一类非线性(3+1)维波动方程的周期解

通过行波约化一类(3+1)维非线性波动方程和建立与立方非线性Klejn-Gordon方程间变换的联系,由此得到其孤立波解和周期解。     

一类非线性(3+1)维波动方程的周期解

通过行波约化一类(3+1)维非线性波动方程和建立与立方非线性Klein-Gordon方程间变换的联系,由此得到其孤立波解和周期解.     

(2+1)维Nizhnik方程的Jacobi椭圆函数周期解

利用最近提出的F-展开法,导出了(2 1)维Nizhnik方程的由Jacobi椭圆函数表示的周期解,并且在极限情况下,可以推得(2 1)维Nizhnik方程的孤波解以及其他形式解。     

(2+1)维色散长波方程组和组合KdV-Burgers方程的新的精确孤立波解

本文在齐次平衡法、双曲正切函数法和辅助方程法的基础上引入一类新的辅助方程,并借助符号计算系统Mathematica来构造了(2 1)维色散长波方程组和组合KdV-Burgers方程的新的精确孤立波解。这种方法也可用于寻找其它非线性发展方程的新的精确孤立波解。     

两类(3+1)维广义BKP方程的Multiple exp-函数方法解北大核心CSCD

本文借助Multiple exp-函数法和齐次平衡原理,求解了两类(3+1)维广义BoussinesqKadomtsev-Petviashvili(BKP)方程,获得了其指数型函数波解.根据参数的任意性,对参数取不同的值,得到了方程不同类型的扭子波解和孤子波解.作为例子,借助Maple分别给出了不同情况下两种特殊类型的波解的图像.通过图像,能够更直观地理解两类广义BKP方程解的特点,这将对后期进行相关方面的研究和涉及广义BKP方程的工程领域的研究有着一定的参考价值.     

Wick型随机WBK浅水波方程的白噪声泛函解

WBK浅水波方程是数学物理中的重要方程之一,对其研究有重要意义。经典方法都是在确定状态下研究波方程的性质和精确求解。随机分析和白噪声理论的建立和发展为波方程的研究提供了新的内容、方法和工具,因此,研究随机状态下波方程就成为可能。本文就是研究随机状态下WBK浅水波方程的精确求解问题:在Kondratiev分布空间(S)-1中,利用Hermite变换和齐次平衡法研究Wick-型随机WBK浅水波方程的精确求解,给出其白噪声泛函解,并给出了该方程在系数F(t)取不同白噪声泛函时的几个例子。     

非线性弦振动的Kdv方程及其速度孤波解

文中用递减微扰法导出了非线性弦振动的 Kdv 方程。文中分析了无色散和无非线性(线性色散)两种特殊情况下的解。然后,详细地讨论了 Kdv 方程的速度孤波解的性态,并指出了非线性弦振动中孤波的主要特征。     

变系数广义KdV方程的精确类孤子解

本文主要研究了变系数广义KdV方程的精确解。利用一种函数变换将变系数KdV方程约化为非线性常微分方程，借助于Mathematica软件求出该类方程的几种精确解。通过数值实例说明了方法的有效性，为变系数Kdv方程在自然科学领域的应用提供了理论依据。     

具任意次非线性项的非线性Klein-Gordon方程孤波解的轨道稳定性

具任意次非线性项的非线性Klein-Gordon方程是一类非常重要的物理模型,它的孤波解的轨道稳定性有着很好的物理意义.本文利用抽象的Grillakis轨道稳定性理论和谱分析,讨论具任意次非线性项的非线性Klein-Gordon方程的孤波解的轨道稳定性.当非线性项的系数以及波速满足一定的条件时,得出了其钟状孤波解总是不...     

(2+1)维mKdV方程dromions相互作用的研究

从两个线孤子解出发，可以得到双线性形式2 1维mKdV方程的某个势函数的dromion解。该dromion解在所有方向都是局域的。两个dromion之间的相互作用通过图形分析的方法进行了详细研究。依据参数的不同选择，该相互作用可以是弹性的也可以是非弹性的。     

一类高阶KdV类型水波方程的多辛Euler-box格式

高阶KdV类型水波方程作为一类重要的非线性方程有着许多广泛的应用前景．本文主要研究高阶KdV类型水波方程的多辛Euler-box格式．首先,通过正则变换,构造了高阶KdV方程的多辛结构,并得到该系统的多辛守恒律、局部能量守恒律和动量守恒律．然后,我们利用Euler-box格式对高阶KdV方程进行离散,并基于Hamilton空间体系的多辛理论研究了该系统的离散Euler-box格式．我们证明该格式满足离散多辛守恒律,并且给出该格式的向后误差分析．最后,数值算例结果表明该多辛离散格式具有较好的长时间数值稳定性．     

Dark solitons and rouge waves for a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential in the Bose–Einstein condensate

Under investigation in this paper is a (2+1)-dimensional Gross–Pitaevskii equation with time-varying trapping potential, which describes the dynamics of the (2+1)-dimensional Bose–Einstein condensate. Employing the Hirota method and symbolic computation, we obtain the dark one-soliton, two-soliton, three-soliton, breather-wave and rouge-wave solutions, respectively. We graphically study the dark solitons with the time-varying harmonic potential and scaled scattering length. Parallel and period solitons are observed. We obtain that when the external trapping potential increases with time, amplitudes of the dark solitons increase and widths of those solitons become narrower; when the external trapping potential is a periodic function, amplitudes and widths of the dark solitons periodically change. Decrease in the scaled scattering length leads to the narrower solitons’ widths, but does not affect the solitons’ amplitudes. Breather waves and rouge waves are also displayed: Rouge waves emerge when the period of the breather waves go to the infinity.     

A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part I theory

This paper introduces a G space theory and a weakened weak form (W²) using the generalized gradient smoothing technique for a unified formulation of a wide class of compatible and incompatible methods. The W² formulation works for both finite element method settings and mesh‐free settings, and W² models can have special properties including softened behavior, upper bounds and ultra accuracy. Part I of this paper focuses on the theory and fundamentals for W² formulations. A normed G space is first defined to include both continuous and discontinuous functions allowing the use of much more types of methods/techniques to create shape functions for numerical models. Important properties and a set of useful inequalities for G spaces are then proven in the theory and analyzed in detail. These properties ensure that a numerical method developed based on the W² formulation will be spatially stable and convergent to the exact solutions, as long as the physical problem is well posed. The theory is applicable to any problems to which the standard weak formulation is applicable, and can offer numerical solutions with special properties including ‘close‐to‐exact’ stiffness, upper bounds and ultra accuracy. Copyright © 2009 John Wiley & Sons, Ltd.     

A G space theory and a weakened weak (W2) form for a unified formulation of compatible and incompatible methods: Part II applications to solid mechanics problems

In part I of this paper, we have established the G space theory and fundamentals for W² formulation. Part II focuses on the applications of the G space theory to formulate W² models for solid mechanics problems. We first define a bilinear form, prove some of the important properties, and prove that the W² formulation will be spatially stable, and convergent to exact solutions. We then present examples of some of the possible W² models including the SFEM, NS‐FEM, ES‐FEM, NS‐PIM, ES‐PIM, and CS‐PIM. We show the major properties of these models: (1) they are variationally consistent in a conventional sense, if the solution is sought in a proper H space (compatible cases); (2) They pass the standard patch test when the solution is sought in a proper G space with discontinuous functions (incompatible cases); (3) the stiffness of the discretized model is reduced compared with the finite element method (FEM) model and possibly to the exact model, allowing us to obtain upper bound solutions with respect to both the FEM and the exact solutions and (4) the W² models are less sensitive to the quality of the mesh, and triangular meshes can be used without any accuracy problems. These properties and theories have been confirmed numerically via examples solved using a number of W² models including compatible and incompatible cases. We shall see that the G space theory and the W² forms can formulate a variety of stable and convergent numerical methods with the FEM as one special case. Copyright © 2009 John Wiley & Sons, Ltd.