REDUCTION OFSYSTEMS OF PARTIAL

DIFFERENTIALEQUATIONS –

VESSIOT’S METHODREVISITED

JuhaPohjanpelto

1

What is New?

• Direct constructive algorithms for:

• Invariant Maurer–Cartan forms

• Structure equations

• Moving frames

• Differential invariants

• Invariant differential forms

• Invariant differential operators

• Recurrence formulas

• Constructive Basis Theorem for differential invariants

• Constructive Syzygy Theorem for differential invariants

2

Further applications:

• Symmetry groups of differential equations

• Vessiot method of group splitting

• Congruence of curves, surfaces, etc. in homogeneous spaces

• Invariant variational bicomplex:

• Calculus of variations

• Gauge theories

• Riemannian submanifolds

• Characteristic classes of foliations, Gelfand-Fuks coho-

mology

3

Diffeomorphism Pseudogroup

Mm m dimensional manifold

D = D(M) pseudogroup of local diffeomorphisms of M

Dn ⊂ Jn(M,M) bundle of nth order jets, 0 ≤ n ≤ ∞

Coordinates on Dn:

gn = jnz ϕ = (za, Zb, Zb

c1, Zb

c1c2, . . . , Zb

c1c2···cn),

where za, Zb are local coordinates of M about the source

and the target, and Zbc1

, Zbc1c2

, . . . stand for the derivative

variables.

4

Pseudogroups

G ⊂ D is a pseudogroup if

1. id ∈ G;

2. ϕ, ψ ∈ G ⇒ ϕ ψ ∈ G where defined;

3. ϕ ∈ G ⇒ ϕ−1 ∈ G.

G is a Lie (or continuous) pseudogroup if, in addition, for all

n ≥ N ,

4. Gn ⊂ Dn is a subbundle;

5. ϕ ∈ G ⇐⇒ jnz ϕ ∈ Gn;

6. GN+k = prk GN , k ≥ 1.

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Examples of Lie pseudogroups.

1. Symmetry groups of Euler, Navier-Stokes, boundary layer,

quasi-geostrophic equations and various other equations

arising in fluid mechanics, magnetohydrodynamics, mete-

orology and geophysics.

2. Symmetry groups in gauge and field theories – Maxwell,

Yang-Mills, conformal field theories, general relativity. Cur-

rent/loop groups.

3. Symmetry groups of integrable equations in 2+1 dimen-

sions – KP, Davey-Stewartson, and their variants.

4. Canonical transformations in Hamiltonian mechanics.

5. Configuration spaces:

a) Diff(Ω) → compressible fluid flow

b) Diffvol(Ω) → incompressible fluid flow

c) Canonical transformations → Maxwell-Vlasov

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6. Transformations preserving a geometric structure:

a) Foliations

b) Symplectic/Poisson structures

c) Contact structures (quantomorphisms)

d) Complex manifolds/real hypersurfaces

e) G-structures

7. Image recognition – shape representation.

8. Finite dimensional Lie group actions.

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Infinitesimal generators

A local vector field v ∈ X (M) is a G vector field, v ∈ g, if

the flow Φvt ∈ G for all fixed t on some interval about 0.

Let Gn be given locally by Fα(z, Z(n)) = 0. Then a G vector

field v satisfies

Fα(z, Φvt

(n)) = 0 =⇒ Lα(z, jnz v) = 0.

These are the infinitesimal determining equations for G.

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Maurer-Cartan forms for G∞

G acts on Gn, n ≥ 0, from both left and right by

Lψjnz ϕ = jn

z (ψ ϕ), Rψjnz ϕ = jn

ψ−1(z)(ϕ ψ).

Horizontal forms: dza

Contact forms: θbc1···cp

= dZac1···cp

−∑mi Za

c1···cpcp+1dzcp+1

Maurer-Cartan forms are represented by G-invariant contact

forms on G∞.

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Construction of Maurer-Cartan forms on D∞:

The target coordinate Zb invariant under Rψ =⇒

ωb =∑

c

Zbcdzc, µb = dZb − Zb

cdzc,

are also invariant under Rψ.

Operators of invariant differentiation:

DZa = W baDzb , where

Dzb =∂

∂zb+

∑p≥0 Zc

c1···cpb

∂

∂Zcc1···cp

and W = (∇Z)−1.

Right invariant coframe on D∞:

ωa, µab1···bp

= LDZb1

· · · LDZ

bpµa, p ≥ 0.

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Example: M = R. As a coordinate space

D∞(R) = z, Z, Zz, Zzz, . . . , Zzn , . . . .

Now

Dz =∂

∂z+ Zz

∂

∂Z+ Zzz

∂

∂Zz+ · · · .

Basic right invariant horizontal form ω = dHZ = Zzdz.

The dual total differentiation DZ =1ZzDz commutes with

the group action.

Right invariant Maurer-Cartan forms:

µ = θ, µZ = LDZµ = (Zz)−1θz,

µZZ = L2DZ

µ = (Zz)−3(Zzθzz − Zzzθz), . . . .

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Invariant coframe for G∞: Simply restrict ωi, µaJ as con-

structed on D∞ to G∞.

Relations: On G∞, the Maurer-Cartan forms µaJ satisfy the

right invariant infinitesimal determining equations

Lα(Z, µaJ ) = 0.

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Extended Jet Bundles

Jn = Jn(M) = n-jets of p-dimensional submanifolds of M.

Local coordinates on Jn: zn = (xi, uα, uαi1

, . . . , uαi1···in

).

D acts on Jn through its action on submanifolds, and this

action factors into an action of Dn (and Gn) on Jn.

Moving Frames

A local moving frame of order n is a G-equivariant mapping

ρn : V → Gn, V ⊂ Jn,

preserving base points.

Hence

ρn(ϕ · zn) = Rϕ−1ρ(zn), ϕ ∈ G.

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Existence of moving frames.

Isotropy subgroup at zn:

Inzn = gn ∈ Gn

z | gn · zn = zn.

G acts freely at zn if Inzn = idn

z and locally freely at zn if

Inzn is a discrete subgroup of Gn

z .

Theorem. A local moving frame of order n exists in a neigh-

borhood of zn ∈ Jn(M) if and only if G acts locally freely at

zn.

Theorem. If Gn acts (locally) freely at zn ∈ Jn, then Gl acts

(locally) freely at any zl ∈ J l with πln(zl) = zn, for l > n.

Construction: Choose a cross-section K for the action of

G on Jn. Define ρ(zn) by the condition ρ(zn) · zn ∈ K.

By the equivariance of ρ and the definition of a cross section,

the components of κ(zn) = ρ(zn) · zn contain a complete set

of local differential invariants for the action of G on Jn.

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This defines the process of invariantization; write

Hi = ι(xi), IαJ = ι(uα

J )

for the xi and uαJ components of κ.

Invariantization can be extended to include forms =⇒existence of an invariant local coframe on J∞.

In particular, this leads to a full set of invariant total deriv-

ative operators on J∞. Now differential invariants can be

constructed either by the means of invariantization or by the

means of computing invariant derivatives of known invari-

ants.

The recurrence formulas intertwine the two types of invari-

ants.

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Recurrence formulas.

These are based on the structure equations: Formally

dι(ω) = ι(dω + Lprvω). (*)(Write Lprvω = ζa

,b1···brωb1···br

a . Then

ι(Lprvω) = ρ∗(µab1···br

) ∧ ι(ωb1···bra ).

)

The invariant Maurer-Cartan forms µab1···br

above are subject

to the right-invariant IDE for G. This can be exploited in

analyzing the structure of invariant objects for G.

Fix a coordinate cross section. Write

Hi = ι(xi), Iαj1···jr

= ι(uαj1···jr

)

for the normalized differential invariants. Call an invariant

phantom if it is a constant.

Let

ωi = πHι(dxi),

βi = πHι(ξi), ψαj1···jr

= πHι(Lprvuαj1···jr

),

and let Di be total differential operators dual to ωi.

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The horizontal component of (*) yields

(DjHi)ωj = ωi + βi,

(Djr+1Iαj1···jr

)ωjr+1 = Iαj1···jrjr+1

ωjr+1 + ψαj1···jr

.

The above equations for phantom invariants can be solved for

the independent horizontal pulled-back Maurer-Cartan forms!

Substitute the expression for these into the above equations

for non-phantom invariants to derive the recurrence formulas

DjHi = δi

j + P ij ,

Djr+1Iαj1···jr

= Iαj1···jrjr+1

+ Mαj1···jr,jr+1

.

Note: only a cross section and the infinitesimal determining

equations are used!

Complication: The correction terms Mαj1···jr,jr+1

may be of

the same order as Iαj1···jrjr+1

.

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One can resolve this issue by employing an alternate basis

for differential invariants:

L — Invariantized infinitesimal determining equations.

p — Invariantized prolongation map.

Z = (p∗)−1(L) — Invariantized annihilator subbundle.

Identify elements of Z with polynomial functions by

duαj1···jr

⇐⇒ sj1 · · · sjrSα.

U — Leading order terms of elements in Z.

The terms in U of degree higher than the order of freeness of

the pseudogroup form a submodule J .

Associate the differential invariant

Iσ =∑

hJα(I(1))Iα

J

to σ =∑

hJα(I(1))sJSα ∈ J .

Theorem. A basis for local differential invariants is given

by

(i) Non-phantom differential invariants IαJ , |J | ≤ n∗, and

(ii) Iσν , where σν forms a Grobner basis for J .

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Syzygies.

Write I1, . . . ,It for the generating set of differential invariants

in the basis theorem. A syzygy is a non-trivial function Z =

Z(w1, . . . , wj,K , . . . ) with

Z(I1, . . . ,DKIj , . . . ) ≡ 0,

where K is an ordered multi-index.

Syzygies can be divided into 3 classes:

(1) Those arising from the commutator relations [Di,Dj ] =∑p

k=1 Y kijDk.

(2) Those obtained by invariantly differentiating low-order

basis invariants of type (i).

(3) Those arising from the algebraic syzygies amongst the

Grobner basis elements σν for J .

Theorem. Any syzygy amongst the basis invariants is a dif-

ferential consequence of the the above three types of syzygies.

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Exterior Differential Systems.

The exterior algebra Ω∗(M) of M consists of differential forms

on M , that is, fields of multilinear, skew-symmetric mappings

from the tangent space of M into the reals.

An exterior differential system I is a differential ideal of the

exterior algebra.

An integral manifold of I is an immersion σ : P → M so that

σ∗(I) = 0.

Any sufficiently regular system ∆ = 0 of PDEs can be asso-

ciated with an exterior differential system I∆ on some Jn so

that solutions of ∆ = 0 correspond to the integral manifolds

of I∆.

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Example: ∆ = y′′ − F (x, y, y′) = 0.

Underlying jet space: J2 = (x, y, y1, y2).Contact forms: θ = dy − y1dx, θ1 = dy1 − y2dx.

Then a mapping σ : R → J2, σ(x) = (x, η(x), η1(x), η2(x)),

is the prolongation of the function y = η(x) if and only if

σ∗(θ) = 0, σ∗(θ1) = 0.

(For example,

0 = σ∗(θ) = dη(x)− η1(x)dx = (η′(x)− η1(x))dx

=⇒ η1(x) = η′(x).

Similarly, 0 = σ∗(θ1) = 0 =⇒ η2(x) = η′′(x).)

Then

M = (x, y, y1, y2 = F (x, y, y1)) ⊂ J2,

I∆ =< θ, θ1, dθ, dθ1 >|M .

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An exterior differential system I on Jn is invariant under the

action of a pseudogroup G if prolonged pseudogroup trans-

formations preserve I.

If G is a symmetry group of ∆ = 0, then I∆ is invariant under

G.

Reduction process:

Let K ⊂ Jn be a cross section to the action of G on Jn with

the associated moving frame ρ, and let I be a G-invariant

exterior differential system.

Reduced exterior differential system on K:

I = ω ∈ Ω∗(K) | (τ ρ)∗(ω) ∈ I,

where τ ρ(z(n)) = ρ(z(n)) · z(n) ∈ K.

Key fact: If σ is an integral manifold of I, then, under suit-

able regularity assumptions, σ = τ ρ σ is one for I.

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I can be computed algebraically!

Algebra of semi-basic forms:

Isb = ω ∈ I |v ω = 0 for all v ∈ prn g.

=⇒I = Isb|K.

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Reconstruction problem:

Since σ(p), σ(p) are on the same orbit of the action of G,

there is a jet gn(p) ∈ Gn so that

σ(p) = gn(p) · σ(p), p ∈ P.

Consequently, every integral manifold of I can be recovered

from the integral manifolds of I by finding all mappings

γ : P → Gn

so that the composition σ = γ · σ is an integral manifold of

I.

Conclusion: we have found an two-step algorithm for con-

structing analytic solutions for systems of PDEs admitting a

symmetry pseudogroup.

Step #1 consists of finding solutions of a reduced system

defined on a lower dimensional space;

Step #2, the reconstruction step, involves solving a system

of differential equations for the pseudogroup jets.

Any solution of the original equation can in principle be found

in this manner!

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Example:

Nonlinear wave equation:

uxy − u−1uxuy = u2

Symmetry group: X = f(x), Y = y, U =u

f ′(x),

where f(x) is a locally defined, smooth invertible function.

Parametrize the pseudogroup jet bundles Gn by

f = f(x), fx = f ′(x), fxx = f ′′(x), . . . , fxn+1 = f (n+1)(x).

Lifted invariant total derivative operators:

DX =1fx

Dx, DY = Dy.

Prolonged action of Gn on Jn:

X = f, Y = y, U =u

fx, UX =

ux

f2x

− ufxx

f3x

,

UY =uy

fx, UXX =

uxx

f3x

− 3uxfxx

f4x

− ufxxx

f4x

+3uf2

xx

f5x

,

UXY =uxy

f2x

− uyfxx

f3x

, UY Y =uyy

fx, . . . .

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Cross section K:

x = 0, u = 1, ux = 0, uxx = 0, . . . , uxn = 0.

Noormalization equations:

X = 0, U = 1, UX = 0, UXX = 0, . . . , UXn = 0.

=⇒

f = 0, fx = u, fxx = ux, . . . , fxn+1 = uxn .

Differential invariants:

I = y, J =uy

u, K =

uuxy − uxuy

u3, L =

uyy

u, . . . .

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Solutions of the equation uxy − u−1uxuy = u2 correspond to

the 2-dimensional integral manifolds of the exterior differen-

tial system I on J2 generated by the contact forms

θ = du− uxdx− uydy, θx = dux − uxxdx− uxydy,

θy = duy − uxydx− uyydy, dθ, dθx dθy,

and the 1-form dK.

Infinitesimal generators g of the group action:

v = a(x)∂

∂x− a′(x)u

∂

∂u,

where a(x) is an arbitrary smooth function.

Prolong:

pr2 v = a(x)∂

∂x− a′(x)u

∂

∂u− (a′′(x)u + 2a′(x)ux)

∂

∂ux

− a′(x)uy∂

∂uy− (a′′′(x)u + 3a′′(x)ux + 3a′(x)uxx)

∂

∂uxx

− (a′′(x)uy + 2a′(x)uxy)∂

∂uxy− a′(x)uyy

∂

∂uyy.

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Semibasic forms: pr2 v ω = 0 for all v ∈ g iff

Xi ω = 0,

where

X0 =∂

∂x,

X1 = −u∂

∂u− 2ux

∂

∂ux− uy

∂

∂uy

− 3uxx∂

∂uxx− 2uxy

∂

∂uxy− uyy

∂

∂uyy

,

X2 = −u∂

∂ux− 3ux

∂

∂uxx− uy

∂

∂uxy,

X3 =∂

∂uxx.

=⇒ Isb =< dK > =⇒ I =< dK >.

Integral manifolds σ(s, t) of I are given by

I = I(s, t), J = J(s, t), K = κ, L = L(s, t).

28

Reconstruction: Find group parameters

f = f(s, t), fx = fx(s, t), fxx = fxx(s, t), fxxx = fxxx(s, t),

so that

x = f, y = I, u =1fx

, ux = −fxx

f3x

, uy =J

fx,

uxx =fxxx

f4x

+ 3f2

xx

f5x

, uxy =κ

f2x

− Jfxx

f3x

, uyy =L

fx,

is an integral manifold of I, that is, a solution of the nonlinear

wave equation. This leads to the system

JdI − fxx

f2x

df +1fx

dfx = 0,

(L− J2) dI + dJ − κ

fxdf = 0,

(2Jfxx + κfx) dI − fxxx

fxdf + dfxx = 0.

Assume

I = s, J = t, K = κ, L = L(s, t).

29

Then

fx = κ∂f

∂t, L = t2 +

κ

fx

∂f

∂s,

∂f

∂s+ (

t2

2+ g(s))

∂f

∂t= 0,

and where g(s) is arbitrary. Consequently, solutions to the

nonlinear wave equation can be constructed from solutions

of a generalized Riccati equation!

With g(s) = 0, g(s) = α2, g(s) = −α2, where α is a constant,

we obtain the respective familes of solutions

u(x, y) = −2G′(x)

(G(x)− y)2,

u(x, y) = −α2

2sec2(

α

2(y −G(x))) G′(x),

u(x, y) = −4α2 e√

2α(y−G(x))

(e√

2α(y−G(x)) − 1)2G′(x),

where G(x) is arbitrary.