浅水中度振幅孤立波解的分支

利用平面动力系统分支方法研究浅水中度振幅方程的定性行为和孤立波解.给出了系统在不同参数条件下的相图.获得了光滑孤立波、cuspon解和周期波解的隐式表达式.对方程的光滑孤立波解、cuspon解和周期波解进行了数值模拟.获得的结果完善了相关文献已有的结果.     

一类描述大气中重力孤立波的新模型

从两层流体浅水波方程出发,运用尺度分析与扰动方法,建立了一类新的模型(mKdV-BO模型)来描述大气中的重力孤立波。前人建立的KdV模型和BO模型适合描述经向和纬向扰动较弱时重力孤立波的生成和演化,而该模型的非线性更强,适合描述经向、纬向扰动较强时重力孤立波的生成与演化。通过运用试探函数法获得了模型的代数孤波解,并分析了孤立波的生成条件与传播速度。新模型的建立对于进一步解释大气中列队雷雨阵的形成机制,探讨大气中的强对流天气如飑线的形成等具有重要意义。 关键词：重力孤立波;试探函数法;列队雷雨阵     

RLW方程的一个新的显式多辛格式

以多辛Euler-box格式为基础对正则长波(RLW)方程的初边值问题进行了讨论,推导了一个新的显式10点格式.模拟孤立波的数值实验表明,这个新的多辛格式是行之有效的,能很好的反映出RLW方程的非弹性性质.     

表面张力对非传播孤立波的影响

本文在研究非传播弧立波时仔细考虑了表面张力的影响,把表面张力和液体深度的参数平面划分为三个区域,发现其中两个区可产生呼吸弧立波。到目前为止,所有理论和实验文章中提到的呼吸弧立波的参数都在一个参数区内,我们首先报道了另一个参数区并被我们的实验证实.在第三个参数区中,理论分析得到的解是纽结孤立波,但是在我们的实验中除了得到纽结孤立波之外,过得到了一种类似于呼吸孤立波的非传播孤立波.     

一个两流体系统中mKdV孤立波的迎撞*

本文从文[2]的基本方程出发,采用约化摄动方法和PLK方法,讨论了三阶非线性和色散效应相平衡的修正的KdV(mKdV)孤立波迎撞问题.这些波在流体密度比等于流体深度比平方的两流体系统界面上传播.我们求得了二阶摄动解,发现在不考虑非均匀相移的情况下,碰撞后孤立波保持原有的形状,这与Fornberg和whitham[6]的追撞数值分析结果一致,但当考虑波的非均匀相移后,碰撞后波形将变化.     

Degasperis-Procesi方程的孤立尖波解

利用动力系统的定性分析理论对D egasperis-P rocesi方程的孤立尖波解进行了研究.给出了D e-gasperis-P rocesi方程对应行波系统的相图分支,利用相图获得了孤立尖波解和周期尖波解的解析表达式,通过数值模拟给出了部分解的图像.     

参数激励下的内波孤立子和扭结 *

在不相容、不等密度的双层液体的参数激励下的Faraday实验中,观察到了非传播界面波孤立子和扭结以及稳定的双孤立子等现象.总体上看,界面波现象和已有的表面波现象基本上一致,只是由于上层液体的存在,使得界面波的模式振动频率明显红移、波形变矮变宽、其稳定性也不如表面波.在理论上,对流体力学方程组及其相应的边界和界面条件进行了约化,得到了一个有阻尼、带驱动的非线性Schr(?)dinger方程.从而,令人满意地同时解释了界面孤立子波和扭结波.无论从实验现象上,还是从理论上看,自由表面波只是界面波的一个特例.     

CH-γ方程的两类新有界波

用动力系统分支方法和微分方程数值方法研究CH-γ方程.发现了两类新的有界波, 一类称为紧孤立子, 另一类称为广义扭波.文中模拟了它们的平面图形, 并给出了其隐函数表达式, 最后展示了理论推导和数值模拟的一致性.     

非线性波方程的精确孤立波解

立了一种求解非线性波方程精确孤立波解的双曲函数方法,并在计算机代数系统上加以实现,推导出了一大批非线性波方程的精确孤立波解.方法的基本原理是利用非线性波方程孤立波解的局部性特点,将方程的孤立波解表示为双曲函数的多项式,从而将非线性波方程的求解问题转化为非线性代数方程组的求解问题.利用吴消元法或Gröbner基方法在计算机代数系统上求解非线性代数方程组, 最终获得非线性波方程的精确孤立波解,其中有很多新的精确孤立波解.     

一类变式Boussinesq系统的行波解

本文研究—类变式Boussinesq系统ηt+((1+αη)w)x-β/6wxxx=0, wt+αwwx+ηx-β/2wxxt=0,其中α和β都是正常数.许多逼近模型都能从此系统中被推导出,比如Boussinesq系统和两分量Camassa-Holm系统等.本文利用平面动力系统方法研究它的行波解及相图,得到了孤立波解,广义扭波解,广义反扭波解,紧孤立波解和周期波解,并给出了这些解的数值模拟.     

A Petrov-Galerkin method for equal width equation

The equal width equation is solved numerically by Petrov-Galerkin method using linear hat function and quadratic B-spline function as trial and test functions respectively. Product approximation has been used in this method. A linear stability analysis of the scheme is shown to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. The Maxwellian initial condition pulse and the development of an undular bore are also studied.     

A Petrov-Galerkin method for solving the generalized equal width (GEW) equation

The generalized equal width (GEW) equation is solved numerically by the Petrov-Galerkin method using a linear hat function as the test function and a quadratic B-spline function as the trial function. Product approximation has been used in this method. A linear stability analysis of the scheme shows it to be conditionally stable. Test problems including the single soliton and the interaction of solitons are used to validate the suggested method, which is found to be accurate and efficient. Finally, the Maxwellian initial condition pulse is studied.     

Soliton solutions for a generalized KdV and BBM equations with time-dependent coefficients

A generalized KdV equation with time-dependent coefficients will be studied. The BBM equation with time-dependent coefficients and linear damping term will also be examined. The wave soliton ansatz will be used to obtain soliton solutions for both equations. The conditions of existence of solitons are presented.     

Numerical solution of a coupled Korteweg–de Vries equations by collocation method

A numerical method for solving the coupled Korteweg‐de Vries (CKdV) equation based on the collocation method with quintic B‐spline finite elements is set up to simulate the solution of CKdV equation. Invariants and error norms are studied wherever possible to determine the conservation properties of the algorithm. Simulation of single soliton, interaction of two solitons, and birth of solitons are presented. A linear stability analysis shows the scheme to be unconditionally stable. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

Stabilization Mechanism for Two-Dimensional Solitons in Nonlinear Parametric Resonance

We consider a simple model system supporting stable solitons in two dimensions. The system is the parametrically driven damped nonlinear Schrodinger equation, and the soliton stabilizes for sufficiently strong damping. We elucidate the stabilization mechanism by reducing the partial differential equation to a finite-dimensional dynamical system and conclude that the negative feedback loop occurs via enslaving the soliton phase, locked to the driver, to its amplitude and width. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 144, No. 2, pp. 226–233, August, 2005.     

Relaxation of solitons in nonlinear Schrödinger equations with potential

In this paper we study dynamics of solitons in the generalized nonlinear Schrödinger equation (NLS) with an external potential in all dimensions except for 2. For a certain class of nonlinearities such an equation has solutions which are periodic in time and exponentially decaying in space, centered near different critical points of the potential. We call those solutions which are centered near the minima of the potential and which minimize energy restricted to L²-unit sphere, trapped solitons or just solitons. In this paper we prove, under certain conditions on the potentials and initial conditions, that trapped solitons are asymptotically stable. Moreover, if an initial condition is close to a trapped soliton then the solution looks like a moving soliton relaxing to its equilibrium position. The dynamical law of motion of the soliton (i.e. effective equations of motion for the soliton's center and momentum) is close to Newton's equation but with a dissipative term due to radiation of the energy to infinity.     

Exact soliton solutions of a D-dimensional nonlinear Schr?dinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schr?dinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

Exact soliton solutions of a D-dimensional nonlinear Schrödinger equation with damping and diffusive terms

In the present study, we apply function transformation methods to the D-dimensional nonlinear Schrödinger (NLS) equation with damping and diffusive terms. As special cases, this method applies to the sine-Gordon, sinh-Gordon, and other equations. Also, the results show that these equations depend on only one function that can be obtained analytically by solving an ordinary differential equation. Furthermore, certain exact solutions of these three equations are shown to lead to the exact soliton solutions of a D-dimensional NLS equation with damping and diffusive terms. Finally, our results imply that the planar solitons, N multiple solitons, propagational breathers, and quadric solitons are solutions to the sine-Gordon, sinh-Gordon, and D-dimensional NLS equations.     

N-soliton solutions,Bäcklund transformation and conservation laws for the integro-differential nonlinear Schröbinger equation from the isotropic inhomogeneous Heisenberg spin magnetic chain

Under investigation in this paper is an integro-differential nonlinear Schröbinger (IDNLS) equation, which is equivalent to the spin evolution equation of a classical in-homogeneous Heisenberg magnetic chain in the continuum limit. Based on the Hirota method, the bilinear form and N-soliton solution for the IDNLS equation are derived with the help of symbolic computation. Moreover, N-soliton solution for the IDNLS equation is expressed in terms of the double Wronskian and testified through the direct substitution into the bilinear form. Besides, the bilinear Bäcklund transformation and infinitely many conservation laws are also obtained for the IDNLS equation. Propagation characteristics and interaction behaviors of the solitons are discussed by analysis of such physical quantities as the soliton amplitude, width, velocity and initial phase. Interactions of the solitons are proved to be elastic through the asymptotic analysis. Effect of inhomogeneity on the interaction of the solitons is studied graphically.     

Soliton perturbation theory for the modified nonlinear Schrödinger’s equation

The soliton perturbation theory is used to study the solitons that are governed by the modified nonlinear Schrödinger’s equation. The adiabatic parameter dynamics of the solitons in presence of the perturbation terms are obtained. In particular, the nonlinear gain (damping) and filters or the coefficient of finite conductivity are treated as perturbation terms for the solitons.