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 $\pi: J^{\infty}(E)\to M$ and it preserves the Cartan distribution. Since they preserve the Cartan distribution they satisfy the prolongation formula, so they can be written as
$$ V= Q_{\alpha} \partial_{u^{\alpha}}+\sum_{i,\sigma}(D_\sigma Q_{\alpha})\frac{\partial}{\partial u^{\alpha}_\sigma}. $$Here $D_\sigma$ is the composition of total derivatives corresponding to the multi-index $\sigma$, and $\{ Q_{\alpha}\}$ is a smooth map usually called the characteristic or generating function (Lychagin).
Any vector field in the jet bundle
$$ \mathbf{v}_Q=\sum_{\alpha=1}^q Q_\alpha[u] \frac{\partial}{\partial u^\alpha} $$has an evolutionary representative, which is the evolutionary vector field with characteristic
$$ Q_\alpha=\phi_\alpha-\sum_{i=1}^p \xi^i u_i^\alpha, \quad \alpha=1, \ldots, q $$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 $\tilde{\theta}_t$, 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 $E=\mathbb R \times \mathbb R$ is a trivial bundle, and sections $s:M\to E$ are identified with smooth functions $f:\mathbb R \to \mathbb R$. In this case an evolutionary vector field as the form
$$ V= Q {\partial u}+D_x( Q) {\partial u_{1}}+\cdots $$with $Q(x,u,u_1,\ldots,u_m)$ a smooth function on $J^m$ for certain integer $m$ (a differential function).
In this context the section $\tilde{s}$ is identified with $(x,f(x),f'(x),f''(x),\ldots)$. Observe that the flow $\tilde{\theta}_t$ is, by definition,
$$ \tilde{r}_t=\tilde{\theta}_t \circ \tilde{s}\equiv(x,g_t(x),g_t'(x),g_t''(x),\ldots) $$satisfying
$$ \frac{d}{dt}|_{t=0} x=0, $$ $$ \frac{d}{dt}|_{t=0} g_t(x)= Q(x,f(x),f'(x),\ldots,f^{m)(x)}), \tag{*} $$ $$ \frac{d}{dt}|_{t=0} g'_t(x)=D_x Q(x,f(x),f'(x),\ldots,f^{m)(x)}), $$ $$ ... $$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
$$ \left\{ \begin{array}{l} \dfrac{\partial}{\partial t}h(x,t)= Q(x,h,\frac{\partial h}{\partial x},\ldots,\frac{\partial^m h}{\partial^m x}),\\ h(x,0)=f(x) \end{array} \right . $$________________________________________
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Author of the notes: Antonio J. Pan-Collantes
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