short), the existence of smooth and non-smooth solitary wave (so called peakon and valleyon) and infinite many periodic wave solutions is shown. Under different regions of parametric spaces, various sufficient conditions to guarantee the existence of above solutions are given. The formulas to compute the travelling waves are also educed. We notice that some results in [Wazwaz AM. Analytic study on nonlinear variants of the RLW and the PHI-four equations. Commun Nonlinear Sci Numer Simul, in press, doi:10.1016/j.cnsns.2005.03.001] are incorrect.