带强迫项的变系数KdV方程的多孤立波解及其应用

本文利用改进的齐次平衡法,首先得到了带强迫项的变系数KdV方程的多孤立波解,然后借助此解得到了强迫KdV方程的多孤立波解.最后作为应用例子,利用图形分析方法分析了Rossby孤立波的相互作用,指出了影响Rossby孤立波相对幅度、相位、传播方向及平衡位置的主要原因.     

任意边界下的极小曲面造型问题

讨论了任意边界下的极小曲面造型问题,提出了一个用B-样条函数做任意有界区域上极小曲面造型的新方法.基本思想是:对B-样条函数加权,权函数为节点到区域边界的距离函数,使用加权的B-样条函数空间作为有限元子空间,从而可以得到极小曲面在任意边界上的B-样条曲面近似解.结果表明,新方法得到的极小曲面具有良好的光顺性,且计算精度高.     

交通流多格点预估格子模型与数值仿真

考虑驾驶员对多格点交通流量预估效应,建立了新的交通流多格点预估格子模型。通过线性稳定性分析获得了改进模型的稳定性条件。通过非线性分析得到了扭结—反扭结密度波解,得到了交通流相空间的三个区域:稳定区域、亚稳定区域和不稳定区域。数值仿真验证了考虑驾驶员对多格点的预估效应,能够进一步提高交通流的稳定性。     

(2+1)维破裂孤立子方程组的精确解

本文针对(2+1)维破裂孤立子方程组,采用指数函数法,借助于数学工具Maple软件得到了该方程的两个新的孤立波解。     

Burgers方程的精确孤立波解的符号计算

借助于符号计算Maple,给出了一种构造非线性波动方程行波解的直接代数方法,该方法的主要特点是充分利用Riccati方程.使用此方法得到Burgers方程的多组精确行波解,其中包括一些新的孤立波解,这种方法也适用于求解其它的非线性波动方程（组）.     

时变系数下耦合KdV和Burgers方程组的孤波解*

在双曲函数展开法和Jacobi椭圆函数展开法的基础上，应用它们的扩展形式来讨论三类时变系数下耦合KdV和Burgers方程组，获得了在不同情形下的一些孤波解，其中包括类孤立子解，类冲击波解和类三角函数周期型解．     

考虑鸣笛效应和驾驶员异质性的新格子模型稳定性分析

道路环境及密集交通流随机波动是交通扰动的诱因, 文中考虑道路环境中的汽车鸣笛效应和驾驶员异质性的影响, 提出鸣笛发生临界密度的概念, 建立了更符合实际的格子流体动力学模型, 并揭示非饱和交通状态下诱发交通流失稳的机理.在线性稳定性分析中利用扰动法得到了该模型的稳定性条件, 并基于还原微扰法对该模型的非线性稳定性问题进行研究, 通过求解mKDV方程获取的扭结-反扭结孤立波描述了在临界点附近密度波的传输规则.仿真结果表明, 考虑有鸣笛效应的新格子模型相比于Nagatani模型的稳定性更强, 而较大的临界密度对交通流稳定性存在消极影响; 与以往微观模型相比, 本文模型能解释鸣笛现象发生的自然条件, 即密度高且流量低的地方, 同时驾驶员特性也对交通流的稳定性存在着显著影响.     

应用SPH方法实现海面孤立波运动的模拟

首先介绍孤立波的Kd V方程,继而讨论了孤立波SPH方法的数值求解过程,选择SPH光滑核函数作为正则化高斯核函数。分析了数值求解过程的时间积分方法,给出了具体计算公式,最后给出相应程序中的具体参数下孤立波运动模拟效果。     

改进同伦分析方法及推广Kuramoto Sivashinsky方程的近似解

首先介绍了带有两个辅助参数的改进同伦分析方法,然后用该方法得到了推广Kuramoto-Sivashin-sky方程的同伦近似解.所得近似解与精确孤立波解进行比较,发现本文得到的近似解更有效地逼近真实解.因为该解包含了两个辅助参数,所以能够更有效地调节和控制其收敛区域和速度.研究表明带有两个辅助参数的改进同伦分析方法对复杂非线性系统的研究更有它的优点.     

(2+1)维广义Borer-Kaup系统的变量分离解和半包局域结构

基于一个特殊的Painlev啨Bcklund变换和多线性变量分离方法,分析了(2+1)维非线性广义BorerKaup(GBK)系统,求得了该系统具有若干任意函数的变量分离严格解.根据得到的变量分离严格解,并通过选择解中的任意函数,引入恰当的局域函数和多值函数,找到了GBK系统一种新的具有实际物理意义的半包局域相干结构,如海洋表面波,并简要地讨论了这种半包局域相干结构的一些特殊的演化性质.结果表明:这种半包局域相干结构相互作用后,完全保持它们原有的速度、波形和波幅,即它们的演化性质是完全弹性的.     

Solitary wave solutions for a generalized KdV–mKdV equation with variable coefficients

In this work, a generalized time-dependent variable coefficients combined KdV–mKdV (Gardner) equation arising in plasma physics and ocean dynamics is studied. By means of three amplitude ansatz that possess modified forms to those proposed by Wazwaz in 2007, we have obtained the bell type solitary waves, kink type solitary waves, and combined type solitary waves solutions for the considered model. Importantly, the results show that there exist combined solitary wave solutions in inhomogeneous KdV-typed systems, after proving their existence in the nonlinear Schrödinger systems. It should be noted that, the characteristics of the obtained solitary wave solutions have been expressed in terms of the time-dependent coefficients. Moreover, we give the formation conditions of the obtained solutions for the considered KdV–mKdV equation with variable coefficients.     

二维空间旋转孤立波的相互作用

利用图形分析方法对(2 1)维频散长波方程的旋转孤立波之间的相互作用进行了详细分析,发现了旋转孤立波相互作用产生的一些新的重要非线性现象.这就是,两个旋转孤立波的碰撞是完全非弹性的,它们碰撞之后可以合并成一个旋转孤立波或一个不旋转孤立波,同时可以发生波形转换及性质改变等现象.这些现象的发现,对非线性水波传播与相互作用规律的进一步认识、对非线性水波的控制与利用都具有重要的理论意义.     

Exact periodic,cusp, solitary and loop wave solutions of the EX-ROE

The extended reduced Ostrovsky equation (EX-ROE) is investigated by using the dynamical system theory. The bifurcation phase portraits are drawn in different regions of parameter plane. The bounded travelling wave solutions such as periodic waves, periodic cusp waves, solitary waves, peakon, solitary loop waves and periodic loop waves are obtained. The dynamic characters of these solutions are investigated.     

Solitary wave families of NLPDES via reversible systems theory

The Ostrovsky equation is an important canonical model for the unidirectional propagation of weakly nonlinear long surface and internal waves in a rotating, inviscid and incompressible fluid. Since solitary wave solutions often play a central role in the long-time evolution of an initial disturbance, we consider such solutions here (via the normal form approach) within the framework of reversible systems theory. Besides confirming the existence of the known family of solitary waves and its reduction to the KdV limit, we find a second family of multihumped (or N-pulse) solutions, as well as a continuum of delocalized solitary waves (or homoclinics to small-amplitude periodic orbits). On isolated curves in the relevant parameter region, the delocalized waves reduce to genuine embedded solitons. The second and third families of solutions occur in regions of parameter space distinct from the known solitary wave solutions and are thus entirely new. Directions for future work, including on other NLPDEs, are also mentioned.     

Numerical Simulation of Cylindrical Solitary Waves in Periodic Media

We study the behavior of nonlinear waves in a two-dimensional medium with density and stress relation that vary periodically in space. Efficient approximate Riemann solvers are developed for the corresponding variable-coefficient first-order hyperbolic system. We present direct numerical simulations of this multiscale problem, focused on the propagation of a single localized perturbation in media with strongly varying impedance. For the conditions studied, we find little evidence of shock formation. Instead, solutions consist primarily of solitary waves. These solitary waves are observed to be stable over long times and to interact in a manner approximately like solitons. The system considered has no dispersive terms; these solitary waves arise due to the material heterogeneity, which leads to strong reflections and effective dispersion.     

Soliton perturbation theory for a higher order Hirota equation

Solitary wave evolution for a higher order Hirota equation is examined. For the higher order Hirota equation resonance between the solitary waves and linear radiation causes radiation loss. Soliton perturbation theory is used to determine the details of the evolving wave and its tail. An analytical expression for the solitary wave tail is derived and compared to numerical solutions. An excellent comparison between numerical and theoretical solutions is obtained for both right- and left-moving waves. Also, a two-parameter family of higher order asymptotic embedded solitons is identified.     

A lumped Galerkin method for the numerical solution of the modified equal-width wave equation using quadratic B-splines

The numerical solution of the one-dimensional modified equal width wave (MEW) equation is obtained by using a lumped Galerkin method based on quadratic B-spline finite elements. The motion of a single solitary wave and the interaction of two solitary waves are studied. The numerical results obtained show that the present method is a remarkably successful numerical technique for solving the MEW equation. A linear stability analysis of the scheme is also investigated.     

Traveling wave solutions to the (n+1)-dimensional sinh–cosh–Gordon equation

Traveling wave solutions for a generalized sinh–cosh–Gordon equation are studied. The equation is transformed into an auxiliary partial differential equation without any hyperbolic functions. By using the theory of planar dynamical system, the existence of different kinds of traveling wave solutions of the auxiliary equation is obtained, including smooth solitary wave, periodic wave, kink and antikink wave solutions. Some explicit expressions of the blow-up solution, kink-like solution, antikink-like solution and periodic wave solution to the generalized sinh–cosh–Gordon equation are given. Planar portraits of the solutions are shown.     

Convergence conditions for iterative methods seeking multi-component solitary waves with prescribed quadratic conserved quantities

We obtain linearized (i.e., non-global) convergence conditions for iterative methods that seek solitary waves with prescribed values of quadratic conserved quantities of multi-component Hamiltonian nonlinear wave equations. These conditions extend the ones found for single-component solitary waves in a recent publication by Yang and the present author. We also show that, and why, these convergence conditions coincide with dynamical stability conditions for ground-state solitary waves.Notably, our analysis applies regardless of whether the number of quadratic conserved quantities, s, equals or is less than the number of equations, S. To illustrate the situation when s < S, we use one of our iterative methods to find ground-state solitary waves in spin-1 Bose-Einstein condensates in a magnetic field (s = 2, S = 3).