一类耦合KdV方程的孤波解和周期波解及其相互关系

On the Solitary Wave Solution and Periodic Wave Solution of a Class of Coupled KdV Equations

运用平面动力系统的理论和方法对一类耦合KdV波动方程所对应的平面动力系统进行了定性分析,给出了该方程在一定条件下存在唯一钟状孤波解和无穷多个周期波解的结论.分别利用待定系数法和首次积分法求得了该方程钟状孤波解和周期波解的精确表达式,并直观地指出了它们所对应的解轨线在全局相图中的位置.进一步讨论了方程孤波解与Jacobi椭圆函数型周期波解的关系,并直观地给出了当模数趋于1时Jacobi椭圆函数周期波解向钟状孤波解演变的三维示意图.

For a class of coupled KdV equations, the theory and method of planar dynamical system were applied to qualitatively analyse the dynamical system which the equation corresponds to. It is concluded that the equation has a unique bell profile solitary wave solution and infinite number of periodic wave solutions. The exact expressions of the bell solitary wave solution and the periodic solutions were provided by using the methods of undetermined coefficients and first integral respectively,and the positions of their orbits on the global phase portrait were pointed out. The relation between the solitary wave solution and the periodic wave solutions was discussed. Finally, 3-dimensional figures were presented to illustrate the evolution process of Jacobi elliptic functional periodic wave solution to bell solitary wave solution when the modulus tends to be 1.