中文摘要：

从KdV方程的谱问题出发，推导出它的孤子方程族，并由前两个非平凡的孤子方程导出一个新的（2＋1）维耦合KdV方程及其对应的Lax对。借助零曲率方程得到三种达布变换，并讨论三种达布变换间的关系。借助达布变换，解出（2＋1）维耦合KdV方程的孤子解及研究解的性态。利用计算机数学软件，画出了孤子解各种碰撞图形。

英文摘要：

Hierarchy of soliton equations of KdV equation is obtained from its spectral problem. Based on the first two nontrivial soliton equations, a new （ 2 ＋ 1 ） dimensional coupled KdV equation and its Lax pair are derived. With the help of zero curvature equation, three Darboux transformations （DTs） are obtained, and further relations among these three DTs are discussed. By an application of DT, the multiple soliton solutions of （2 ＋ 1 ） dimensional coupled KdV equation are given, and properties of solutions are discussed. Using mathematical software, various collision graphics of the soliton solutions are given.