Symbolic Computation of
Conservation Laws of Nonlinear
Partial Differential Equations
in Multiple Space Dimensions
Willy Hereman
Department of Mathematical and Computer Sciences
Colorado School of Mines
Golden, Colorado
Symmetry Plus Integrability 2010
South Padre Island, Texas
Monday, June 14, 2010, 11:30a.m.
Acknowledgements
Douglas Poole (Ph.D. Dissertation, CSM)
Mark Hickman (Univ. of Canterbury, New Zealand)
Bernard Deconinck (Univ. of Washington, Seattle)
Several undergraduate and graduate students
Research supported in part by NSF
under Grant No. CCF-0830783
This presentation was made in TeXpower
Outline
•
Why compute conservation laws?
•
The Korteweg-de Vries and Zakharov-Kuznetsov
equations
•
Demonstration of ConservationLawsMD.m
•
Algorithm for computing conservation laws
•
Tools from the calculus of variations
•
Applicaton to the Zakharov-Kuznetsov equation
•
Conclusions and future work
•
Additional examples
I
Manakov-Santini system
I
Camassa-Holm equation in (2+1) dimensions
I
Khoklov-Zabolotskaya equation
I
Shallow water wave model for atmosphere
I
Kadomtsev-Petviashvili equation
I
Potential Kadomtsev-Petviashvili equation
I
Generalized Zakharov-Kuznetsov equation
Conservation Laws for Nonlinear PDEs
•
System of evolution equations of order M
ut = F(u(M ) (x))
with u = (u, v, w, . . .) and x = (x, y, z).
•
Conservation law in (1+1)-dimensions
Dt ρ + D x J = 0
evaluated on the PDE.
Conserved density ρ and flux J.
•
Conservation law in (2+1)-dimensions
D t ρ + ∇ · J = D t ρ + D x J1 + D y J2 = 0
evaluated on the PDE.
Conserved density ρ and flux J = (J1 , J2 ).
•
Conservation law in (3+1)-dimensions
Dt ρ + ∇ · J = Dt ρ + Dx J1 + Dy J2 + Dz J3 = 0
evaluated on the PDE.
Conserved density ρ and flux J = (J1 , J2 , J3 ).
Reasons for Computing Conservation Laws
•
Conservation of physical quantities (linear
momentum, mass, energy, electric charge, ... )
•
Testing of complete integrability and application
of Inverse Scattering Transform
•
Testing of numerical integrators
•
Study of quantitative and qualitative properties
of PDEs (Hamiltonian structure, recursion
operators, . . . )
•
Verify the closure of a model
Examples of PDEs with Conservation Laws
•
Korteweg-de Vries (KdV) equation models shallow
water waves, ion-acoustic waves in plasmas, etc.
∂u
∂u
∂3u
+u
+
=0
3
∂t
∂x
∂x
Diederik Korteweg
or ut + uux + uxxx = 0
Gustav de Vries
•
First six (of infinitely many) conservation laws:
Dt u + Dx 12 u2 + uxx = 0
Dt u2 + Dx 32 u3 − ux 2 + 2uuxx = 0
Dt u3 − 3u2x
+ Dx 43 u4 − 6uu2x + 3u2 uxx + 3u2xx − 6ux uxxx = 0
Dt u4 − 12uu2x +
36 2
u
5 xx
− 18uu2x + 4u3 uxx
2
36 2
72
+12u2x uxx + 96
uu
−24uu
u
−
u
+
u u =0
x
xxx
xx
5
5 xxx
5 xx 4x
+ Dx
4 5
u
5
2
Dt u5 − 30 u2 u2x + 36 uu2xx − 108
u
xxx
7
+ Dx 56 u6 − 40u3 u2x − . . . − 216
uxxx u5x = 0
7
Dt u6 − 60 u3 u2x − 30 u4x + 108 u2 u2xx
3
2
648
216 2
−
+
u
uu
u4x
+ 720
xx
xxx
7
7
7
+ Dx 76 u7 − 75u4 u2x − . . . + 432
u4x u6x = 0
7
•
Third conservation law: Gerald Whitham, 1965
•
Fourth and fifth: Norman Zabusky, 1965-66
Seventh (sixth thru tenth): Robert Miura, 1966
•
Robert Miura
•
Conservation law explicitly dependent on t and x:
Dt tu2 − 2xu
+Dx 23 tu3 − xu2 + 2ux − tu2x + 2tuuxx − 2xuxx = 0
First eleven densities: Control Data Computer
CDC-6600 computer (2.2 seconds)
−→ large integers problem!
Control Data CDC-6600
•
Zakharov-Kuznetsov (ZK) equation
models ion-sound solitons in a low pressure
uniform magnetized plasma
ut + αuux + β(uxx + uyy )x = 0
•
Conservation laws:
Dt u + Dx α2 u2 + βuxx + Dy βuxy = 0
3
2
2
Dt u2 + Dx 2α
u
−
β(u
−
u
x
y ) + 2βu(uxx + uyy )
3
− Dy 2βux uy = 0
More conservation laws (ZK equation):
2
2
2
2
2
3α 4
+
u
)
+
D
+
u
Dt u3 − 3β
(u
u
+
3βu
u
−
6βu(u
x
xx
x
y
x
y)
α
4
2
2
+ 3βα (u2xx − u2yy ) − 6βα (ux (uxxx + uxyy ) + uy (uxxy + uyyy ))
2
+ Dy 3βu2 uxy + 6βα uxy (uxx + uyy ) = 0
3
2
2
Dt tu2 − α2 xu + Dx t( 2α
u
−
β(u
−
u
x
y ) + 2βu(uxx + uyy ))
3
2β
1
− x(u2 + 2β
u
)
+
u
−
D
2β(tu
u
+
xuxy ) = 0
y
x y
α xx
α x
α
Methods for Computing Conservation Laws
•
Use the Lax pair L and A, satifying [L, A] = 0.
If L = Dx + U, A = Dt + V then Vx − Ut + [U, V ] = 0.
L̂ = T LT −1 gives the densities, Â = T AT −1 gives
the fluxes.
•
Use Noether’s theorem (Lagrangian formulation)
to generate conservation laws from symmetries
(Ovsiannikov, Olver, Rosenhaus, and many
others).
•
Integrating factor methods (Anderson, Bluman,
Anco, Cheviakov, Wolf, etc.) require solving
ODEs (or PDEs).
Proposed Algorithmic Method
•
Density is linear combination of scaling invariant
terms (in the jet space) with undetermined
coefficients.
•
Compute Dt ρ with total derivative operator.
•
Use variational derivative (Euler operator) to
express exactness.
•
Solve a (parametrized) linear system to find the
undetermined coefficients.
•
Use the homotopy operator to compute the flux
(invert Dx or Div).
•
Work with linearly independent pieces in finite
dimensional spaces.
•
Use linear algebra, calculus, and variational
calculus (algorithmic).
•
Implement the algorithm in Mathematica.
Software Demonstration
Notation – Computations on the Jet Space
•
Independent variables x = (x, y, z)
•
Dependent variables u = (u(1) , u(2) , . . . , u(j) , . . . , u(N ) )
In examples: u = (u, v, θ, h, . . .)
•
∂k u
Partial derivatives ukx = ∂xk ,
∂5u
Examples: uxxxxx = u5x = ∂x5
∂6u
uxx yyyy = u2x4y = ∂x2 y4
•
u(M ) represents all components of u and all its
partial derivatives up to order M.
•
Differential functions
Example: f = uvvx + x2 u3x vx + ux vxx
ukx ly =
∂ k+l u
,
∂xk y l
etc.
Tools from the Calculus of Variations
•
Definition:
A differential function f is a exact iff f = DivF.
Special case (1D): f = Dx F.
•
Question: How can one test that f = DivF ?
•
Theorem (exactness test):
f = Div F iff Lu(j) (x) f ≡ 0, j = 1, 2, . . . , N.
N is the number of dependent variables.
The Euler operator annihilates divergences
•
Euler operator in 1D (variable u(x)):
M
X
k ∂
Lu(x) =
(−Dx )
∂ukx
k=0
∂
∂
∂
2 ∂
3
=
− Dx
+ Dx
− Dx
+ ···
∂u
∂ux
∂uxx
∂uxxx
•
Euler operator in 2D (variable u(x, y)):
Lu(x,y) =
My
Mx X
X
k=0
∂
(−Dx ) (−Dy )
∂u
kx `y
`=0
k
`
∂
∂
∂
=
− Dx
− Dy
∂u
∂ux
∂uy
∂
∂
2 ∂
2 ∂
3
+ Dx
+Dx Dy
+Dy
−Dx
···
∂uxx
∂uxy
∂uyy
∂uxxx
•
Question: How can one compute F = Div−1 f ?
•
Theorem (integration by parts):
• In 1D: If f is exact then
F = D−1
x f =
Z
f dx = Hu(x) f
• In 2D: If f is a divergence then
(x)
(y)
F = Div−1 f = (Hu(x,y) f, Hu(x,y) f )
The homotopy operator inverts total
derivatives and divergences!
•
Homotopy Operator in 1D (variable x):
Z
Hu(x) f =
0
N
1X
dλ
(Iu(j) f )[λu]
λ
j=1
with integrand
(j)
Mx
Iu(j) f =

X
k−1
X

k=1

(j)
uix
k−(i+1) 
(−Dx )
i=0
∂f
(j)
∂ukx
(Iu(j) f )[λu] means that in Iu(j) f one replaces
u → λu, ux → λux , etc.
More general: u → λ(u − u0 ) + u0
ux → λ(ux − ux 0 ) + ux 0
etc.
Application to Zakharov-Kuznetsov Equation
ut + αuux + β(uxx + uyy )x = 0
•
Step 1: Compute the dilation invariance
ZK equation is invariant under scaling symmetry
t x y 2
(t, x, y, u) → ( 3 , , , λ u)= (t̃, x̃, ỹ, ũ)
λ λ λ
λ is an arbitrary parameter.
•
Assign weights to each variable
W (u) = 2, W (Dt ) = 3, W (Dx ) = 1, W (Dy ) = 1.
•
Rank of a monomial is the sum of the weights of
the variables.
Example: Rank(αuux ) = 2W (u) + W (Dx ) = 5.
•
Key observation: A conservation law is invariant
under the scaling symmetry of the PDE
W (u) = 2, W (Dt ) = 3, W (Dx ) = 1, W (Dy ) = 1.
For example,
2
2
2
2
2
3α 4
+u
)
+
D
+
u
Dt u3 − 3β
(u
u
+3βu
u
−6βu(u
x 4
xx
x
y
x
y)
α
2
2
+ 3βα (u2xx −u2xy ) − 6βα (ux (uxxx +uxyy ) + uy (uxxy +uyyy ))
2
6β
2
+ Dy 3βu uxy + α uxy (uxx + uyy ) = 0
Rank(ρ) = 6, Rank(J) = 8.
Rank (conservation law) = 9.
Rank of the density needs to be selected!
•
Step 2: Construct the candidate density
For example, construct a density of rank 6.
Make a list of all terms with rank 6:
{u3 , u2x , uuxx , u2y , uuyy , ux uy , uuxy , u4x , u3xy , u2x2y , ux3y , u4y }
Remove divergences and divergence-equivalent
terms.
Candidate density of rank 6:
ρ = c1 u3 + c2 u2x + c3 u2y + c4 ux uy
•
Step 3: Compute the undetermined coefficients
Compute
∂ρ
Dt ρ =
+ ρ0 (u)[ut ]
∂t
My
Mx X
X
∂ρ
∂ρ
=
+
Dkx D`y ut
∂t
∂ukx `y
k=0 `=0
= 3c1 u2 I + 2c2 ux Dx + 2c3 uy Dy + c4 (uy Dx + ux Dy ) ut
Substitute ut = − αuux + β(uxx + uyy )x .
E = −Dt ρ = 3c1 u2 (αuux + β(uxx + uxy )x )
+ 2c2 ux (αuux + β(uxx + uyy )x )x + 2c3 uy (αuux
+ β(uxx + uyy )x )y + c4 (uy (αuux + β(uxx + uyy )x )x
+ ux (αuux + β(uxx + uyy )x )y )
Apply the Euler operator (variational derivative)
My
Mx X
X
k
` ∂E
Lu(x,y) E =
(−Dx ) (−Dy )
∂ukx `y
k=0 `=0
= −2 (3c1 β + c3 α)ux uyy + 2(3c1 β + c3 α)uy uxy
+2c4 αux uxy +c4 αuy uxx +3(3c1 β +c2 α)ux uxx
≡0
Solve a parameterized linear system for the ci :
3c1 β + c3 α = 0, c4 α = 0, 3c1 β + c2 α = 0
Solution:
3β
,
c
=
−
, c4 = 0
c1 = 1, c2 = − 3β
3
α
α
Substitute the solution into the candidate density
ρ = c1 u3 + c2 u2x + c3 u2y + c4 ux uy
Final density of rank 6:
3β 2
ρ=u −
(ux + u2y )
α
3
•
Step 4: Compute the flux
Use the homotopy operator to invert Div:
(x)
(y)
J = Div−1 E = Hu(x,y) E, Hu(x,y) E
Z 1
where
dλ
(x)
(x)
(Iu E)[λu]
Hu(x,y) E =
λ
0
with
Iu(x) E =
My k−1 `
Mx X
X
XX
k=1 `=0
(−Dx )
i+j k+`−i−j−1
k−i−1
uix jy i
k+`
i=0 j=0
k
k−i−1
`−j ∂E
−Dy
∂ukx `y
(y)
(y)
Similar formulas for Hu(x,y) E and Iu E.
Let A = αuux + β(uxxx + uxyy ) so that
E = 3u2 A −
6β
u A
α x x
−
6β
u A
α y y
Then,
(x)
(y)
J = Hu(x,y) E, Hu(x,y) E
= 43 αu4 + βu2 (3uxx + 2uyy ) − βu(6u2x + 2u2y )
+
−
+
−
3β 2
β2
7
u(u
+
u
)
−
u
(
u
+ 6uxxx )
2x2y
4y
4α
α x 2 xyy
β2
β2
2
3
5 2
3 2
u
(4u
+
u
)
+
(3u
+
u
+
u )
y
xxy
yyy
xx
α
2
α
2 xy
4 yy
5β 2
2
u
u
,
βu
uxy − 4βuux uy
xx
yy
4α
3β 2
β2
u(ux3y + u3xy ) − 4α ux (13uxxy + 3uyyy )
4α
− 4α uy (uxxx + 3uxyy ) + 4α uxy (uxx + uyy )
5β 2
9β 2
However, Div−1 E is not unique.
Indeed, J = J̃ + K, where K = (Dy θ, −Dx θ)
is a curl term.
For example,
2
β
θ = 2βu2 uy + 4α
3u(uxxy +uyyy )+10ux uxy +5uy (3uyy +uxx )
Shorter flux:
J̃ = J − K
4
2
2
2
u
+
3βu
u
−
6βu(u
+
u
= 3α
xx
x
y) +
4
−
6β 2
(ux (uxxx
α
3βu2 uxy
3β 2
α
u2xx − u2yy
+ uxyy ) + uy (uxxy + uyyy )),
2
+ 6βα uxy (uxx + uyy )
Additional Examples
•
Manakov-Santini system
utx + uyy + (uux )x + vx uxy − uxx vy = 0
vtx + vyy + uvxx + vx vxy − vy vxx = 0
•
Conservation laws for Manakov-Santini system:
Dt f ux vx + Dx f (uux vx − ux vx vy − uy vy )
− f 0 y(ut + uux − ux vy ) + Dy f (ux vy + uy vx + ux vx2 )
+ f 0 (u − yuy − yux vx ) = 0
where f = f (t) is arbitrary.
Conservation laws – continued:
Dt f (2u + vx2 − yux vx ) + Dx f (u2 + uvx2 + uy v
− vy2 − vx2 vy − y(uux vx − ux vx vy − uy vy ))
− f 0 y(vt + uvx − vx vy ) − (2f x − f 0 )y 2 (ut + uux − ux vy )
− Dy f (ux v − 2vx vy − vx3 + y(ux vx2 + ux vy + uy vx ))
−f 0 (v − y(2u + vy + vx2 )) + (2f x − f 0 y 2 )(ux vx + uy ) = 0
where f = f (t) is arbitrary.
There are three additional conservation laws.
•
(2+1)-dimensional Camassa-Holm equation
(αut + κux − utxx + 3βuux − 2ux uxx − uuxxx )x + uyy = 0
Interchange t with y
(αuy + κux − uxxy + 3βuux − 2ux uxx − uuxxx )x + utt = 0
Set v = ut to get
ut = v
vt = −αuxy − κuxx + u3xy − 3βu2x − 3βuuxx + 2u2xx
+ 3ux uxxx + uu4x
•
Conservation laws for the Camassa-Holm equation
Dt f u + Dx
2
1
3
f
(
βu
α
2
+ κu − 12 u2x − uuxx − utx )
− ( α1 f x − 12 f 0 y 2 )(αut + κux + 3βuux − 2ux uxx − uuxxx
− utxx ) − Dy uy ( α1 f x − 12 f 0 y 2 ) + f 0 yu = 0
Dt f yu + Dx α1 f y( 32 βu2 + κu − 21 u2x − uuxx − utx )
− y( α1 f x − 16 f 0 y 2 )(αut + κux + 3βuux − 2ux uxx − uuxxx
− utxx ) − Dy yuy ( α1 f x − 61 f 0 y 2 ) − u( α1 f x − 12 f 0 y 2 ) = 0
where f = f (t) is an arbitrary function.
•
Khoklov-Zabolotskaya equation
describes e.g. sound waves in nonlinear media
(ut − uux )x − uyy − uzz = 0
Conservation
law: Dt f u − Dx 12 f u2 + (f x + g)(ut − uux )
− Dy (fy x + gy )u − (f x + g)uy
− Dz (fz x + gz )u − (f x + g)uz = 0
under the constraints ∆f = 0 and ∆g = ft
where f = f (t, y, z) and g = g(t, y, z).
•
Shallow water wave model (atmosphere)
ut + (u·∇)u + 2 Ω × u + ∇(θh) − 12 h∇θ = 0
θt + u·(∇θ) = 0
ht + ∇·(uh) = 0
where u(x, y, t), θ(x, y, t) and h(x, y, t).
•
In components:
ut + uux + vuy − 2 Ωv + 12 hθx + θhx = 0
vt + uvx + vvy + 2 Ωu + 21 hθy + θhy = 0
θt + uθx + vθy = 0
ht + hux + uhx + hvy + vhy = 0
•
First few conservation laws of
 SWW
 model:
u
(1)

ρ(1) = h
J =h
v
 
u
(2)
ρ(2) = h θ
J = hθ 
v
 
u
2
(3)
2
ρ(3) = h θ
J = hθ  
v


2 + v 2 + 2hθ)
u
(u

ρ(4) = h (u2 + v 2 + hθ)
J(4) = h 
v (v 2 + u2 + 2hθ)
ρ(5) = θ (2Ω + vx − uy )


4Ωu − 2uuy + 2uvx − hθy
1 
(5)

J =2θ
4Ωv + 2vvx − 2vuy + hθx
•
More general conservation laws for SWW model:
Dt f (θ)h + Dx f (θ)hu + Dy f (θ)hv = 0
Dt g(θ)(2Ω + vx − ux )
+ Dx 21 g(θ)(4Ωu − 2uuy + 2uvx − hθy )
+ Dy 12 g(θ)(4Ωv − 2uy v + 2vvx + hθx ) = 0
for any functions f (θ) and g(θ).
•
Kadomtsev-Petviashvili (KP) equation
(ut + αuux + uxxx )x + σ 2 uyy = 0
parameter α ∈ IR and σ 2 = ±1.
Equation be written as a conservation law
Dt (ux ) + Dx (αuux + uxxx ) + Dy (σ 2 uy ) = 0.
Exchange y and t and set ut = v
ut = v
vt
1
= − 2 (uxy + αu2x + αuuxx + uxxxx )
σ
•
Examples of conservation laws for KP equation
(explicitly dependent on t, x, and y)
Dt xux +Dx 3u2 −uxx −6xuux +xuxxx +Dy αxuy = 0
Dt yux +Dx y(αuux + uxxx ) +Dy σ 2 (yuy − u) = 0
√ √
2y2
2y2
√
σ
σ
Dt
tu + Dx 12 α tu2 + tuxx + √ ut + √ uxxx
4 t
4 t
√
√
√
ασ 2 y 2
+ √ uux − x tut − αx tuux − x tuxxx
4 t
√ yu
y 2 uy
+Dy − √ + √ + x tuy = 0
2 t
4 t
•
More general conservation laws for KP equation:
Dt f (t)u + Dx f (t)( 12 αu2 + uxx )
+( 21 σ 2 f 0 (t)y 2 − f (t)x)(ut + αuux + u3x )
+Dy uy ( 21 f 0 (t)y 2 − σ 2 f (t)x) − f 0 (t)yu = 0
Dt f (t)yu + Dx f (t)y( 21 αu2 + uxx )
+( 61 σ 2 f 0 (t)y 3 − f (t)xy)(ut + αuux + u3x )
+Dy uy ( 61 f 0 (t)y 3 − σ 2 f (t)xy) −u( 21 f 0 (t)y 2 − σ 2 f (t)x) = 0
where f (t) is arbitrary function.
•
Potential KP equation
Replace u by ux and integrate with respect to x.
uxt + αux uxx + uxxxx + σ 2 uyy = 0
•
Examples of conservation laws
(not explicitly dependent on x, y, t):
Dt ux + Dx 12 αu2x + uxxx + Dy σ 2 uy = 0
Dt u2x + Dx 23 αu3x − u2xx + 2ux uxxx − σ 2 uyy
+Dy 2σ 2 ux uy = 0
Conservation laws for pKP equation – continued:
Dt ux uy + Dx αu2x uy + ut uy + 2uxxx uy − 2uxx uxy
+Dy σ 2 u2y − 31 u3x − ut ux + u2xx = 0
Dt 2αuux uxx + 3uu4x − 3σ 2 u2y + Dx 2αut u2x + 3u2t
− 2αuux utx − 3utx uxx + 3ut uxxx + 3ux ut xx − 3uut xxx
+Dy 6σ 2 ut uy = 0
Various generalizations exist.
•
Generalized Zakharov-Kuznetsov equation
ut + αun ux + β(uxx + uyy )x = 0
where n is rational, n 6= 0.
Conservation laws:
α
Dt u + Dx n+1
un+1 + βuxx + Dy βuxy = 0
2α n+2
Dt u2 + Dx n+2
u
− β(u2x − u2y ) + 2βu(uxx + uyy )
− Dy 2βux uy = 0
•
Third conservation law for gZK equation:
(n+1)(n+2)β
n+2
2
2
−
Dt u
(u
+
u
x
y)
2α
(n+2)α 2(n+1)
+ (n + 2)βun+1 uxx
+ Dx 2(n+1) u
(u2x
+
u2y )
(n+1)(n+2)β 2
2
(u
xx
2α
− u2yy )
2
− (n+1)(n+2)β
(ux (uxxx + uxyy ) + uy (uxxy + uyyy ))
α
2
+ Dy (n + 2)βun+1 uxy + (n+1)(n+2)β
uxy (uxx + uyy ) = 0.
α
− (n + 1)(n + 2)βu
n
+
Conclusions and Future Work
•
•
The power of Euler and homotopy operators:
I
Testing exactness
I
−1
Integration by parts: D−1
and
Div
x
Integration of non-exact expressions
Example: f = ux v + uvx + u2 uxx
R
R 2
f dx = uv + u uxx dx
•
Use other homotopy formulas (moving terms
amongst the components of the flux; prevent curl
terms)
•
Broader class of PDEs (beyond evolution type)
Example: short pulse equation (nonlinear optics)
uxt = u + (u3 )xx = u + 6uu2x + 3u2 uxx
with non-polynomial conservation law
p
p
Dt
1 + 6u2x − Dx 3u2 1 + 6u2x = 0
•
Continue the implementation in Mathematica
•
Software: http://inside.mines.edu/∼whereman
Thank You
Gelukkige Verjaardag Yuji