KdV方程的直接微扰方法

DIRECT PERTURBATION THEORY FOR THE KdV EQUATION 

系统地讨论了含修正项的ＫｄＶ方程的直接微扰方法．从反散射变换所得的不含修正项方程的严格多孤子解出发，导出了线性化算子的零本征值的所有本征函数———平方Ｊｏｓｔ函数．引入了它们所对应的伴随函数和定义了内积．计算了应有的正交关系，并自然得到单位元的平方Ｊｏｓｔ函数的展开式．利用广义的Ｍａｒｃｈｅｎｋｏ方程，证明了平方Ｊｏｓｔ函数的完备性．同时得到展开式中的积分是沿实轴从－∞到∞，但在原点附近将从上方绕过．这不同于过去所得的Ｃａｕｃｈｙ主值积分．为最明确显示这一差别，在单孤子情况下又用平方Ｊｏｓｔ函数的显式，直接作了验证．同时指出，以前由于取Ｃａｕｃｈｙ主值积分导出的ＫｄＶ方程所特有的孤子尾，在采用从上方绕过原点的积分时，则事实上并不存在．

Abstract An exact direct perturbation theory of the KdV equation with corrections is developed for multi soliton case.After showing that the derivatives of the squared Jost functions with respect to x are the eigenfunctions of the linearized operator,suitable definitions of the adjoint functions and the inner product are introduced.Orthogonality relations are derived and the expansion of the unity in terms of the squared Jost functions is implied.The completeness relation of the squared Jost functions is shown by the generalized Marchenko equation.The final result indicates that in the expression of the completeness relation,the integral path is along the real axis from -∞ to ∞ but runs over near the origin,which is contrary to the Cauchy principal value appearing in previous works.This leads to the disappear once of the shelf behind the soliton due to perturbations,which was considered as a characterized effect in the previous theories.