【作者】 王作杰；

【导师】 彭彦泽；

【作者基本信息】 华中科技大学 ， 应用数学， 2007， 硕士

【摘要】 求解非线性方程的行波解是一种非常重要和有意义的工作,不管行波解表达式是以显函数的形式给出还是以隐函数的行式给出,我们感兴趣的是它的应用.这些形式的波在其传播过程中将不改变它的形状,也比较容易观察.当然我们最关心的是三种形式的行波解:首先是孤立波解,它是一个局部的行波解,此种行波解在充分远处波形将趋于零;其次是周期波解;最后是冲击波解,这种波从一端到另一端单调减少或者单调增加.近来出现了很多求解非线性波方程精确行波解的方法,一些重要的方法比如,齐次平衡法、双曲正切函数法、试探函数法、非线性变换法和sine-cosine法等等.但是这些方法大部分只能求得非线性波方程的冲击波解和孤立波解,不能求得非线性波方程的周期解.本文首先介绍和应用Tanh函数法和推广的Tanh函数法去求解变形Boussinesq方程;接着介绍了另一种代数方法叫映射法,并且也介绍了改进的映射法和推广的映射法去获得偏微分方程的精确行波解,并应用椭圆函数展开法去求解Boussinesq方程;最后重点讨论应用一种统一的方法----映射法去求解大量的偏微分方程的精确行波解.映射法包括几种直接的方法作为特例比如:双曲正切函数展开法、Jacobi椭圆函数法等待.总而言之,如果行波解存在用此方法同时可获得冲击波解、周期波解和孤立波解.通过映射法、改进的映射法和推广的映射法可以去获得大量的偏微分方程的精确行波解,但是此时要求所研究的非线性偏微分方程中不能同时存在奇数阶导数和偶数阶导数.更多还原

【Abstract】 The exact traveling wave solution is one of most important solutions during solving the nonlinear partial differential equations. Traveling waves, whether their solutions are in explicit or implicit forms, are very interesting form the point of view of applications. These types of waves will not change their shapes during propagation and are thus easy to detect, of particular interesting are three types of traveling waves: the solitary waves, which are localized traveling waves, asymptotically zero at large distances; the periodic waves; the kink waves, which rise or decent from one asymptotic state to another. Recently, there many methods for obtain the exact traveling wave solutions of a nonlinear partial differential equation. Some of the most important methods, for instance, tanh-function method, nonlinear transformation method, sine-cosine method, trial function method and so on. However, using these methods does not obtain the periodic wave solutions of the nonlinear wave equations, only get the solutions of the solitary wave and the kink wave solutions. This paper will firstly introduce and apply tanh-function method and extended tanh-function method to obtain the traveling wave solutions of the Boussinesq equation; secondly, we will introduce and apply Jacobi elliptic function expansion method get the traveling wave solutions of the Boussinesq equation. Finally we introduce and apply a unified algebraic method called the mapping method to obtain exact traveling wave solutions for a large variety of nonlinear partial differential equations. This method includes several direct methods as special case, such as tanh-function method, sech-function method and Jacobi elliptic function method. Above all, by means of this method, the solitary wave, the periodic wave and the kink wave solution can, if they exist, be obtained simultaneously to the equation in question without extra efforts. A large variety of exact traveling solutions of nonlinear partial differential equations are obtained by means of the mapping method, the modified mapping method and the extended mapping method, as long as odd-and even-order derivative terms do not coexist in nonlinear partial differential equations under consideration.更多还原