(@olver86 page 96)
A system $\mathscr{S}$ of $n$-th order differential equations in $p$ independent and $q$ dependent variables is given by a system of equations
$$ \Delta_{\nu}(x,u_{(n)})=0 $$with $\nu=1,\ldots,\ell$ and being
$$ x=\left(x^{1}, \ldots, x^{p}\right), u=\left(u^{1}, \ldots, u^{q}\right) $$We will denote by $X\times U$ the space for $(x,u)$, which is a trivial vector bundle.
The function $\Delta=(\Delta_1,\ldots, \Delta_{\ell})$ will be assumed to be smooth
$$ \Delta: J^n(\mathbb{R}^p,\mathbb{R}^q)\rightarrow \mathbb{R}^{\ell} $$The system can be identified with a subvariety of the jet space also denoted by $\mathscr{S} \subseteq J^n(\mathbb{R}^p,\mathbb{R}^q)$. The jet space is also denoted by $J^n(X\times U)$.
A solution is a smooth function $u=f(x)$ such that
$$ \Delta_{\nu}(x,pr^{(n)}f(x))=0 $$for $\nu=1,\ldots,\ell$ and for every $x$ in the domain of $f$. The symbol $pr^{(n)}f$ refers to the prolongation of $f$ up to order $n$.
This can be restated as: the graph of the prolonged function lies entirely inside the subvariety $\mathscr{S}$.
And also in this way: the solutions are given by the integral submanifolds of the distribution $\iota^*(\mathcal{E})$, being $\iota:\mathcal{S}\to J^n$ the inclusion and $\mathcal{E}$ the Cartan distribution.
See this.
See conservation laws
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Author of the notes: Antonio J. Pan-Collantes
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