${\mathcal{C}}^{\mathrm{\infty }}$-structure-based method

Based on the result: integration by Pfaffian equations.

Distributions version

This is in this paper.
Just like a solvable structure, a cinf-structure $⟨X_1,\dots ,X_{n-r}⟩$ of an involutive distribution $\mathcal{Z}=\{Z_1,\dots ,Z_r\}$ can be used to find their integral manifolds by following a procedure which is similar to the case of solvable structures. This algorithm is as follows:

• First, observe that the set $\{Z_1,\dots ,Z_r,X_1,\dots ,X_{n-r}\}$ constitute a frame on $U$, so we can consider the 1-forms

where ${\hat{X}}_i$ denotes omission.

• Thus we obtain a sequence of Pfaffian systems

associated to the sequence of distributions $\mathcal{Z}\oplus \mathcal{S}(\{X_1,\dots ,X_i\})$, $i=1,\dots ,n-r-1$. These Pfaffian systems are completely integrable, since the corresponding distributions are involutive.

• Then we look for a solution $I_{n-r}$ of the completely integrable Pfaffian equation ${\mathcal{P}}_{n-r-1}$, that is ${\omega }_{n-r}\equiv 0$, by solving the system of homogeneous linear first order PDEs given by

• We fix an arbitrary constant $C_{n-r}\in \mathbb{R}$ and consider the integral manifold of ${\mathcal{P}}_{n-r-1}$ given by the level set $I_{n-r}=C_{n-r}$, which will be denoted by ${\mathrm{\Sigma }}_{(C_{n-r})}$. Then we obtain the restriction of the Pfaffian systems ${\mathcal{P}}_{n-r-2}$ to ${\mathrm{\Sigma }}_{(C_{n-r})}$,

by pulling back with a suitable embedding ${\iota }_{n-r}:V_{n-1}\to {\mathrm{\Sigma }}_{(C_{n-r})}$, with domain an open subset $V_{n-1}$ of ${\mathbb{R}}^{n-1}$. Observe than both ${\iota }_{n-r}$ and $V_{n-1}$ depend on the constant $C_{n-r}$.

• The integral manifolds of the restricted Pfaffian system give rise to integral manifolds of the original ${\mathcal{P}}_{n-r-2}$ after composition with the embedding ${\iota }_{n-r}$. But now observe that ${\mathcal{P}}_{n-r-2}{|}_{{\mathrm{\Sigma }}_{(C_{n-r})}}$ is a new Pfaffian equation (since ${\omega }_{n-r}{|}_{{\mathrm{\Sigma }}_{(C_{n-r})}}=0$) defined on a space of one less variable. So we proceed as in step 2 by solving ${\omega }_{n-r-1}{|}_{{\mathrm{\Sigma }}_{(C_{n-r})}}\equiv 0$, and using a solution to define the level sets ${\mathrm{\Sigma }}_{(C_{n-r-1},C_{n-r})}$ with $C_{n-r-1}\in \mathbb{R}$.

• In every iteration we solve the Pfaffian equation

A solution $I_i$ gives rise to level sets

which are integral manifolds of the restriction of the Pfaffian system ${\mathcal{P}}_{i-1}$ to ${\mathrm{\Sigma }}_{(C_{i+1},\dots ,C_{n-r})}$. The embedding of ${\mathrm{\Sigma }}_{(C_i,C_{i+2},\dots ,C_{n-r})}$ into $U$ are integral manifolds of ${\mathcal{P}}_{i-1}$.
Eventually, after $n-r$ iterations we arrive to integral manifolds of ${\mathcal{P}}_0={\mathcal{Z}}^{\circ }$, which are the integral manifolds of $\mathcal{Z}$.

ODEs version

Consider an $m$th-order ODE

where $\varphi$ is a smooth function defined on an open subset $U$ of the jet space $J^{m-1}(\mathbb{R},\mathbb{R})$, with coordinates $(x,u,u_1,\dots ,u_{m-1})$. We can associate to this equation the (trivially) involutive distribution $\mathcal{S}(\{Z\})$, generated by the vector field

The integral manifolds of this distribution correspond to the prolongation of solutions of the ODE. The ${\mathcal{C}}^{\mathrm{\infty }}$-structure-based method of integration allows us to find these integral manifolds by looking for a ${\mathcal{C}}^{\mathrm{\infty }}$-structure for the equation.

The main result concerning ${\mathcal{C}}^{\mathrm{\infty }}$-structures is that they can be used to integrate the ODE by means of the resolution of $m$ Pfaffian equations which are completely integrable (see \cite[Theorem 4.1]{pancinf-sym}). This integration can be performed as follows:
This integration can be performed as follows:

1. Consider the standard volume form Ω on $U$:

   $$\mathbf{\Omega }=dx\wedge du\wedge d{u}_1\wedge \cdots \wedge d{u}_{m-1}.$$

   We introduce the 1-forms:

   $${\omega }_i=X_m⌟\dots ⌟\widehat{X_i}⌟\dots ⌟X_1⌟A⌟\mathbf{\Omega },1\le i\le m,$$

   where $\widehat{X_i}$ indicates omission of $X_i$ and $⌟$ denotes the interior product.

2. The Pfaffian equation ${\omega }_m\equiv 0$ is completely integrable [1, 2, 3]. A first integral $I_m=I_m(x,u,u_1,\dots ,u_{m-1})$ can be found by solving the system of linear homogeneous first-order PDEs arising from the condition $d{I}_m\wedge {\omega }_m=0$. The level sets ${\mathrm{\Sigma }}_{(C_m)}$, implicitly defined by $I_m(x,u,u_1,\dots ,u_{m-1})=C_m$, where $C_m\in \mathbb{R}$, are the solutions to ${\omega }_m\equiv 0$.

3. Given a local parametrization ${\iota }_m$ of ${\mathrm{\Sigma }}_{(C_m)}$, defined on an open subset $V_m\subseteq {\mathbb{R}}^m$, we calculate:

   $${\omega }_i{|}_{{\mathrm{\Sigma }}_{(C_m)}}:={\iota }_m^{\ast }({\omega }_i),1\le i\le m.$$

   We repeat the process for the Pfaffian equation ${\omega }_{m-1}{|}_{{\mathrm{\Sigma }}_{(C_m)}}\equiv 0$, which is completely integrable. Similarly, we denote:

   $${\omega }_i{|}_{{\mathrm{\Sigma }}_{(C_{m-1},C_m)}}:={\iota }_{m-1}^{\ast }({\omega }_i{|}_{{\mathrm{\Sigma }}_{(C_m)}}),1\le i\le m,$$

   and we proceed with ${\omega }_{m-2}{|}_{{\mathrm{\Sigma }}_{(C_{m-1},C_m)}}\equiv 0$, and so on.

4. At each step, we solve the completely integrable Pfaffian equation ${\omega }_k{|}_{{\mathrm{\Sigma }}_{(C_{k+1},\dots ,C_m)}}\equiv 0$, obtaining a first integral:

   $$I_k=I_k(x,u,u_1,\dots ,u_{k-1};C_{k+1},\dots ,C_m).$$

   Then, a local parametrization ${\iota }_k$, defined on an open subset $U_k\subseteq {\mathbb{R}}^k$, of the level set ${\mathrm{\Sigma }}_{(C_k,\dots ,C_m)}$ is found. The process finalizes when:

   $$I_1=I_1(x,u;C_2,\dots ,C_m)$$

   is obtained. The solutions are implicitly defined by the level sets ${\mathrm{\Sigma }}_{(C_1,\dots ,C_m)}$.

References

1. @pancinf-sym
2. @pancinf-struct
3. @pan23integration