广义 KdV 方程的达布变换及精确解

孤立子理论的迅速发展，使得众多学者对其研究产生浓厚兴趣。研究孤立子理论中的一个重要问题，就是非线性偏微分方程的求解。本文主要讨论了利用达布变换解决偏微分方程的精确解问题，达布变换是求解非线性偏微分方程的一个有效方法。它通过寻找一种保持相应的Lax对不变的规范变换，最终找到方程解之间关系的变换。本文首先从广义KdV方程的AKNS系统的谱问题出发，经过一系列分类讨论，得到该方程的三类达布变换，并给出证明。然后适当的选取该方程的平凡解，进而求出该方程新的精确解。广义KdV方程在流体力学、等离子体物理、气体动力学领域有重要的实践和理论应用，因此对广义KdV方程的研究具有重大意义。

Darboux transformation and exact solutions of the generalized KdV equation

The rapid development of the soliton theory has evoked a heated discussion among many scholars who are interested in it .The most important issue in the theory of soliton is the solutions to partial differential e -quations .This paper mainly discusses the issue of exact solutions , which solved by the Darboux transformation . Darboux transformation is an effective method for solving nonlinear partial differential equations , the method that find the relationship between the solutions to equation through finding a gauge transformation to keep the corre -sponding Lax pair invariant .The author of this paper ,firstly, from the point of spectrum in the AKNS system of generalized KdV equation ,through a series of classified discussions , having got three kinds of Darboux transfor-mations of the equation and then proved them .Then we selected the suitable trivial solutions to the equation , the new exact solutions are obtained .The generalized KdV equation has practical and theoretical role in fluid me-chanics , plasma physics and aerodynamics , so the study of the generalized KdV equation is of great significance .