Coupled scalar field equations for nonlinear wave modulations in dispersive media

A review of the generic features as well as the exact analytical solutions of a class of coupled scalar field equations governing nonlinear wave modulations in dispersive media like plasmas is presented. The equations are derivable from a Hamiltonian function which, in most cases, has the unusual property that the associated kinetic energy is not positive definite. To start with, a simplified derivation of the nonlinear Schrödinger equation for the coupling of an amplitude modulated high-frequency wave to a suitable low-frequency wave is discussed. Coupled sets of time-evolution equations like the Zakharov system, the Schrödinger-Boussinesq system and the Schrödinger-Korteweg-de Vries system are then introduced. For stationary propagation of the coupled waves, the latter two systems yield a generic system of a pair of coupled, ordinary differential equations with many free parameters. Different classes of exact analytical solutions of the generic system of equations are then reviewed. A comparison between the various sets of governing equations as well as between their exact analytical solutions is presented. Parameter regimes for the existence of different types of localized solutions are also discussed. The generic system of equations has a Hamiltonian structure, and is closely related to the well-known Hénon-Heiles system which has been extensively studied in the field of nonlinear dynamics. In fact, the associated generic Hamiltonian is identically the same as the generalized Hénon-Heiles Hamiltonian for the case of coupled waves in a magnetized plasma with negative group dispersion. When the group dispersion is positive, there exists a novel Hamiltonian which is structurally same as the generalized Hénon-Heiles Hamiltonian but with indefinite kinetic energy. The above correspondence between the two systems has been exploited to obtain the parameter regimes for the complete integrability of the coupled waves. There exists a direct one-to-one correspondence between the known integrable cases of the generic Hamiltonian and the stationary Hamiltonian flows associated with the only integrable nonlinear evolution equations (of polynomial and autonomous type) with a scale-weight of seven. The relevance of the generic system to other equations like the self-dual Yang-Mills equations, the complex Korteweg-de Vries equation and the complexified classical dynamical equations has also been discussed.     

Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics.Various nonlinear excitations of the Boussinesq equation have been found by many methods.However,it is very difcult to find interaction solutions among diferent types of nonlinear excitations.In this peper,two equivalent very simple methods,the truncated Painlev′e analysis and the generalized tanh function expansion approaches,are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

近海波-流相互作用的缓坡方程理论体系

在对近海一大类水波模型——缓坡方程的总体考察、分析和判断基础上,引进一个表征近海普遍波-流相互作用机制的算子,进而借助普适的Hamilton水波变分原理,建立了一个关联于时间的近海波-流相互作用缓坡方程理论体系,从形式到内容上达到了一种高度协调和统一.     

在不同扩散系数下反应扩散平面波的折射

在二维复金兹堡-朗道方程描述的反应扩散振荡系统中,就扩散对平面波折射率的影响进行了数值研究,从Snell定律出发导出了折射率的解析表达式,数值和理论结果表明：在纯扩散情况下,平面波的折射满足Snell折射定律,扩散只影响着平面波折射率的大小;在同时存在反应扩散情况下,只有在适当的扩散系数和系统参数下,平面波的折射才满足Snell折射定律.这些结果表明扩散系数对折射规律和折射率都有影响. 关键词： 复金兹堡-朗道方程 扩散 折射率     

Interactions Between Solitons and Cnoidal Periodic Waves of the Boussinesq Equation

The Boussinesq equation is one of important prototypic models in nonlinear physics. Various nonlinear excitations of the Boussinesq equation have been found by many methods. However, it is very difficult to find interaction solutions among different types of nonlinear excitations. In this peper, two equivalent very simple methods, the truncated Painlevé analysis and the generalized tanh function expansion approaches, are developed to find interaction solutions between solitons and any other types of Boussinesq waves.     

Evolution equation for short surface waves on water of finite depth

We address the question of determining the evolution equation for surface waves propagating in water whose depth is much larger than the typical wavelength of the surface disturbance. We avoid making the usual approximation of supposing the evolution to be given in the form of a modulated wave-packet. We treat the problem by means of a conformal transformation allowing to explicitly find the Dirichlet-to-Neumann operator for the problem together with asymptotic expansions in parameters measuring the nonlinearity and depth. This allows us to obtain an equation in physical variables valid in the weakly nonlinear, deep-water regime. The equation is an integro-differential equation, which reduces to known cases for infinite depth. We discuss solutions in a perturbative setting and show that the evolution equation describes Stokes-like waves.     

Early detection of rogue waves in a chaotic wave field

The ability to unveil growing rogue waves in the ocean is essential for safe marine travel in stormy conditions. This vital problem has not been adequately addressed so far. We show that the specific triangular spectra of rogue waves can be detected at early stages of their development in a chaotic wave field. Continuously measuring the spectra of various parts of the wave field allows us to find a rogue wave before the dangerous peak appears. This possibility of early detection is a necessary part of a rogue wave early-warning system.     

CRE Method for Solving m Kd V Equation and New Interactions Between Solitons and Cnoidal Periodic Waves

In nonlinear physics, the modified Korteweg de-Vries(m Kd V) as one of the important equation of nonlinear partial differential equations, its various solutions have been found by many methods. In this paper, the CRE method is presented for constructing new exact solutions. In addition to the new solutions of the m Kd V equation, the consistent Riccati expansion(CRE) method can unearth other equations.     

The symmetries of wave equations on new lattices

This paper focuses on studying the symmetry of a practical wave equation on new lattices. It is a new step in that the new lattice equation is applied to reduce the discrete problem of motion of an elastic thin homogeneous bar. The equation of motion of the bar can be changed into a discrete wave equation. With the new lattice equation, the translational and scaling invariant, not only is the infinitesimal transformation given, but the symmetry and Lie algebras are also calculated. We also give a new form of invariant called the ratio invariant, which can reduce the process of the computing invariant with the characteristic equation.     

蜂巢光子晶格中光波的无衍射和反常折射

空间频率模式的光子带隙反映了光波在周期性结构中的线性传输特性.以这种线性传输特性为基础,研究了蜂巢光子晶格中光波的无衍射和反常折射.通过详细分析带隙结构第一通带上的衍射与折射特性,得出了光波发生反常衍射和折射的入射条件.匹配不同的入射条件,数值模拟了光波的无衍射传输和反常折射现象.结果表明:将入射光束的波矢设置在蜂巢晶格布里渊区中正常、反常衍射区的交界处,可使高斯光束沿x轴、y轴方向的衍射得到有效抑制;以多光束干涉场作为入射光场,可对蜂巢晶格进行模式匹配,激发第二布里渊区的传输模式;进一步将模式匹配后入射光场的波矢设置在反常折射区,可实现光波的反常折射.     

双元场和光子的波动方程专题介绍

把电场和磁场组合成双元场,用来表示光子的状态,使得光子的波动方程成为一阶导数型的.该方程既具备宇称变换下的协变性,也体现出电和磁间的对称性,且保持了态函数的统计诠释.文中给出了光子的一些常用算符,并分别在p^和H^,p^z、J^z和H^以及J^2、J^z和H^同时确定的情况下求出了方程的解.不需要给态函数附加Lorentz条件,即能用该波动方程得到只存在两种横光子的结论.     

New solitary wave solutions for nonlinear evolution equations

Three important nonlinear evolution equations are solved with the aid of the symbolic manipulation system.Maple,using the direct algebraic method proposed recently,We explicitly obtain several new solutions of physical interest in addition to rederiving all the known solutions.     

Whittaker方程的场方法

用场方法来求解Whittaker方程.将一个场变量取作为其余场变量和时间的函数并对这个函数建立基本偏微方程.如能求得它的完全积分，那么Whittaker方程的解可由解代数方程来得到. 关键词： 场变量 基本偏微分方程 场方法 积分     

Third-order Stokes wave solutions for interfacial internalwaves in three-layer dendity-stratified fluid

<正>Interfacial internal waves in a three-layer density-stratified fluid are investigated using a singular perturbation method,and third-order asymptotic solutions of the velocity potentials and third-order Stokes wave solutions of the associated elevations of the interfacial waves are presented based on the small amplitude wave theory.As expected,the third-order solutions describe the third-order nonlinear modification and the third-order nonlinear interactions between the interfacial waves.The wave velocity depends on not only the wave number and the depth of each layer but also on the wave amplitude.     

Angular spectrum analysis of near-field diffraction of a small aperture

The near-field diffraction of a nanometer aperture is studied in this paper according to the angular spectrum theory. The mathematic expressions of the near-field diffraction are presented, and the influences of the polarization of the electromagnetic field and the evanescent wave on the diffraction in near field are analyzed by means of vector and scalar diffraction theory, respectively. The transverse and longitudinal diffraction intensity distributions of a small aperture are numerically calculated, and the results show the near-field diffraction of a small aperture is influenced strongly by the vector property of the electromagnetic field and the evanescent component.     

磁控溅射制备ZnO薄膜及其声表面波特性

通过射频磁控溅射技术在GaAs，Au/GaAs，Si和玻璃基片上成功制备了ZnO多晶薄膜，利用X射线衍射对ZnO薄膜的取向、结晶性进行了表征，结果表明ZnO薄膜呈完全c轴取向，Au缓冲层可以有效地改善ZnO薄膜的晶体质量，X射线摇摆曲线结果表明ZnO(002)衍射峰的半高宽仅为2.41°，同时发现Au缓冲层的结晶质量对ZnO薄膜的c轴取向度有很大影响，通过扫描电子显微镜对ZnO/GaAs和ZnO/Au/GaAs薄膜的表面形貌进行了观测，利用网络分析仪对IDT/ZnO/GaAs薄膜的声表面波特性进行了测量. 关键词： ZnO薄膜 X射线衍射 声表面波     

三层密度分层流体毛细重力波二阶Stokes波解

以小振幅波理论为基础,利用摄动方法研究了三层密度分层流体的毛细重力波,给出了三层成层状态下各层流体速度势的二阶渐近解及毛细重力波波面位移的二阶Stokes波解.结果表明:一阶解及二阶解除了依赖于各层流体的厚度及密度,与表面张力也有很重要的关系.     

关于转动弦的波动方程

数学物理方法的一些学习指导书或教科书上给出了两种转动弦的波动方程,但没有考虑Coriolis力的影响,其讨论方法存在不正确(竖直转动重弦)或不全面(水平转动轻弦)之处.重新讨论了两种转动弦的波动方程及其求解问题.     

Generalization of the Brewster phenomenon for the wave diffraction process

Polarization anomalies of the Brewster type are considered for processes such as wave reflection, wave scattering, wave diffraction by surface grating and surface plasmon waves. Generalized conditions for Brewster anomalies in wave diffraction are written in analytical form. The application of the Brewster phenomenon for integrated optical polarizers is investigated numerically.