symbolic computation of conservation laws of nonlinear
TRANSCRIPT
Symbolic Computation ofConservation Laws of Nonlinear
Partial Differential Equations
Willy Hereman
Department of Applied Mathematics and Statistics
Colorado School of Mines
Golden, Colorado
http://inside.mines.edu/∼whereman/
Department of Mathematics and Statistics
Jordan University of Science and Technology, Irbid, Jordan
Tuesday, October 22, 2019, 13:30
Acknowledgements
Douglas Poole (Chadron State College, Nebraska)
Unal Goktas (Turgut Ozal University, Turkey)
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• Examples of conservation laws
• The Korteweg-de Vries equation
• Other scaling invariant quantities (KdV equation)
• The Zakharov-Kuznetsov equation
• Methods for computing conservation laws
• Tools from the calculus of variations
• The variational derivative (testing exactness)
• The homotopy operator (inverting Dx and Div)
• Application 1: The KdV equation
• Application 2: The Zakharov-Kuznetsov equation
• Demonstration of ConservationLawsMD.m
• 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
Examples of Conservation Laws
Example 1: Traffic Flow
u
x
a b
= density of cars
Modeling the density of cars (Bressan, 2009)
u(x, t) density of cars on a highway (e.g., number of
cars per 100 meters).
s(u) mean (equilibrium) speed of the cars (depends
on the density).
u
x
a b
= density of cars
Change in number of cars in segment [a, b] equals the
difference between cars entering at a and leaving at b
during time interval [t1, t2]:∫ b
a
(u(x, t2)− u(x, t1)
)dx =
∫ t2
t1
(J(a, t)− J(b, t)
)dt∫ b
a
(∫ t2
t1
ut(x, t) dt
)dx = −
∫ t2
t1
(∫ b
aJx(x, t) dx
)dt
where J(x, t) = u(x, t)s(u(x, t)) is the traffic flow
(e.g., in cars per hour) at location x and time t.
Then,∫ ba
∫ t2t1
(ut + Jx) dt dx = 0 holds ∀(a, b, t1, t2)
Yields the conservation law:
ut + [s(u)u]x = 0 or Dtρ+ DxJ = 0
ρ = u is the conserved density;
J(u) = s(u)u is the associated flux.
A simple Lighthill-Whitham-Richards model:
s(u) = smax
(1−
u
umax
), 0 ≤ u ≤ umax
smax is posted road speed, umax is the jam density.
Example 2: Fluid and Gas Dynamics
Euler equations for a compressible, non-viscous fluid:
ρt +∇.(ρu) = 0
(ρu)t +∇.(u⊗ (ρu)
)+∇p = 0
Et +∇.((E + p) u
)= 0
or, in components
ρt +∇.(ρu) = 0
(ρ ui)t +∇.(ρ ui u + p ei) = 0 (i = 1, 2, 3)
Et +∇.((E + p) u
)= 0
Express conservation of mass, momentum, energy.
⊗ is the dyadic product.
ρ is the mass density.
u = u1 e1 + u2 e2 + u3 e3 is the velocity.
p is the pressure p(ρ, e).
E is the energy density per unit volume:
E = 12ρ|u|2 + ρe.
e is internal energy density per unit of mass
(related to temperature).
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, . . .)
• Partial derivatives ukx = ∂ku∂xk
, ukx ly = ∂k+lu∂xkyl
, etc.
Examples: uxxxxx = u5x = ∂5u∂x5
uxx yyyy = u2x 4y = ∂6u∂x2y4
• Differential functions
Example: f = uvvx + x2u3xvx + uxvxx
• Total derivatives: Dt, Dx,Dy, . . .
Example: Let f = uvvx + x2u3xvx + uxvxx
Then
Dxf =∂f
∂x+ ux
∂f
∂u+ uxx
∂f
∂ux
+vx∂f
∂v+ vxx
∂f
∂vx+ vxxx
∂f
∂vxx
= 2xu3xvx + ux(vvx) + uxx(3x
2u2xvx + vxx)
+vx(uvx) + vxx(uv + x2u3x) + vxxx(ux)
= 2xu3xvx + vuxvx + 3x2u2
xvxuxx + uxxvxx
+uv2x + uvvxx + x2u3
xvxx + uxvxxx
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
Dtρ+ DxJ = 0
where the dot means evaluated on the PDE.Conserved density ρ and flux J.
P =
∫ ∞−∞
ρ dx = constant in time
if J vanishes at ±∞.
• Conservation law in (2+1)-dimensions
Dtρ+∇·J = Dtρ+ DxJ1 + DyJ2 = 0
Conserved density ρ and flux J = (J1, J2).
• Conservation law in (3+1)-dimensions
Dtρ+∇·J = Dtρ+ DxJ1 + DyJ2 + DzJ3 = 0
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
Example 1: KdV Equation
∂u
∂t+ 6u
∂u
∂x+∂3u
∂x3= 0 or ut + 6uux + uxxx = 0
shallow water waves, ion-acoustic waves in plasmas
Diederik Korteweg Gustav de Vries
Dilation Symmetry
ut + 6uux + uxxx = 0
has dilation (scaling) symmetry (x, t, u)→ (xλ, tλ3, λ2u)
λ is an arbitrary parameter.
Notion of weight: W (x) = −1, thus, W (Dx) = 1.
W (t) = −3, hence, W (Dt) = 3.
W (u) = 2.
Notion of rank (total weight of a monomial).
Examples: Rank(u3) = Rank(3u2x) = 6.
Rank(u3uxx) = 10.
Key Observation: Scaling Invariance
Every term in a density has the same fixed rank.
Every term in a flux has some other fixed rank.
The conservation law
Dtρ+ DxJ = 0
is uniform in rank.
Hence,
Rank(ρ) + Rank(Dt) = Rank(J) + Rank(Dx)
• First six (of infinitely many) conservation laws:
Dt
(u)
+ Dx
(3u2 + uxx
)= 0
Dt
(u2)
+ Dx
(4u3 − ux2 + 2uuxx
)= 0
Dt
(u3 −
1
2u2x
)+ Dx
(92u4 − 6uu2
x + 3u2uxx +1
2u2xx − uxuxxx
)= 0
Dt
(u4 − 2uu2
x + 15u2xx
)+ Dx
(245u5 − 18uu2
x + 4u3uxx
+2u2xuxx + 16
5uu2
xx −4uuxuxxx − 15u2xxx + 2
5uxxu4x
)= 0
Dt
(u5 − 5u2u2
x + uu2xx − 1
14u2xxx
)+ Dx
(5u6 − 40u3u2
x − . . . − 17uxxxu5x
)= 0
Dt
(u6 − 10u3u2
x −5
6u4x + 3u2u2
xx
+1021u3xx − 3
7uu2
xxx + 142u2
4x
)+ Dx
(367u7 − 75u4u2
x − . . . + 121u4xu6x
)= 0
• 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 −
1
3xu)
+Dx
(4tu3 − xu2 +
1
3ux − tu2
x + 2tuuxx −1
3xuxx
)= 0
• First five: IBM 7094 computer with FORMAC
(1966) −→ storage space problem!
IBM 7094 Computer
• First eleven densities: Control Data Computer
CDC-6600 computer (2.2 seconds)
−→ large integers problem!
Control Data CDC-6600
Other Scaling Invariant Quantities
• Generalized Symmetries:
G(x, t, u(N)) is an Nth order generalized symmetry
iff it leaves ut = F (x, t, u(M)) invariant for the
replacement u→ u+ εG, uix → uix + εDixG, within
order ε:
Dt(u+ εG) = F (u+ εG)
must hold up to order ε.
• Defining equation:
DtG = F ′(u)[G]
F (u)′[G] is the Frechet derivative of F (u) in the
direction of G:
F ′(u)[G] =∂
∂εF (u+ εG)|ε=0 =
M∑i=0
(DixG)
∂F
∂uix
• First 4 generalized symmetries of ut = 6uux + u3x
G(1) = ux
G(2) = 6uux + uxxx
G(3) = 30u2ux + 20uxuxx + 10uuxxx + u5x
G(4) = 140u3ux + 70u3x + 280uuxuxx + 70u2uxxx
+70uxxuxxx + 42uxu4x + 14uu5x + u7x
Generalized symmetries are invariant under thescaling symmetry.
• Recursion Operator:
A recursion operator R connects symmetries
G(j+s) = RG(j), j = 1, 2, . . .
s is seed (s = 1 in simplest case).
• Defining equation:
DtR+ [R, F ′(u)] = 0
Explicitly,
∂R∂t
+R′[F ] +R F ′(u)− F ′(u) R = 0
where stands for composition, and R′[F ] is theFrechet derivative of R in the direction of F :
R′[F ] =
n∑i=0
(DixF )
∂R∂uix
• Recursion operator (KdV equation):
R = D2x + 2uI + 2DxuD−1
x = D2x + 4uI + 2uxD
−1x
• For example,
Rux=(D2x + 2uI + 2DxuD−1
x )ux=6uux + u3x
R(6uux + u3x)=(D2x + 2uI + 2DxuD−1
x )(6uux + u3x)
=30u2ux + 20uxu2x + 10uu3x + u5x
Recursion operator is invariant under the scalingsymmetry.
• Lax Pair:
• Key idea: Replace ut + 6uux + uxxx = 0 with a
compatible linear system (Lax pair):
ψxx + (u− λ)ψ = 0
ψt + 4ψxxx + 6uψx + 3uxψ = 0
ψ is eigenfunction; λ is constant eigenvalue
(λt = 0) (isospectral)
• Lax Pair (L,M) in Operator Form:
Lψ = λψ and Dtψ =Mψ
• Require compatibility of both equations
Ltψ + (LM−ML)ψ = 0
• Defining Equation: Lt + [L,M] = O
with commutator [L,M] = LM−ML.
Furthermore, Ltψ = [Dt,L]ψ = Dt(Lψ)− LDtψ.
• Lax pair for the KdV equation:
L = D2x + u I
M = −(
4 D3x + 6uDx + 3ux I
)Lax pair is invariant under the scaling symmetry.
Example 2:
The Zakharov-Kuznetsov Equation
ut + αuux + β(uxx + uyy)x = 0
models ion-sound solitons in a low pressure uniform
magnetized plasma.
• Conservation laws:
Dt
(u)
+ Dx
(α2u2 + βuxx
)+ Dy
(βuxy
)= 0
Dt
(u2)
+ Dx
(2α3u3 − β(u2
x − u2y) + 2βu(uxx + uyy)
)+ Dy
(− 2βuxuy
)= 0
• More conservation laws (ZK equation):
Dt
(u3 − 3β
α(u2
x + u2y))
+ Dx
(3u2(α
4u2 + βuxx)− 6βu(u2
x + u2y)
+ 3β2
α(u2
xx − u2yy)−
6β2
α(ux(uxxx + uxyy) + uy(uxxy + uyyy))
)+ Dy
(3βu2uxy + 6β2
αuxy(uxx + uyy)
)= 0
Dt
(tu2 − 2
αxu)
+ Dx
(t(2α
3u3 − β(u2
x − u2y) + 2βu(uxx + uyy))
− x(u2 + 2βαuxx) + 2β
αux)
+ Dy
(− 2β(tuxuy + 1
αxuxy)
)= 0
Methods for Computing Conservation Laws
• Use the Lax pair L and A, satisfying [L,A] = 0.
If L = Dx + U, A = Dt + V then Vx − Ut + [U, V ] = 0.
L = TLT−1 gives the densities, A = TAT−1 gives
the fluxes.
• Use Noether’s theorem (Lagrangian formulation)
to generate conservation laws from symmetries
(Ovsiannikov, Olver, Mahomed, Kara, etc.).
• Integrating factor methods (Anderson, Bluman,
Anco, Cheviakov, Mason, Naz, 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 Dtρ 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.
Tools from the Calculus of Variations
• Definition:
A differential function f is a exact iff f = Div F.
Special case (1D): f = Dx F.
• Question: How can one test that f = Div F ?
• 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)):
Lu(x) =M∑k=0
(−Dx)k ∂
∂ukx
=∂
∂u−Dx
∂
∂ux+ D2
x
∂
∂uxx−D3
x
∂
∂uxxx+ · · ·
• Euler operator in 2D (variable u(x, y)):
Lu(x,y) =
Mx∑k=0
My∑`=0
(−Dx)k(−Dy)
` ∂
∂ukx `y
=∂
∂u−Dx
∂
∂ux−Dy
∂
∂uy
+ D2x
∂
∂uxx+DxDy
∂
∂uxy+D2
y
∂
∂uyy−D3
x
∂
∂uxxx· · ·
• Question: How can one compute F = Div−1 f ?
• Theorem (integration by parts):
• In 1D: If f is exact then
F = D−1x f =
∫f dx = Hu(x)f
• In 2D: If f is a divergence then
F = Div−1 f = (H(x)u(x,y)f, H
(y)u(x,y)f)
The homotopy operator inverts total
derivatives and divergences!
• Homotopy Operator in 1D (variable x):
Hu(x)f =
∫ 1
0
N∑j=1
(Iu(j)f)[λu]dλ
λ
with integrand
Iu(j)f =
M(j)x∑
k=1
k−1∑i=0
u(j)ix (−Dx)k−(i+1)
∂f
∂u(j)kx
(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 − ux0) + ux0 etc.
• Homotopy Operator in 2D (variables x and y):
H(x)u(x,y)f =
∫ 1
0
N∑j=1
(I(x)
u(j)f)[λu]
dλ
λ
H(y)u(x,y)f =
∫ 1
0
N∑j=1
(I(y)
u(j)f)[λu]
dλ
λ
where for dependent variable u(x, y)
I(x)u f =
Mx∑k=1
My∑`=0
k−1∑i=0
∑j=0
uix jy
(i+ji
)(k+`−i−j−1k−i−1
)(k+`k
)(−Dx)
k−i−1 (−Dy
)`−j) ∂f
∂ukx `y
Application 1: The KdV Equation
ut + 6uux + uxxx = 0
• Step 1: Compute the dilation symmetry
Set (x, t, u)→ (xλ, tλa, λbu) = (x, t, u)
Apply change of variables (chain rule)
λ−(a+b)ut+λ−(2b+1)uux+λ
−(b+3)u3x = 0
Solve a+ b = 2b+ 1 = b+ 3.
Solution: a = 3 and b = 2
(x, t, u)→ (x
λ,t
λ3, λ2u)
Compute the density of selected rank, say, 6.
• Step 2: Determine the form of the density
List powers of u, up to rank 6 : [u, u2, u3]
Differentiate with respect to x to increase
the rank
u has weight 2 → apply D4x
u2 has weight 4 → apply D2x
u3 has weight 6 → no derivatives needed
Apply the Dx derivatives
Remove total and highest derivative terms:
D4xu → u4x → empty list
D2xu
2 → ux2, uuxx → ux2
since uuxx=(uux)x−ux2
D0xu
3 → u3 → u3
Linearly combine the “building blocks”
Candidate density: ρ = c1u3 + c2ux
2
• Step 3: Compute the coefficients ci Compute
Dtρ =∂ρ
∂t+ ρ′(u)[ut]
=∂ρ
∂t+
M∑k=0
∂ρ
∂ukxDkxut
= (3c1u2I + 2c2uxDx)ut
Substitute ut by −(6uux + uxxx)
E = −Dtρ = (3c1u2I + 2c2uxDx)(6uux + uxxx)
= 18c1u3ux + 12c2u
3x + 12c2uuxuxx
+3 c1u2uxxx + 2c2uxu4x
Apply the Euler operator (variational derivative)
Lu(x) =δ
δu=
m∑k=0
(−Dx)k ∂
∂ukx
Here, E has order m = 4, thus
Lu(x)E =∂E
∂u−Dx
∂E
∂ux+D2
x
∂E
∂uxx−D3
x
∂E
∂u3x+D4
x
∂E
∂u4x
= −18(c1 + 2c2)uxuxx
This term must vanish!
So, c2 = −12c1. Set c1 = 1 then c2 = −1
2.
Hence, the final form density is ρ = u3 −1
2ux
2
• Step 4: Compute the flux J
Method 1: Integrate by parts (simple cases)
Now,
E=18u3ux + 3u2uxxx − 6u3x − 6uuxuxx − uxuxxxx
Integration of DxJ = E yields the flux
J =9
2u4 − 6uux
2 + 3u2uxx +1
2u2xx − uxuxxx
Method 2: Use the homotopy operator
J = D−1x E =
∫E dx = Hu(x)E =
∫ 1
0(IuE)[λu]
dλ
λ
with integrand
IuE =M∑k=1
k−1∑i=0
uix (−Dx)k−(i+1)
∂E
∂ukx
Here M = 4, thus
IuE = (uI)(∂E
∂ux) + (uxI− uDx)(
∂E
∂uxx)
+(uxxI− uxDx + uD2x)(
∂E
∂uxxx)
+(uxxxI− uxxDx + uxD2x − uD3
x)(∂E
∂u4x)
= (uI)(18u3 + 18u2x − 6uuxx − uxxxx)
+(uxI− uDx)(−6uux)
+(uxxI− uxDx + uD2x)(3u
2)
+(uxxxI− uxxDx + uxD2x − uD3
x)(−ux)
= 18u4 − 18uu2x + 9u2uxx + u2
xx − 2uxuxxx
Note: correct terms but incorrect coefficients!
Finally,
J =Hu(x)E =
∫ 1
0(IuE)[λu]
dλ
λ
=
∫ 1
0
(18λ3u4 − 18λ2uu2
x + 9λ2u2uxx + λu2xx
−2λuxuxxx) dλ
=9
2u4 − 6uu2
x + 3u2uxx +1
2u2xx − uxuxxx
Final form of the flux:
J =9
2u4 − 6uu2
x + 3u2uxx +1
2u2xx − uxuxxx
Application 2: 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, u)→ (t
λ3,x
λ,y
λ, λ2u)= (t, x, y, u)
λ is an arbitrary parameter.
• Hence, the weights of the variables are
W (u) = 2, W (Dt) = 3, W (Dx) = 1, W (Dy) = 1.
• 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,
Dt
(u3 − 3β
α(u2
x +u2y))
+ Dx
(3u2(α
4u2 +βuxx)−6βu(u2
x + u2y)
+ 3β2
α(u2
xx −u2xy)−
6β2
α(ux(uxxx +uxyy) + uy(uxxy +uyyy))
)+ Dy
(3βu2uxy + 6β2
αuxy(uxx + uyy)
)= 0
Rank (ρ) = 6, Rank (J) = 8.
Rank (conservation law) = 9.
Compute the density of selected rank, say, 6.
• 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, u
2y, uuyy, uxuy, uuxy, u4x, u3xy, u2x2y, ux3y, u4y
Remove divergences and divergence-equivalentterms.
Candidate density of rank 6:
ρ = c1u3 + c2u
2x + c3u
2y + c4uxuy
• Step 3: Compute the undetermined coefficients
Compute
Dtρ=∂ρ
∂t+ ρ′(u)[ut]
=∂ρ
∂t+
Mx∑k=0
My∑`=0
∂ρ
∂ukx `yDkx D`
y ut
=(
3c1u2I + 2c2uxDx + 2c3uyDy + c4(uyDx + uxDy)
)ut
Substitute ut = −(αuux + β(uxx + uyy)x
).
E = −Dtρ = 3c1u2(αuux + β(uxx + uxy)x)
+ 2c2ux(αuux + β(uxx + uyy)x)x + 2c3uy(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)
Lu(x,y)E =
Mx∑k=0
My∑`=0
(−Dx)k(−Dy)
` ∂E
∂ukx `y
=−2(
(3c1β + c3α)uxuyy + 2(3c1β + c3α)uyuxy
+2c4αuxuxy +c4αuyuxx +3(3c1β +c2α)uxuxx)
≡ 0
Solve a parameterized linear system for the ci:
3c1β + c3α = 0, c4α = 0, 3c1β + c2α = 0
Solution:
c1 = 1, c2 = −3βα, c3 = −3β
α, c4 = 0
Substitute the solution into the candidate density
ρ = c1u3 + c2u
2x + c3u
2y + c4uxuy
Final density of rank 6:
ρ = u3 −3β
α(u2
x + u2y)
• Step 4: Compute the flux
Use the homotopy operator to invert Div:
J=Div−1E=(H(x)u(x,y)E, H
(y)u(x,y)E
)where
H(x)u(x,y)E =
∫ 1
0(I(x)u E)[λu]
dλ
λ
with
I(x)u E =
Mx∑k=1
My∑`=0
( k−1∑i=0
∑j=0
uix jy
(i+ji
)(k+`−i−j−1k−i−1
)(k+`k
)(−Dx)
k−i−1 (−Dy
)`−j ) ∂E
∂ukx `y
Similar formulas for H(y)u(x,y)E and I(y)
u E.
Let A = αuux + β(uxxx + uxyy) so that
E = 3u2A− 6βαuxAx − 6β
αuyAy
Then,
J =(H(x)u(x,y)E, H
(y)u(x,y)E
)=(
3α4u4 + βu2(3uxx + 2uyy)− 2βu(3u2
x + u2y)
+ 3β2
4αu(u2x2y + u4y)− β2
αux(
72uxyy + 6uxxx)
− β2
αuy(4uxxy + 3
2uyyy) + β2
α(3u2
xx + 52u2xy + 3
4u2yy)
+ 5β2
4αuxxuyy, βu2uxy − 4βuuxuy
− 3β2
4αu(ux3y + u3xy)− β2
4αux(13uxxy + 3uyyy)
− 5β2
4αuy(uxxx + 3uxyy) + 9β2
4αuxy(uxx + uyy)
)
However, Div−1E is not unique.
Indeed, J = J + K, where K = (Dyθ, −Dxθ)
is a curl term.
For example,
θ=2βu2uy+ β2
4α
(3u(uxxy +uyyy)+10uxuxy+5uy(3uyy+uxx)
)Shorter flux:
J = J−K
=(
3u2(α4u2 + βuxx)− 6βu(u2
x + u2y) + 3β2
α
(u2xx − u2
yy
)− 6β2
α(ux(uxxx + uxyy) + uy(uxxy + uyyy)),
3βu2uxy + 6β2
αuxy(uxx + uyy)
)
Software Demonstration
Software packages in Mathematica
Codes are available via the Internet:
URL: http://inside.mines.edu/∼whereman/
Additional Examples
• Manakov-Santini system
utx + uyy + (uux)x + vxuxy − uxxvy = 0
vtx + vyy + uvxx + vxvxy − vyvxx = 0
• Conservation laws for Manakov-Santini system:
Dt
(fuxvx
)+ Dx
(f(uuxvx − uxvxvy − uyvy)
− f ′y(ut + uux − uxvy))
+ Dy
(f(uxvy + uyvx + uxv
2x)
+ f ′(u− yuy − yuxvx))
= 0
where f = f(t) is arbitrary.
Conservation laws – continued:
Dt
(f(2u+ v2
x − yuxvx))
+ Dx
(f (u2 + uv2
x + uyv
− v2y − v2
xvy − y(uuxvx − uxvxvy − uyvy))
− f ′y(vt + uvx − vxvy) + (f ′ − 2fx)y2(ut + uux − uxvy))
+ Dy
(f(v3
x + 2vxvy − uxv − y(uxv2x + uxvy + uyvx))
+f ′(v − y(2u+ vy + v2x)) + (f ′y2 − 2fx)(uxvx + uy)
)=0
where f = f(t) is arbitrary.
There are three additional conservation laws.
• (2+1)-dimensional Camassa-Holm equation
(αut + κux − utxx + 3βuux − 2uxuxx − uuxxx)x + uyy = 0
Interchange t with y
(αuy + κux− uxxy + 3βuux− 2uxuxx− uuxxx)x + utt = 0
Set v = ut to get
ut = v
vt = −αuxy − κuxx + u3xy − 3βu2x − 3βuuxx + 2u2
xx
+ 3uxuxxx + uu4x
• Conservation laws for the Camassa-Holm equation
Dt
(fu)
+ Dx
(1αf(3β
2u2 + κu− 1
2u2x − uuxx − utx)
+ (12f ′y2 − 1
αfx)(αut+ κux+ 3βuux − 2uxuxx − uuxxx
− utxx))
+ Dy
((1
2f ′y2 − 1
αfx)uy − f ′yu
)= 0
Dt
(fyu
)+ Dx
(1αfy(3β
2u2 + κu− 1
2u2x − uuxx − utx)
+ y(16f ′y2 − 1
αfx)(αut + κux + 3βuux − 2uxuxx − uuxxx
− utxx))
+ Dy
(y(1
6f ′y2 − 1
αfx)uy + ( 1
αfx− 1
2f ′y2)u
)= 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
(fu)−Dx
(12fu2 + (fx+ g)(ut − uux)
)+ Dy
((fx+ g)uy − (fyx+ gy)u
)+ Dz
((fx+ g)uz − (fzx+ gz)u
)= 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)− 12h∇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 + 12hθx + θhx = 0
vt + uvx + vvy + 2 Ωu+ 12hθy + θhy = 0
θt + uθx + vθy = 0
ht + hux + uhx + hvy + vhy = 0
• First few conservation laws of SWW model:
ρ(1) = h J(1) = h
u
v
ρ(2) = h θ J(2) = h θ
u
v
ρ(3) = h θ2 J(3) = h θ2
u
v
ρ(4) = h (u2 + v2 + hθ) J(4) = h
u (u2 + v2 + 2hθ)
v (v2 + u2 + 2hθ)
ρ(5) = θ (2Ω + vx − uy)
J(5) = 12θ
4Ωu− 2uuy + 2uvx − hθy4Ωv + 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
(12g(θ)(4Ωu− 2uuy + 2uvx − hθy)
)+ Dy
(12g(θ)(4Ωv − 2uyv + 2vvx + hθx)
)= 0
for any functions f(θ) and g(θ).
• Kadomtsev-Petviashvili (KP) equation
(ut + αuux + uxxx)x + σ2uyy = 0
parameter α ∈ IR and σ2 = ±1.
Equation be written as a conservation law
Dt(ux) + Dx(αuux + uxxx) + Dy(σ2uy) = 0.
Exchange y and t and set ut = v
ut = v
vt = −1
σ2(uxy + αu2
x + αuuxx + uxxxx)
• 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
(σ2(yuy − u)
)=0
Dt
(√tu)
+ Dx
(α2
√tu2 +
√tuxx +
σ2y2
4√tut +
σ2y2
4√tuxxx
+ασ2y2
4√tuux − x
√tut − αx
√tuux − x
√tuxxx
)+Dy
(x√tuy +
y2uy
4√t−
yu
2√t
)= 0
• More general conservation laws for KP equation:
Dt
(fu)
+ Dx
(f(α
2u2 + uxx)
+(σ2
2f ′y2 − fx)(ut + αuux + u3x)
)+Dy
((1
2f ′y2 − σ2fx)uy − f ′yu
)= 0
Dt
(fyu
)+ Dx
(fy(α
2u2 + uxx)
+y(σ2
6f ′y2 − fx)(ut + αuux + u3x)
)+Dy
(y(1
6f ′y2 − σ2fx)uy +(σ2fx− 1
2f ′y2)u
)=0
where f(t) is arbitrary function.
• Potential KP equation
Replace u by ux and integrate with respect to x.
uxt + αuxuxx + uxxxx + σ2uyy = 0
• Examples of conservation laws
(not explicitly dependent on x, y, t):
Dt
(ux)
+ Dx
(α2u2x + uxxx
)+ Dy
(σ2uy
)= 0
Dt
(u2x
)+ Dx
(2α3u3x − u2
xx + 2uxuxxx − σ2uyy)
+Dy
(2σ2uxuy
)= 0
Conservation laws for pKP equation – continued:
Dt
(uxuy
)+ Dx
(αu2
xuy + utuy + 2uxxxuy − 2uxxuxy)
+Dy
(σ2u2
y − 13u3x − utux + u2
xx
)= 0
Dt
(2αuuxuxx + 3uu4x − 3σ2u2
y
)+ Dx
(2αutu
2x + 3u2
t
− 2αuuxutx − 3utxuxx + 3utuxxx + 3uxut xx − 3uut xxx)
+Dy
(6σ2utuy
)= 0
Various generalizations exist.
• Generalized Zakharov-Kuznetsov equation
ut + αunux + β(uxx + uyy)x = 0
where n is rational, n 6= 0.
Conservation laws:
Dt
(u)
+ Dx
(αn+1
un+1 + βuxx)
+ Dy
(βuxy
)= 0
Dt
(u2)
+ Dx
(2αn+2
un+2 − β(u2x − u2
y) + 2βu(uxx + uyy))
+ Dy
(− 2βuxuy
)= 0
• Third conservation law for gZK equation:
Dt
(un+2 − (n+1)(n+2)β
2α(u2
x + u2y))
+ Dx
((n+2)α2(n+1)
u2(n+1) + (n+ 2)βun+1uxx
− (n+ 1)(n+ 2)βun(u2x + u2
y) + (n+1)(n+2)β2
2α(u2
xx − u2yy)
− (n+1)(n+2)β2
α(ux(uxxx + uxyy) + uy(uxxy + uyyy))
)+ Dy
((n+ 2)βun+1uxy + (n+1)(n+2)β2
αuxy(uxx + uyy)
)= 0.
Conclusions and Future Work
• The power of Euler and homotopy operators:
I Testing exactness
I Integration by parts: D−1x and Div−1
• Integration of non-exact expressions
Example: f = uxv + uvx + u2uxx∫fdx = uv +
∫u2uxx 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 + 3u2uxx
with non-polynomial conservation law
Dt
(√1 + 6u2
x
)−Dx
(3u2√
1 + 6u2x
)= 0
• Continue the implementation in Mathematica
• Software: http://inside.mines.edu/∼whereman
Thank You
Publications
1. D. Poole and W. Hereman, Symbolic computation
of conservation laws for nonlinear partial
differential equations in multiple space dimensions,
Journal of Symbolic Computation (2011), 26
pages, in press.
2. W. Hereman, P. J. Adams, H. L. Eklund, M. S.
Hickman, and B. M. Herbst, Direct Methods and
Symbolic Software for Conservation Laws of
Nonlinear Equations, In: Advances of Nonlinear
Waves and Symbolic Computation, Ed.: Z. Yan,
Nova Science Publishers, New York (2009),
Chapter 2, pp. 19-79.
3. W. Hereman, M. Colagrosso, R. Sayers, A.
Ringler, B. Deconinck, M. Nivala, and M. S.
Hickman, Continuous and Discrete Homotopy
Operators and the Computation of Conservation
Laws. In: Differential Equations with Symbolic
Computation, Eds.: D. Wang and Z. Zheng,
Birkhauser Verlag, Basel (2005), Chapter 15, pp.
249-285.
4. W. Hereman, B. Deconinck, and L. D. Poole,
Continuous and discrete homotopy operators:
A theoretical approach made concrete, Math.
Comput. Simul. 74(4-5), 352-360 (2007).
5. W. Hereman, Symbolic computation of
conservation laws of nonlinear partial differential
equations in multi-dimensions, Int. J. Quan.
Chem. 106(1), 278-299 (2006).