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


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.


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 .


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.)


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.


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!


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.



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.



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.


Construction of lower-degree conservation laws

Definition

A lower-degree conservation law in Rn: degree r < n − 1.(Degree n: divergence-type).


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.


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 .


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!


Some Applications and Examples


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)


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)


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)



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.)



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)


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)



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.



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)



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.


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).



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)



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.


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].


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.


nonlocally related pde systems in multi-dimensions...

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