广义变系数Hirota-Satsuma方程组的等价变换和守恒定律

Equivalence Transformation and Conservation Law of the Generalized Hirota-Satsuma Equations with Variable Coefficients

针对变系数非线性偏微分方程的研究,提出了一种新的关于求解变系数非线性偏微分方程的守恒定律的方法;通过研究广义变系数Hirota-Satsuma方程组,运用李群分析法,求出了方程组的李点对称,并证明这个方程组是非线性伴随的,也就是Ibragimov定理,从而构建了一般的守恒定律公式;运用Hirota-Satsuma方程组的等价变换,即增广空间上的一个非退化点变换,从而得到方程组的等价代数;由于广义变系数Hirota-Satsuma方程组的守恒定律公式中含有任意元素,所以方程组中含有无穷个守恒定律。

With respect to the study of nonlinear partial differential equations（ PDEs） with variable coefficients,a new method for solving the conservation laws of nonlinear PDEs is proposed. By studying the generalized Hirota-Satsuma equations with variable coefficients,we use Lie group analysis method to get the Lie point symmetries,and prove that the system is nonlinearly self-adjoint,namely Ibragimov＇s theorem,thus construct the general conservation law formula. Then its equivalence transformation which is a nondegenerate point transformation is used in the augmented space to obtain the equivalence algebras. Since the conservation law formula involves in any arbitrary elements,we can draw an infinite number of the conservation laws.