KdV方程的精确多重波解研究

The famous Korteweg de Vries (KdV) equation is a shallow water wave equation early derived by Korteweg de and Vries, its first application was discovered in the study of collision_free hydromagnetics waves in 1960. Following the further studies of new application, it has attracted much attention. Many researchers have carried out thorough investigations into the solitary wave solutions to the KdV equation in recent years, then a large number of linear one_soliton solutions and some many_wave solutions have been obtained. But most papers relative to many_wave solution only provided formal solutions which can be expressed in terms of either superposition of some exponential functions or Wronskian determinants, and these solutions are not true exact solution in its rigorous sense, thus the obtained exact solution of the KdV equation are not much. As a result, a few linear multi_soliton solutions including one_lump solution that were solved by American mathematicians M.J. Ablowitz and P.A. Clarkson with the inverse scattering transformation method have been well known for a long time. However, as we know, nonlinear exact many_wave solutions for the KdV equation have hardly appeared up to now, this result is caused by the complexity of nonlinear many_wave solution,the suitability of chosen method, tedious calculation and so on. Thanks to the KdV equation has arisen in a number of physical contexts, the exact many_wave solutions play an important role in some areas of research such as stratified internal waves, ion_acoustic waves, plasma physics and lattice dynamics, and some new application of the equation in these sides have not discovered today. We attempt to creatively apply the method of variation of constants which can always be used to find particular solution of the nonhomogeneous linear ordinary differential equation to the Darboux transformation that derives from the study of Mr Chaohao Gu, and further research nonlinear and liner exact multi_soliton solutions of the KdV equation by means of it. At the same time, the method of constructing multi_soliton solution is extensively used for constructing exact two_periodic solutions, three_periodic solutions, stationary periodic_soliton solutions, stationary soliton_periodic solutions for the KdV equation. Actually, the Darboux transformation method is one of the main methods to construct line multi_soliton solutions, and one often starts from a constant solution in order to avoid solving a partial differential system which stems from the lax pair on the given solution, in many cases, this system is more complicated than the original evolution equation, and it is too difficult to solve. However, with the aid of combining the Darboux transformation method and the variation of constants method, we can start from some non_trivial solutions of the KdV equation, not only many multi_soliton solutions are available, but also a group of multi_periodic solutions are obtained by the analogous technique.

近年来，有不少研究者对KdV方程的孤立波解做了深入的研究，得到了大量的线状单孤子解和一些多重波解。但涉及多波解的大多数研究论文只是提供了由多个指数函数迭加或由Wronskian行列式表示的形式解，不是真正意义上的精确解，最终得到的精确多重波解并不多见。所以，由美国数学家M.J. Ablowitz和P.A. Clarkson利用逆散射法求出的几个线状多孤子解一直备受关注。而非线性的多波解几乎没有出现过，这是由于非线状多波解的复杂性和方法的适用性以及计算太繁琐等原因造成的。由于KdV方程的精确多重波解在分层内波、离子声波、等离子物理学和晶格力学等研究领域发挥着重要的作用。我们试图把常微分方程中常用的常数变易法创造性地运用于谷超豪先生关于达布变换的研究成果，进一步研究KdV方程的非线性精确多孤子解。并把研究孤子的方法推广性地运用于精确双周期解、三周期解、静态周期-孤子解和静态孤子-周期解的研究。

KdV类方程具有丰富的物理背景, 为了更好地解释和理解它们背后的物理现象及物理问题, 研究它们的精确多重波解是很有必要的。在过去的几十年里，研究者提出了各种有效的方法来研究KdV类方程的精确解，但有关精确多重波解的研究成果非常少。与单波解相比，多波解的结构更为复杂，也更接近于问题的本质。尽管Backlund变换法和Darboux变换法为非线性发展方程的多孤子解的产生铺平了道路，但值得注意的是， AKNS系统是一个超定线性偏微分方程，当它的系数依赖于自变量时，由于缺少行之有效的方法，要找到该系统的精确解是非常困难、甚至是不可能的事，只有反复试探，才有可能找到解决问题的特殊方法和特殊技巧。. 借助Khater , Wadati 和谷超豪等中外数学家有关Backlund变换和Darboux变换的研究成果，从给定的行波解和静态解出发，综合运用常数变易法和猜测待定法求出了解变系数的AKNS系统的一般解，最终得到了Liouville方程的呼吸孤子解和非线状双波解，并构造了KdV方程和Schrodinger方程的一大批新的显式多重波解。. 常数变易法是求解常微分方程的一种传统方法，而Backlund变换法和Darboux变换法主要用于构造线状多孤子解。但是，当常数变易法被拓展性地用于求解变系数的AKNS系统的一般解时，许多比较复杂的双周期-孤子解和双孤子-周期解也可以由Backlund变换法和Darboux变换法得到。. 基于方法的积累，我们把研究范围从常系数领域扩大到极为困难的变系数领域。例如，利用Darboux变换法求出了带扰动项的变系数的mKdV方程的双孤子解和双周期解。我们注意到，有关五阶变系数的mKdV方程的多波解的研究少到几乎没有。. 最重要的是, 我们的研究思路可借鉴性地用于其它偏微分系统的研究，常数变易法的拓展性应用也为变系数问题的研究提供了新的技巧和方法。