REDUCTION OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS { VESSIOT
Transcript of REDUCTION OF SYSTEMS OF PARTIAL DIFFERENTIAL EQUATIONS { VESSIOT
REDUCTION OFSYSTEMS OF PARTIAL
DIFFERENTIALEQUATIONS –
VESSIOT’S METHODREVISITED
JuhaPohjanpelto
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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
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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
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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.
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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).
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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).
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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.