cinf-structure-based method

Distributions version

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Just like a solvable structure, a cinf-structure ⟨X1,…,Xn−r⟩ of an involutive distribution Z={Z1,…,Zr} 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:

−Thusweobtainasequenceof[[01CONCEPTS/GEOMETRY/Pfaffiansystem∥Pfaffiansystem]]s

\begin{array}{l}
\mathcal{P}0=\mathcal{Z}^{\circ}=\mathcal{S}({\omega_1,\ldots,\omega}),\
\mathcal{P}1=\left(\mathcal{Z}\oplus \mathcal{S}({X_1})\right)^{\circ}=\mathcal{S}({\omega_2,\ldots,\omega}), \
\cdots\
\mathcal{P}{i}=\left(\mathcal{Z}\oplus \mathcal{S}({X_1,\ldots,X_i})\right)^{\circ}=\mathcal{S}({\omega,\ldots,\omega_{n-r}}), \
\cdots \
\mathcal{P}{n-r-1}=\left(\mathcal{Z}\oplus \mathcal{S}({X_1,\ldots,X})\right)^{\circ}=\mathcal{S}({\omega_{n-r}} ),\
\end

associatedtothesequenceofdistributions$Z⊕S({X1,…,Xi})$,$i=1,…,n−r−1$.ThesePfaffiansystemsarecompletelyintegrable,sincethecorrespondingdistributionsareinvolutive.−Thenwelookforasolution$In−r$ofthecompletelyintegrablePfaffianequation$Pn−r−1$,thatis$ωn−r≡0$,bysolvingthesystemofhomogeneouslinearfirstorderPDEsgivenby

dI_{n-r}\wedge \omega_{n-r}=0.

−Wefixanarbitraryconstant$Cn−r∈R$andconsidertheintegralmanifoldof$Pn−r−1$givenbythelevelset$In−r=Cn−r$,whichwillbedenotedby$Σ(Cn−r)$.ThenweobtaintherestrictionofthePfaffiansystems$Pn−r−2$to$Σ(Cn−r)$,

\mathcal{P}{n-r-2}|{\Sigma_{(C_{n-r})}}=\mathcal{S}({\omega_{n-r-1}|{\Sigma)}},\omega_{n-r}|{\Sigma)}}}),

bypullingbackwithasuitableembedding$ιn−r:Vn−1→Σ(Cn−r)$,withdomainanopensubset$Vn−1$of$Rn−1$.Observethanboth$ιn−r$and$Vn−1$dependontheconstant$Cn−r$.−TheintegralmanifoldsoftherestrictedPfaffiansystemgiverisetointegralmanifoldsoftheoriginal$Pn−r−2$aftercompositionwiththeembedding$ιn−r$.Butnowobservethat$Pn−r−2|Σ(Cn−r)$isanewPfaffianequation(since$ωn−r|Σ(Cn−r)=0$)definedonaspaceofonelessvariable.Soweproceedasinstep2bysolving$ωn−r−1|Σ(Cn−r)≡0$,andusingasolutiontodefinethelevelsets$Σ(Cn−r−1,Cn−r)$with$Cn−r−1∈R$.−IneveryiterationwesolvethePfaffianequation

\omega_{i}|{\Sigma,\ldots,C_{n-r})}}\equiv 0.

Asolution$Ii$givesrisetolevelsets

\Sigma_{(C_{i},C_{i+1},\ldots,C_{n-r})}={p\in \Sigma_{(C_{i+1},\ldots,C_{n-r})}:I_{i}(p)=C_{i}}, \text{ with } C_i\in \mathbb R,

which are integral manifolds of the restriction of the Pfaffian system $\mathcal{P}_{i-1}$ to $\Sigma_{(C_{i+1},\ldots,C_{n-r})}$. The embedding of $\Sigma_{(C_{i},C_{i+2},\ldots,C_{n-r})}$ 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

u_m=\phi(x,u,u_1,\ldots,u_{m-1}),

where$ϕ$isasmoothfunctiondefinedonanopensubset$U$ofthejetspace$Jm−1(R,R)$,withcoordinates$(x,u,u1,…,um−1)$.Wecanassociatetothisequationthe(trivially)involutivedistribution$S({Z})$,generatedbythevectorfield

Z=\partial_x+u_1\partial_u+\cdots+\phi \partial_{u_{m-1}}.

TheintegralmanifoldsofthisdistributioncorrespondtotheprolongationofsolutionsoftheODE.The$C∞$−structure−basedmethodofintegrationallowsustofindtheseintegralmanifoldsbylookingfora$C∞$−structurefortheequation.Themainresultconcerning$C∞$−structuresisthattheycanbeusedtointegratetheODEbymeansoftheresolutionof$m$Pfaffianequationswhicharecompletelyintegrable(see\cite[Theorem4.1]pancinf−sym).Thisintegrationcanbeperformedasfollows:Thisintegrationcanbeperformedasfollows:1.Considerthestandardvolumeform∗∗Ω∗∗on$U$:

\boldsymbol{\Omega} = dx \wedge du \wedge du_1 \wedge \cdots \wedge du_{m-1}.

Weintroducethe1−forms:

\omega_i = {X_{m},\lrcorner,\ldots,\lrcorner,\widehat{X_i},\lrcorner,\ldots,\lrcorner,X_{1},\lrcorner,A,\lrcorner,\boldsymbol{\Omega}}, \quad 1\leq i\leq m,

where$Xi^$indicatesomissionof$Xi$and$⌟$denotestheinteriorproduct.2.ThePfaffianequation$ωm≡0$iscompletelyintegrable[1,2,3].Afirstintegral$Im=Im(x,u,u1,…,um−1)$canbefoundbysolvingthesystemoflinearhomogeneousfirst−orderPDEsarisingfromthecondition$dIm∧ωm=0$.Thelevelsets$Σ(Cm)$,implicitlydefinedby$Im(x,u,u1,…,um−1)=Cm$,where$Cm∈R$,arethesolutionsto$ωm≡0$.3.Givenalocalparametrization$ιm$of$Σ(Cm)$,definedonanopensubset$Vm⊆Rm$,wecalculate:

\omega_i|{\Sigma{(C_m)}} := \iota_m^*(\omega_i), \quad 1\leq i\leq m.

WerepeattheprocessforthePfaffianequation$ωm−1|Σ(Cm)≡0$,whichiscompletelyintegrable.Similarly,wedenote:

\omega_{i}|{\Sigma,C_m)}} := \iota_{m-1}^*(\omega_{i}|{\Sigma{(C_m)}}), \quad 1\leq i\leq m,

andweproceedwith$ωm−2|Σ(Cm−1,Cm)≡0$,andsoon.4.Ateachstep,wesolvethecompletelyintegrablePfaffianequation$ωk|Σ(Ck+1,…,Cm)≡0$,obtainingafirstintegral:

I_k = I_k(x,u,u_1,\ldots,u_{k-1};C_{k+1},\ldots,C_{m}).

Then,alocalparametrization$ιk$,definedonanopensubset$Uk⊆Rk$,ofthelevelset$Σ(Ck,…,Cm)$isfound.Theprocessfinalizeswhen:

I_1 = I_1(x,u;C_{2},\ldots,C_{m})

is obtained. The solutions are implicitly defined by the level sets $\Sigma_{(C_1,\ldots,C_m)}$. ### References 1. Pancinf-Sym 2. Pancinf-Struct 3. Pan23Integration