RDEs (1) admit CLBSs (8) if and only if [eta]'E = 0 whenever u satisfies (1) and [eta] = 0, where the prime denotes the Gatea[u.sub.x] derivative, that is, [eta]'E = (d/d[epsilon])|[sub.[epsilon]=0][eta][u + [epsilon]E] and E = -[u.sub.xx] - Q(x)[u.sup.2.sub.x] - P(x)u - R(x).
Solving this system, we can obtain the unknown functions in (1) and the corresponding CLBSs (8).
Thus we have obtained 21 classes of equations (11)-(36) with form (1) which admit certain second-order CLBSs. To reduce and solve equations by means of corresponding CLBSs, one solves [eta] = 0 to obtain u as a function of x with x-independent integration constants and then substitutes this solution into (1) to determine the time evolution of these constants.
Equations of the form (4) admitting CLBSs and the corresponding ISs are classified in Section 3.
In view of Theorem 4, it suffices to consider CLBSs (15) of (14) with 2 [les than or equal to] l [les than or equal to] 5.We first consider the casel = 2.
The transformed equations (14) admitting CLBSs (15) are listed in Tables 1, 2, and 3.