二阶多项式系统的可积条件

本文推广了 Liouville关于方程可积性的定义 ,定义二阶多项式系统 ( * )的可积性为首次积分可由P( x,y) ,Q( x,y)通过有限次代数运算 ,积分 ,微分 ,指数运算和解代数方程得到 .证明了与二阶多项式系统相对应的一阶算子具有由定理给出的某种“特征”是该系统可积的充分条件 .最后 ,利用此结果给出了Burgers-K-d V方程的行波解的首次积分 .

Integrability condition of Second Order Polynomial System

The notation of integrability of differential equation defined by Liouville is extended in this paper to that of second order polynomial system ( *). By the definition, the system is integrable if the first integral of it can be obtained from P(x,y), Q(x,y) through finite operation, including arithmetic operation, integration, differentiation, exponentiation and solving algebraic equations. It is proved in this paper that the condition that the first order differential operater corresponding to the second order system has some what ″characteristic″ listed in the orm is sufficient for integrating the system. Even more, the first integral of travelling wave solution of Burgers-K-dV equation is presented.