Nonlocally Related PDE Systems in Multi-dimensions...
Transcript of Nonlocally Related PDE Systems in Multi-dimensions...
Nonlocally Related PDE Systems in Multi-dimensions:Construction and Applications
Alexei Cheviakov (USask), George Bluman (UBC)
December 4, 2010
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 1 / 19
Outline
1 Nonlocally related PDE systems in two dimensions
2 Nonlocally related PDE systems in Multi-dimensions: introductionDivergence-type conservation laws and resulting potential systemsNonlocally related subsystems in multi-dimensions
3 Conservation laws of degree rConservation laws of degree rConstruction of lower-degree conservation lawsPotential equations
4 Applications and ExamplesA nonlocal symmetry arising from a nonlocally related subsystem in 3DNonlocal symmetries/C.L.’s from a lower-degree potential systemNonlocal symmetries of helical Euler equations of fluid flowNonlocal symmetries of the three-dimensional MHD equilibrium equationsOther applications
5 Conclusions
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 2 / 19
Local conservation laws
Consider a PDE system Rx , t ; u of order k, with m dependent variablesu = (u1, . . . , um) and two independent variables (x1, x2) = (x , t):
Rσ[u] = Rσ(x , t, u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N. (1)
Local conservation laws
DtΨ[u] + DxΦ[u] = 0 (2)
One seeks nontrivial linearly independent conservation laws.
Theorem
Suppose each PDE of the given system is written in a solved form
Rσ[u] = ujσiσ,1...iσ,s
− Gσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N. (3)
Then each local conservation law (2) of the PDE system (1), up to conservation lawequivalence, arises from the characteristic form
DiΦi [U] = Λσ[U]
(U jσ
iσ,1...iσ,s− Gσ[U]
)= 0, (4)
in terms of a set of local multipliers Λσ[U]Nσ=1.Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 3 / 19
Nonlocally related systems in 2D
Local conservation law: DtΨ[u] + DxΦ[u] = 0
Potential system:
Rσ[u] = Rσ(x , t, u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N,vx = Ψ[u],vt = −Φ[u].
(5)
(Redundant equations can be excluded from (5) or kept as needed.)
Combination potential system:
Suppose q linearly independent local conservation laws are known for a given PDEsystem Rx , t ; u. In terms of the resulting potential variables v 1, . . . , vq, the set of allcorresponding 2q − 1 potential systems is called a combination potential system Pv1...vq .
Nonlocally related subsystems:
Exclude dependent variables by cross-differentiation.
All of the above can be useful to obtain, e.g., nonlocal symmetries, nonlocalconservation laws, additional exact solutions.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 4 / 19
Nonlocal conservation laws, trees, nonlocal symetries
Nonlocal conservation laws:
Those not expressible as a linear combination of the local conservation laws of thegiven system.
Nonlocal conservation laws can arise only from multipliers that depend on potentials.
Trees of nonlocally related systems:
Use nonlocal conservation laws to produce further potentials and subsystems.
Nonlocal symmetries:
Symmetries
X = ξS(x , t, u, v)∂
∂x+ τS(x , t, u, v)
∂
∂t+ ηµS (x , t, u, v)
∂
∂uµ+ ζpS (x , t, u, v)
∂
∂vp.
whose infinitesimals (ξS(x , t, u, v), τS(x , t, u, v), ηS(x , t, u, v)) have an essentialdependence on v .
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 5 / 19
Divergence-type conservation laws and resulting potential systems in Rn
Consider a system Rx ; u of N PDEs of order k with n independent variablesx = (x1, . . . , xn) and m dependent variables u(x) = (u1(x), . . ., um(x)), given by
Rσ[u] = Rσ(x , u, ∂u, . . . , ∂ku) = 0, σ = 1, . . . ,N. (6)
Divergence-type conservation laws:
divΦ[u] ≡ DiΦi (x , u, ∂u, . . . , ∂ru) = 0. (7)
(Still follow from multipliers for systems in solved form.)
Potential equations are under-determined:
Γ3y [u]− Γ2
z [u] = Φ1[u],
Γ1z [u]− Γ3
x [u] = Φ2[u],
Γ2x [u]− Γ1
y [u] = Φ3[u].
(8)
Gauge symmetry:Γ[u]→ Γ[u] + gradφ(x , y , z), (9)
No nonlocal symmetries can arise in the presence of a gauge symmetry!(Nonlocal conservation laws can.)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 6 / 19
Divergence-type conservation laws and resulting potential systems
Possible gauges:
divergence (Coulomb) gauge: div Γ ≡ Γ1x + Γ2
y + Γ3z = 0,
spatial gauge: Γk = 0, k = 1 or 2 or 3,
Poincare gauge: xΓ1 + yΓ2 + zΓ3 = 0.
In time-dependent systems:
If one of the coordinates in a given PDE system is time t, special gauges are frequentlyused, such as:
Lorentz gauge (in (2+1) dimensions, x = (t, x , y)): Γ1t − Γ2
x − Γ3y = 0,
Cronstrom gauge (in (2+1) dimensions, x = (t, x , y)): tΓ1 − xΓ2 − yΓ3 = 0.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 7 / 19
Nonlocally related subsystems in multi-dimensions
Euler equations of an inviscid, constant density equilibrium fluid flow in 3D:
v × (curlv) = grad(p
ρ+ 1
2|v|2),
divv = 0.(10)
v = (v 1, v 2, v 3) is the fluid velocity vector and ρ = const the fluid density.The pressure p can be excluded by taking the curl:
curl[v × (curlv)
]= 0,
divv = 0.(11)
The subsystem (11) is equivalent and nonlocally related to the Euler system.
No gauge freedom!
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 8 / 19
Conservation laws of degree r
A differential form of order r (an r -form) is given by
ω(r) =1
r !ωµ1...µr dxµ1 ∧ . . . ∧ dxµr . (12)
Example:
in R3 with coordinates (x , y , z), 0- to 3-forms are given by
ω(0) = f , ω(1) = ω1dx + ω2dy + ω3dz ,
ω(2) = ω12dx ∧ dy + ω23dy ∧ dz + ω31dz ∧ dx , ω(3) = ω123dx ∧ dy ∧ dz ,
where f , ωi , ωij , ωijk are functions of x , y , z .
Note: usual conservation laws in 3D:
Degree two: consider a 2-form
ω(2)[U] = Φ1[U]dy ∧ dz + Φ2[U]dz ∧ dx + Φ3[U]dx ∧ dy . (13)
Thendω(2)[u] = (DxΦ1[u] + DyΦ2[u] + DzΦ3[u])dx ∧ dy ∧ dz = 0,
equivalent to divΦ = 0.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 9 / 19
Conservation laws of degree r
A differential form of order r (an r -form) is given by
ω(r) =1
r !ωµ1...µr dxµ1 ∧ . . . ∧ dxµr . (12)
Example:
in R3 with coordinates (x , y , z), 0- to 3-forms are given by
ω(0) = f , ω(1) = ω1dx + ω2dy + ω3dz ,
ω(2) = ω12dx ∧ dy + ω23dy ∧ dz + ω31dz ∧ dx , ω(3) = ω123dx ∧ dy ∧ dz ,
where f , ωi , ωij , ωijk are functions of x , y , z .
Note: usual conservation laws in 3D:
Degree three: consider a 1-form
ω(1)[U] = ω1[U]dx + ω2[U]dy + ω3[U]dz , (13)
Then dω(1)[u] = 0⇔ curlΦ = 0:
Dyω3[u]−Dzω2[u] = 0, Dzω1[u]−Dxω3[u] = 0, Dxω2[u]−Dyω1[u] = 0.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 9 / 19
Conservation laws of degree r
Definition
A conservation law of degree r (1 ≤ r ≤ n − 1) is given by an r -form ω(r)[U], such thatits exterior derivative
Ω(r+1)[u] = dω(r)[u] = 0 (14)
on all solutions U = u of a given PDE system Rx ; u:
Ωνµ1...µr [u]dxν ∧dxµ1 ∧ . . .∧dxµr ≡(
∂
∂xνωµ1...µr [u]
)dxν ∧dxµ1 ∧ . . .∧dxµr = 0. (15)
Example: conservation law of degree one in R4
Consider a 1-form ω(1)[U] = ω1[U]dx + ω2[U]dy + ω3[U]dz + ω4[U]dw ,
Conservation law: six divergence expressions, components of
dω(1)[u] = (Dxω2[u]−Dyω1[u])dx ∧ dy + (Dxω3[u]−Dzω1[u])dx ∧ dz+(Dxω4[u]−Dwω1[u])dx ∧ dw + (Dyω3[u]−Dzω2[u])dy ∧ dz+(Dyω4[u]−Dwω2[u])dy ∧ dw + (Dzω4[u]−Dwω3[u])dz ∧ dz = 0.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 10 / 19
Construction of lower-degree conservation laws
Definition
A lower-degree conservation law in Rn: degree r < n − 1.(Degree n: divergence-type).
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 11 / 19
Construction of lower-degree conservation laws
Construction:
Need: Ω(r+1)[u] = dω(r)[u] = 0.
Each component: still a divergence expression.
Seek in the form
Ωµ1...µr+1 [U] = Λ(i)σ [U]Rσ[U], i = 1, . . . ,
( nr + 1
). (16)
Determining equations:(∂
∂xλ(Λ(i)σ [U]Rσ[U])
)dxλ ∧ dxµ1 ∧ . . . ∧ dxµr+1 ≡ 0. (17)
For 1 ≤ r ≤ n − 2, determining equations have the form (17).
For r = n − 1, determining equations involve Euler operators.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 11 / 19
Potential equations
Conservation law of degree r : dω(r)[u] = 0.
Hence locally,ω(r)[u] = dω(r−1)[u]
for some (r − 1)-form ω(r−1)[u].
Potential equations
Can show: potential equations can be explicitly written as
ωµ1...µr [u] =r∑
i=1
(−1)i−1 ∂
∂xµiωµ1...µi ...µr [u]. (18)
... Obtain a potential system of degree r .
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 12 / 19
Potential equations
Table 1: Numbers of conservation law components, r potential equations and potentialvariables, for a conservation law (15) of degree r (1 ≤ r ≤ n − 1).
CL degree # of CL components # of potential equations # of potential variables
r (divergence expressions) (= # of CL fluxes)
1n(n − 1)
2n 1
2n(n − 1)(n − 2)
6
n(n − 1)
2n
.
.
....
.
.
....
r
(n
r + 1
) (n
r
) (n
r − 1
)...
.
.
....
.
.
.
n − 2 nn(n − 1)
2
n(n − 1)(n − 2)
6
n − 1 1 nn(n − 1)
2
Only potential systems of degree 1 are determined!
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 12 / 19
Some Applications and Examples
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 13 / 19
A nonlocal symmetry arising from a nonlocally related subsystem in 3D
Consider the PDE systemvt = grad u,
ut = K(|v|) divv.(18)
A nonlocally related subsystem:
vtt = grad [K(|v|)divv] . (19)
Consider the one-parameter class of constitutive functions
K(|v|) = |v|2m =(
(v 1)2 + (v 2)2)m. (20)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 14 / 19
A nonlocal symmetry arising from a nonlocally related subsystem in 3D
Symmetries of the given system:
X1 =∂
∂t, X2 =
∂
∂x, X3 =
∂
∂y, X4 =
∂
∂u,
X5 = t∂
∂t+ x
∂
∂x+ y
∂
∂y,
X6 = −y∂
∂x+ x
∂
∂y− v 2 ∂
∂v 1+ v 1 ∂
∂v 2,
X7 = m
(x∂
∂x+ y
∂
∂y
)+ (m + 1)u
∂
∂u+ v 1 ∂
∂v 1+ v 2 ∂
∂v 2.
Symmetries of the subsystem:
Y1 = X1, Y2 = X2, Y3 = X3, Y4 = X5, Y5 = X6,
Y6 = m
(x∂
∂x+ y
∂
∂y
)+ v 1 ∂
∂v 1+ v 2 ∂
∂v 2.
In the case m = −2, the subsystem has an additional symmetry(nonlocal symmetry of the given system)
Y7 = t2 ∂
∂t+ tv 1 ∂
∂v 1+ tv 2 ∂
∂v 2. (18)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 14 / 19
Nonlocal symmetries/C.L.’s from a lower-degree potential system
Consider a PDE system in R3: (x , y , z), given by
curl(
K(|h|)(curlh)× h)
= 0, div h = 0. (19)
LetK(|h|) = |h|2m ≡
((h1)2 + (h2)2 + (h3)2
)m,
where m is a parameter.
A potential system (degree 1):
K(|h|)(curlh)× h = gradw , div h = 0 (20)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 15 / 19
Nonlocal symmetries/C.L.’s from a lower-degree potential system
Symmetries:
1 Given system, arbitrary m 6= −1:
X1 =∂
∂x, X2 =
∂
∂y, X3 =
∂
∂z, X4 = x
∂
∂x+ z
∂
∂z+ y
∂
∂y,
X5 = −z∂
∂x+ x
∂
∂z− h3 ∂
∂h1+ h1 ∂
∂h3, X6 = y
∂
∂x− x
∂
∂y+ h2 ∂
∂h1− h1 ∂
∂h2,
X7 = z∂
∂y− y
∂
∂z+ h3 ∂
∂h2− h2 ∂
∂h3, X8 = h1 ∂
∂h1+ h2 ∂
∂h2+ h3 ∂
∂h3,
2 Potential system, arbitrary m 6= −1:
Yi = Xi , i = 1, . . . , 7; Y8 = X8 + 2(m + 1)w∂
∂w, Y9 =
∂
∂w.
3 Potential system, m = −1:
Y∞ = F (w)( ∂
∂w+ h1 ∂
∂h1+ h2 ∂
∂h2+ h3 ∂
∂h3
).
(Infinite nonlocal symmetries.)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 15 / 19
Nonlocal symmetries/C.L.’s from a lower-degree potential system
Conservation laws:
Potential system: in terms of an arbitrary function G(W ), an infinite family ofmultipliers:
Λi = H iG ′(W ), σ = 1, 2, 3; Λ4 = G(W ),
The corresponding divergence-type conservation laws:
3∑i=1
∂
∂x i[(G(w) + 2(m + 1)wG ′(w))hi ] = 0. (19)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 15 / 19
Nonlocal symmetries of helical Euler equations of fluid flow
Consider the Euler equations describing the motion for an incompressible inviscid fluid inR3, which in Cartesian coordinates are given by
divu = 0, (20a)
ut + (u · ∇)u + grad p = 0, (20b)
where the fluid velocity vector u = u1ex + u2ey + u3ez and fluid pressure p are functionsof x , y , z , t.
The fluid vorticity is a local vector variable defined by
ω = curlu. (21)
The vorticity subsystem (nonlocally related):
divu = 0, (22a)
ωt + curl (ω × u) = 0, (22b)
ω = curlu. (22c)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 16 / 19
Nonlocal symmetries of helical Euler equations of fluid flow
Helical reduction:Helical coordinates (r , η, ξ) in R3:
ξ = az + bϕ, η = aϕ− bz/r 2, a, b = const, a2 + b2 > 0.
u = urer + uηeη + uξeξ, ω = ωrer + ωηeη + ωξeξ,
Euler equations:ur
r+∂ur
∂r+
1
B(r)
∂uξ
∂ξ= 0, (20a)
(ur )t + ur (ur )r +1
B(r)uξ(ur )ξ −
B2(r)
r
(b
ruξ + auη
)2
+ pr = 0, (20b)
(uη)t + ur (uη)r +1
B(r)uξ(uη)ξ +
a2B2(r)
ruruη = 0, (20c)
(uξ)t + ur (uξ)r +1
B(r)uξ(uξ)ξ +
2abB2(r)
r 2uruη +
b2B2(r)
r 3uruξ +
1
B(r)pξ = 0. (20d)
B(r) =r√
a2r 2 + b2.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 16 / 19
Nonlocal symmetries of helical Euler equations of fluid flow
Vorticity equations:
ωr = − (uη)ξB(r)
, (20a)
ωη = −1
r
∂
∂r(ruξ)− 2
abB2(r)
r 2uη +
a2B2(r)
ruξ +
1
B(r)(ur )ξ, (20b)
ωξ =a2B2(r)
ruη + (uη)r . (20c)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 16 / 19
Nonlocal symmetries of helical Euler equations of fluid flow
Symmetries of the Euler system:
X1 =∂
∂t, X2 =
∂
∂ξ,
X3 = t∂
∂t− ur ∂
∂ur− uη
∂
∂uη− uξ
∂
∂uξ− 2p
∂
∂p− ωr ∂
∂ωr− ωη ∂
∂ωη− ωξ ∂
∂ωξ,
X4 = t∂
∂ξ− bB(r)
ar
∂
∂uη+ B(r)
∂
∂uξ,
X5 = F (t)∂
∂p,
(20)
Symmetries of the Vorticity subsystem:
Y1 = X1, Y2 = X3 − ωr ∂
∂ωr− ωη ∂
∂ωη− ωξ ∂
∂ωξ,
Y3 = G(t)∂
∂ξ− bB(r)
arG ′(t)
∂
∂uη+ B(r)G ′(t)
∂
∂uξ.
(21)
The full Galilei group in the direction of ξ: a nonlocal symmetry of the helicallysymmetrically reduced Euler system.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 16 / 19
Nonlocal symmetries of the three-dimensional MHD equilibrium equations
The PDE system of ideal magnetohydrodynamics (MHD) equilibrium equations in threespace dimensions given by
div(ρv) = 0, divb = 0, (22a)
ρv × curlv − b× curl b− grad p − 12ρ grad |v|2 = 0, (22b)
curl v × b = 0. (22c)
Dependent variables: plasma density ρ, plasma velocity v = (v 1, v 2, v 3), pressure p andmagnetic field b = (b1, b2, b3).Independent variables: spatial coordinates (x , y , z).
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 17 / 19
Nonlocal symmetries of the three-dimensional MHD equilibrium equations
A simplified example: ρ = 1.
Given system:divv = 0, divb = 0, (22a)
v × curlv − b× curl b− grad p − 12grad |v|2 = 0, (22b)
curl (v × b) = 0. (22c)
Potential system of degree 1:
divv = 0, divb = 0, v × b = gradψ, (23a)
v × curlv − b× curl b− grad p − 12grad |v|2 = 0. (23b)
Infinite nonlocal symmetries:
X∞ = M(ψ)
(v i ∂
∂bi+ bi ∂
∂v i− (b · v)
∂
∂p
). (24)
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 17 / 19
Nonlocal symmetries of the three-dimensional MHD equilibrium equations
Generalization for the given MHD system
div(ρv) = 0, divb = 0,
ρv × curlv − b× curl b− grad p − 12ρ grad |v|2 = 0,
curl v × b = 0 :
Infinite interchange/scaling symmetries [Bogoyavlenskij, 2000].
x ′ = x , y ′ = y , z ′ = z ,
B′ = b(Ψ)B + c√ρV, V′ =
c(Ψ)
a(Ψ)√ρ
B +b(Ψ)
a(Ψ)V,
ρ′ = a2(Ψ)ρ,P ′ = CP + 12
(C |B|2 − |B′|2
).
(23)
Here a(Ψ), b(Ψ) are arbitrary functions constant on magnetic surfaces Ψ = const, andb2(Ψ)− c2(Ψ) = C = const.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 17 / 19
Other applications
Other important results following from nonlocally related PDE systems inmulti-dimensions include:
Nonlocal symmetries /conservation laws of the linear wave equation and Maxwell’sequations in (2+1) and (3+1) dimensions [Bluman and Anco; Anco and The].
Additional nonlocal conservation laws of Maxwell’s equations arising from algebraicand divergence gauges [C. and Bluman].
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 18 / 19
Some conclusions
Nonlocal symmetries need determined nonlocally related systems; nonlocalconservation laws don’t.
Subsystems are always determined.
Lower-degree conservation laws can be constructed for multi-D PDE systems in analgorithmic manner.
Lower-degree potential systems can be less underdetermined or even determined.
Nonlocal symmetries and conservation laws have been found for multi-dimensionalnonlinear and linear PDE systems.
How does one choose useful gauges in underdetermined potential systems?
More examples of use of lower-degree CLs are needed...
Main references:
A.C.& G.Bluman, Nonlocally Related Multi-dimensional PDE Systems. Parts I,II.JMP, 2010.
Alexei Cheviakov (USask), George Bluman (UBC) () Nonlocal PDE Systems in Multi-dimensions December 4, 2010 19 / 19