变系数Benjamin-Bona-Mahony-Burgers方程的微分不变量和精确解

通过应用经典李群方法,得到了变系数的Benjamin-Bona-Mahony-Burgers(BBMB)方程的连续等价变换。从等价代数着手,讨论了该方程的微分不变量,发现此方程不存在零阶微分不变量,但是具有8个相互独立的一阶不变量。利用已经求得的一阶微分不变量对方程进行了群分类。在此过程中,进一步应用上述微分不变量将一般的变系数BBMB方程映射为常系数BBMB方程、Burgers方程、Benjamin-Bona-Mahony(BBM)方程,进而得到了变系数BBMB方程的一些新的精确解,并且作出了特殊变系数BBM方程、Burgers方程的精确解的图像。

Differential invariants and exact solutions of variable coefficients Benjamin-Bona-Mahony-Burgers equation

The Lie symmetry method is performed for the variable coefficients Benjamin-Bona-Mahony-Burgers(BBMB)equation and the continuous equivalence transformations are obtained. Starting with the equivalent algebra, the differential invariants of order one are constructed. It is found that there is no zero-order differential invariant for this equation, but there are eight first-order invariants that are independent of each other. Using the obtained first-order differential invariants, we make the group classification. Finally, the general variable coefficient BBMB equations are mapped to the constant coefficient BBM equation or Burgers equation or BBMB equation by the given equivalent transformation. And then a series of new exact solutions of those variable coefficient equations are obtained. The images of the exact solution of the special BBM equation with variable coefficients and Burgers equation of the exact solution are made.