System of DEs

Geometric formulation

(@olver86 page 96)
A system $\mathcal{S}$ of $n$th-order differential equations in $p$ independent and $q$ dependent variables is given by a system of equations

with $\nu =1,\dots ,\ell$ and being

We will denote by $X\times U$ the space for $(x,u)$, which is a trivial vector fiber bundle.

The function $\mathrm{\Delta }=({\mathrm{\Delta }}_1,\dots ,{\mathrm{\Delta }}_{\ell })$ will be assumed to be smooth

The system can be identified with a subvariety of the jet bundle also denoted by $\mathcal{S}\subseteq J^n({\mathbb{R}}^p,{\mathbb{R}}^q)$. The jet space is also denoted by $J^n(X\times U)$.

Solutions

A solution is a smooth function $u=f(x)$ such that

for $\nu =1,\dots ,\ell$ and for every $x$ in the domain of $f$. The symbol $p{r}^{(n)}f$ 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 $\mathcal{S}$.

And also in this way: the solutions are given by the integral submanifolds of the distribution ${\iota }^{\ast }(\mathcal{E})$, being $\iota :\mathcal{S}\to J^n$ the inclusion and $\mathcal{E}$ the Cartan distribution. This distribution is Vessiot distribution.

Example: The second-order scalar case ($J^2$)

We begin with a scalar Partial Differential Equation for a function $u(x,t)$.

• The Space: The second-order jet space $J^2(E)\cong {\mathbb{R}}^8$ with local coordinates:

• The Equation Manifold: The PDE $F(x,t,u,\dots )=0$ defines a hypersurface (submanifold) $S\subset J^2$ of dimension 7.
• The Vessiot Distribution:
  • We start with the Cartan Distribution $\mathcal{E}$ on $J^2$ (spanned by total derivative operators $D_x,D_t$ or generated by the contact forms).
  • We restrict it to the equation manifold $S$: $\mathcal{V}=\mathcal{C}{|}_S\cap TS$.
  • Dimension: In this case, $\text{dim}(\mathcal{V})=4$.
• The Problem: We seek a solution surface (dimension 2) integral to this distribution. Since $2<4$, the solution is not unique; the "excess" dimensions correspond to the infinite-dimensional freedom (arbitrary functions) typical of PDEs.
  

Symmetry groups

See this.

Conservation laws

See conservation laws

Particular cases:

• ODE
• PDE
• system of ODEs
• system of first order ODEs
• system of linear first order homogeneous PDEs

See also

• visualization of DE systems.
• differential constraint method.