I am working through Peter Olver's book Application of Lie groups to Differential Equations and I came across this example on page 311 dealing with the criteria for Recursion Operators, now the theorem I am using is
Where $\text{D}$ is the Frechet derivative. After this he goes on to do an example I don't understand how he got the highlighted part:
I am pretty sure there is some rule I am missing somewhere because I have been at this problem all day and am just about ready to put in a hole in my wall with how frustrated I am.
For the second highlighted part I simplified (5.42) $\text{D}_{\Delta} \cdot R (Q) = R \cdot \text{D}_{\Delta}(Q)$, where $Q$ is an arbitrary differential function. (I used $R$ instead of the cursive $R$ for simplicity) into $$prv_{R(Q)}(u_t) - prv_Q(R(u_t)) + prv_Q(R(K)) - prv_{R(Q)}(K) + prv_Q(R)(u_t-K) = 0$$
where $prv_Q$ is the prolongation of the evolutionary vector field $v_Q$, with characteristic $Q$ and the last term $prv_Q(R)(u_t-k)$ is the Lie derivative of the operator $R$ with respect to time evaluated at $u_t - K$.
I have done this computation over and over again in a variety of manners and I can never get what the author got. I was thinking there has to be a way to cancel the terms relating to $u_t$, but the only property I have require the differential operator $R$ to be constant coefficiant but that doesn't work in this case. Below is some extra info on the Lie derivative. I was thinking therr must be a way to relate some of those prolongations but the Lie bracket doesn't work and neither does any other properties I know of. Please if somebody could help me, that would be wondereful sos.
