Evolutionary vector field

See Definition 5.4 @olver86.
A evolutionary vector field $V$ is a vector field on the infinite jet bundle of a bundle $E \to M$ , such that is vertical with respect to the projection $π : J^{\infty} (E) \to M$ and it preserves the Cartan distribution. Since they preserve the Cartan distribution they satisfy the prolongation formula, so they can be written as

V = Q_{α} \partial_{u^{α}} + \sum_{i, σ} (D_{σ} Q_{α}) \frac{\partial}{\partial u_{σ}^{α}} .

Here $D_{σ}$ is the composition of total derivatives corresponding to the multi-index $σ$ , and ${Q_{α}}$ is a smooth map usually called the characteristic or generating function (see @lychagin2007contact).

Any vector field in the jet bundle

v_{Q} = \sum_{α = 1}^{q} Q_{α} [u] \frac{\partial}{\partial u^{α}}

has an evolutionary representative, which is the evolutionary vector field with characteristic

Q_{α} = ϕ_{α} - \sum_{i = 1}^{p} ξ^{i} u_{i}^{α}, α = 1, \dots, q

How can we interpret this?
Consider a section $s : M \to E$ of the original bundle. It can be prolonged to a section $\tilde{s} : M \to J^{\infty} (E)$ . The obtained section represents the original section together with "all its derivatives", and it is tangent to the Cartan distribution. Since $V$ preserves this distribution, the flow of $V$ , let's call it ${\tilde{θ}}_{t}$ , sends $\tilde{s}$ to another section ${\tilde{r}}_{t}$ also tangent to Cartan distribution, so we can consider that it comes from a section $r_{t} : M \to E$ .

For simplicity, suppose $E = R \times R$ is a trivial bundle, and sections $s : M \to E$ are identified with smooth functions $f : R \to R$ . In this case an evolutionary vector field as the form

V = Q \partial u + D_{x} (Q) \partial u_{1} + \dots

with $Q (x, u, u_{1}, \dots, u_{m})$ a smooth function on $J^{m}$ for certain integer $m$ (a differential function).
In this context the section $\tilde{s}$ is identified with $(x, f (x), f^{'} (x), f^{″} (x), \dots)$ . Observe that the flow ${\tilde{θ}}_{t}$ is, by definition,

{\tilde{r}}_{t} = {\tilde{θ}}_{t} \circ \tilde{s} \equiv (x, g_{t} (x), g_{t}^{'} (x), g_{t}^{″} (x), \dots)

satisfying

\frac{d}{d t} |_{t = 0} x = 0,

\begin{matrix} (*) & \frac{d}{d t} |_{t = 0} g_{t} (x) = Q (x, f (x), f^{'} (x), \dots, f^{m) (x)}), \end{matrix}

\frac{d}{d t} |_{t = 0} g_{t}^{'} (x) = D_{x} Q (x, f (x), f^{'} (x), \dots, f^{m) (x)}),

. . .

But all this equations are satisfied if and only if the single equation $(*)$ is satisfied, as it can be checked.

In conclusion, an evolutionary vector field can be identified with the function $Q$ . If you think of $f$ as a point in an infinite-dimensional space, $Q$ is like a tangent direction for this point to flow through. The flow is obtained by solving the evolution equation

{\begin{cases} \frac{\partial}{\partial t} h (x, t) = Q (x, h, \frac{\partial h}{\partial x}, \dots, \frac{\partial^{m} h}{\partial^{m} x}), \\ h (x, 0) = f (x) \end{cases}