.
Symbolic Computation of Conserved Densities
of Nonlinear Evolution Equations and
Differential-Difference Equations
WILLY HEREMAN
Mathematical and Computer Sciences
Colorado School of Mines
Golden, CO 80401-1887, USA
Centre de Recherches Mathématiques
Montréal, Québec, Canada
Friday, May 23, 1997
Time: 14:00
1
• Purpose
Design and implement an algorithm to compute polynomial conservation laws for nonlinear systems of evolution equations, differentialdifference equations, and dynamical systems
• Motivation
– Conservation laws describe the conservation of fundamental
physical quantities such as linear momentum and energy.
Compare with constants of motion (first integrals) in mechanics
– For nonlinear PDEs and DDEs, the existence of a sufficiently large
(in principal infinite) number of conservation laws assures complete
integrability
– Conservation laws provide a simple and efficient method to study
both quantitative and qualitative properties of equations and
their solutions, e.g. Hamiltonian structures
– Conservation laws can be used to test numerical integrators
2
PART I:
Evolution Equations
• Conservation Laws for PDEs
Consider a single nonlinear evolution equation
ut = F(u, ux, uxx, ..., unx)
or a system of N nonlinear evolution equations
ut = F(u, u 0, ..., u(n))
where u = [u1, . . . , uN ]T , or, component-wise,
0
00
(n)
ui,t + Fi(uj , uj , uj , . . . , uj ) = 0,
i, j = 1, 2, . . . , N,
where
n
∂ui
def ∂ (uj )
(n)
ui,t =
,
uj = uj,nx =
∂t
∂xn
All components of u depend on x and t.
def
Conservation law:
Dt ρ + Dx J = 0
ρ is the density, J is the flux
Both are polynomial in u, ux, u2x, ..., unx
Consequently
P =
Z +∞
−∞
ρ dx = constant
if J vanishes at infinity
3
• Example: Korteweg-de Vries (KdV) equation
ut + uux + u3x = 0
Conserved densities:
u2
(u)t + ( + u2x)x = 0
2
ρ1 = u,
2u3
(u )t + (
+ 2uu2x − ux2)x = 0
3
2
2
ρ2 = u ,
ρ3 = u3 − 3ux2,
3
u −3ux
...
2
3
 u4 −6uu 2 +3u2 u +3u 2 −6u u  = 0
+
x
2x
2x
x 3x
t
4
x


ρ6 = u6 − 60u3ux2 − 30ux4 + 108u2u2x2
+
720 3 648
216 2
u2x −
uu3x2 +
u4x ,
7
7
7
...... long ......
...
Note: KdV equation is invariant under the scaling symmetry
(x, t, u) → (λx, λ3t, λ−2u)
u (resp. t) carry the weight of 2 (resp. 3) derivatives with respect to x
∂2
u ∼ 2,
∂x
∂
∂3
∼
∂t ∂x3
4
• The Euler Operator (calculus of variations)
Useful tool to verify if an expression is a total derivative.
Theorem:
If
(n)
(n)
f = f (x, y1, . . . , y1 , . . . , yN , . . . , yN )
then
Ly (f ) ≡ 0
if and only if
f = Dx g
where
(n−1)
g = g(x, y1, . . . , y1
(n−1)
, . . . , yN , . . . , yN
)
Notations:
y = [y1, . . . , yN ]T
Ly (f ) = [Ly1 (f ), . . . , LyN (f )]T
0 = [0, . . . , 0]T
(T for transpose)
and Euler Operator:
n
∂
d ∂
d2 ∂
∂
n d
Lyi =
− ( 0 ) + 2 ( 00 ) + · · · + (−1)
(
)
∂yi dx ∂yi
dx ∂yi
dxn ∂yi(n)
5
• Key Steps of the Algorithm
1. Determine weights (scaling properties) of variables & parameters
2. Construct the form of the density (building blocks)
3. Determine the unknown constant coefficients
• Example: For the KdV equation, compute the density of rank 6
∂
(i) Take all the variables, except ( ∂t
), with positive weight.
Here, only u with w(u) = 2
List all possible powers of u, up to rank 6 :
[u, u2, u3]
Introduce x derivatives to ‘complete’ the rank
u has weight 2, introduce
u2 has weight 4, introduce
∂4
∂x4
∂2
∂x2
u3 has weight 6, no derivatives needed
(ii) Apply the derivatives
Remove terms that are total derivatives with respect to x
or total derivative up to terms kept earlier in the list
[u4x] → [ ] empty list
[ux2, uu2x] → [ux2] since uu2x = (uux)x − ux2
[u3] → [u3]
6
Combine the ‘building blocks’
ρ = c1 u3 + c 2 ux 2
(iii) Determine the coefficients c1 and c2
1. Compute Dtρ = 3c1u2ut + 2c2uxuxt
2. Replace ut by −(uux + u3x) and uxt by −(uux + u3x)x
3. Integrate the result with respect to x
Use the Euler operator (or carry out the integrations by parts)
3
Dtρ = −[ c1u4 −(3c1 −c2)uu2x + 3c1u2u2x −c2u2x2 + 2c2uxu3x]x
4
−(3c1 + c2)ux3
4. The non-integrable (last) term must vanish. Thus, c1 = − 13 c2.
Set c2 = −3, hence, c1 = 1
Result:
ρ = u3 − 3ux2
Expression [. . .] yields
3
J = u4 − 6uux2 + 3u2u2x + 3u2x2 − 6uxu3x
4
7
• Example: Boussinesq equation
utt − u2x + 3uu2x + 3ux2 + αu4x = 0
with nonzero parameter α. Can be written as
ut + vx = 0
vt + ux − 3uux − αu3x = 0
The terms ux and α u3x are not uniform in rank
Introduce auxiliary parameter β with weight.
Replace the system by
ut + v x = 0
vt + βux − 3uux − αu3x = 0
The system is invariant under the scaling symmetry
(x, t, u, v, β) → (λx, λ2t, λ−2u, λ−3v, λ−2β)
Hence,
w(u) = 2, w(β) = 2, w(v) = 3 and w(
or
∂2
u ∼ β ∼ 2,
∂x
Form ρ of rank 6
∂3
v ∼ 3,
∂x
∂
)=2
∂t
∂
∂2
∼
∂t ∂x2
ρ = c1 β 2u + c2 βu2 + c3 u3 + c4 v 2 + c5 uxv + c6 u2x
Compute the ci. At the end set β = 1
ρ = u2 − u3 + v 2 + αu2x
which is no longer uniform in rank!
8
• Application: A Class of Fifth-Order Evolution Equations
ut + αu2ux + βuxu2x + γuu3x + u5x = 0
where α, β, γ are nonzero parameters, and u ∼
∂2
∂x2
Special cases:
α
α
α
α
=
=
=
=
30
5
20
2
β = 20
β=5
β = 25
β=6
γ = 10
γ=5
γ = 10
γ=3
Lax
Sawada − Kotera
Kaup−Kupershmidt
Ito
Under what conditions for the parameters α, β and γ does this equation
admit a density of fixed rank?
– Rank 2:
No condition
ρ=u
– Rank 4:
Condition: β = 2γ
(Lax and Ito cases)
ρ = u2
9
– Rank 6:
Condition:
10α = −2β 2 + 7βγ − 3γ 2
(Lax, SK, and KK cases)
ρ = u3 +
15
ux 2
(−2β + γ)
– Rank 8:
1. β = 2γ (Lax and Ito cases)
ρ = u4 −
2. α = − 2β
2 −7βγ−4γ 2
6γ
6
uux2 + u2x2
α
α
(SK, KK and Ito cases)
45
ρ = u4 −
675
135
2
uux2 +
u
2x
2β + γ
(2β + γ)2
– Rank 10:
Condition:
β = 2γ
and
10α = 3γ 2
(Lax case)
ρ = u5 −
50 2 2 100
500
u ux + 2 uu2x2 − 3 u3x2
γ
γ
7γ
10
What are the necessary conditions for the parameters α, β and γ for
this equation to admit infinitely many polynomial conservation laws?
– If α =
3 2
10 γ
and β = 2γ then there is a sequence
(without gaps!) of conserved densities (Lax case)
– If α = 51 γ 2 and β = γ then there is a sequence
(with gaps!) of conserved densities (SK case)
– If α = 51 γ 2 and β = 25 γ then there is a sequence
(with gaps!) of conserved densities (KK case)
– If
2β 2 − 7βγ + 4γ 2
α=−
45
or
β = 2γ
then there is a conserved density of rank 8
Combine both conditions: α =
11
2γ 2
9
and β = 2γ (Ito case)
• More Examples
• Nonlinear Schrödinger Equation
iqt − q2x + 2|q|2q = 0
Program can not handle complex equations
Replace by
ut − v2x + 2v(u2 + v 2) = 0
vt + u2x − 2u(u2 + v 2) = 0
where q = u + iv
Scaling properties
∂
u∼v∼
,
∂x
First seven conserved densities:
∂
∂2
∼
∂t ∂x2
ρ1 = u 2 + v 2
ρ2 = vux
ρ3 = u4 + 2u2v 2 + v 4 + ux2 + vx2
1
1
ρ4 = u2vux + v 3ux − vu3x
3
6
12
1
1
3
3
1
5
ρ5 = − u 6 − u 4 v 2 − u 2 v 4 − v 6 − u 2 u x 2 − v 2 u x 2
2
2
2
2
2
2
3
5
1
1
− u2vx2 − v 2vx2 + uv 2u2x − u2x2 − v2x2
2
2
4
4
3
1
1
3
1
ρ6 = − u4vux − u2v 3ux − v 5ux + vux3 − vuxvx2
4
2
20
4
4
1
1
1
+uvuxu2x + u2vu3x + v 3u3x − vu5x
4
12
40
ρ7 =
15
5
35
5 8
u + 5u6v 2 + u4v 4 + 5u2v 6 + v 8 + u4ux2
4
2
4
2
5
7
15
−5u2v 2ux2 + v 4ux2 − ux4 + u4vx2 + 25u2v 2vx2
2
4
2
+
35 4 2 5 2 2 7 4
v vx − ux vx − vx − 10u3v 2u2x − 5uv 4u2x
2
2
4
7
1
5
−5uvx2u2x + u2u2x2 + v 2u2x2 + u2v2x2
2
2
2
7
1
1
+ v 2v2x2 − v 2uxu3x + u3x2 + v3x2 + uv 2u4x
2
4
4
13
• The Ito system
ut − u3x − 6uux − 2vvx = 0
vt − 2uxv − 2uvx = 0
∂2
u ∼ 2,
∂x
∂2
v∼ 2
∂x
ρ1 = c 1 u + c 2 v
ρ2 = u 2 + v 2
ρ3 = 2u3 + 2uv 2 − ux2
ρ4 = 5u4 + 6u2v 2 + v 4 − 10uux2 + 2v 2u2x + u2x2
ρ5 = 14u5 + 20u3v 2 + 6uv 4 − 70u2ux2 + 10v 2ux2
−4v 2vx2 + 20uv 2u2x + 14uu2x2 − u3x2 + 2v 2u4x
and more conservation laws
14
• The dispersiveless long-wave system
ut + vux + uvx = 0
vt + ux + vvx = 0
u ∼ 2v
choose u ∼
ρ1
ρ2
ρ3
ρ4
ρ5
=
=
=
=
=
ρ6 =
w(v) is free
∂
∂x
and
∂
∂x
v
u
uv
u2 + uv 2
3u2v + uv 3
1 3
1
u + u2v 2 + uv 4
3
6
ρ7 = u 3 v + u 2 v 3 +
ρ8 =
2v ∼
1 5
uv
10
1
1 4
u + 2u3v 2 + u2v 4 + uv 6
3
15
and more
Always the same set irrespective the choice of weights
15
• A generalized Schamel equation
2
n2ut + (n + 1)(n + 2)u n ux + u3x = 0
where n is a positive integer
ρ1 = u,
ρ 2 = u2
1 2 n2 2+ 2
ρ3 = u x − u n
2
2
For n 6= 1, 2 no further conservation laws
16
• Three-Component Korteweg-de Vries Equation
ut − 6uux + 2vvx + 2wwx − u3x = 0
vt − 2vux − 2uvx = 0
wt − 2wux − 2uwx = 0
Scaling properties
∂2
u ∼ v ∼ w ∼ 2,
∂x
∂
∂3
∼
∂t ∂x3
First five densities:
ρ1 = c 1 u + c 2 v + c 3 w
ρ2 = u 2 − v 2 − w 2
ρ3 = −2u3 + 2uv 2 + 2uw2 + ux2
5
1
1
ρ4 = − u4 + 3u2v 2 − v 4 + 3u2w2 − v 2w2 − w4
2
2
2
1
+5uux2 + v 2u2x + w2u2x − u2x2
2
ρ5 = −
7 5
3
3
3
u + u3v 2 − uv 4 + u3w2 − uv 2w2 − uw4
10
10
5
10
7
1
1
1
+ u2 ux 2 + v 2 ux 2 + w 2 ux 2 + v 2 vx 2
2
2
2
5
1
1
7
− w2vx2 + w2wx2 + uv 2u2x + uw2u2x − uu2x2
5
5
10
1
1
1
1
− vw2v2x + u3x2 + v 2u4x + w2u4x
5
20
10
10
17
• The Deconinck-Meuris-Verheest equation
Consider the modified vector derivative NLS equation:
∂B⊥
∂
∂B⊥
∂ 2B⊥
2
=0
+ (B⊥B⊥) + αB⊥0B⊥0 ·
+ ex ×
∂t
∂x
∂x
∂x2
Replace the vector equation by
2
2
ut + u(u + v ) + βu − vx
vt + v(u2 + v 2) + ux
x
x
=0
=0
u and v denote the components of B⊥ parallel
2
and perpendicular to B⊥0 and β = αB⊥0
∂
∂
∂
,
v2 ∼
,
β∼
u2 ∼
∂x
∂x
∂x
First 6 conserved densities
ρ1 = c 1 u + c 2 v
ρ2 = u 2 + v 2
ρ3 =
1 2
(u + v 2)2 − uvx + uxv + βu2
2
1
β
1
ρ4 = (u2 + v 2)3 + (ux2 + vx2) − u3vx + v 3ux + (u4 − v 4)
4
2
4
18
ρ5 =
2
4
1 2
(u + v 2)4 − (uxv2x − u2xvx) + (uux + vvx)2
4
5
5
6
+ (u2 + v 2)(ux2 + vx2) − (u2 + v 2)2(uvx − uxv)
5
β
β2 4
2
3
6
4 2
6
+ (2ux − 4u vx + 2u + 3u v − v ) + u
5
5
ρ6 =
1
7 2
2
(u + v 2)5 + (u22x + v2x
)
16
2
−
5 2
(u + v 2)(uxv2x −u2xvx) + 5(u2 + v 2)(uux + vvx)2
2
+
15 2
35
(u + v 2)2(ux2 + vx2) − (u2 + v 2)3(uvx − uxv)
4
16
+
β 8
(5u + 10u6v 2 − 10u2v 6 − 5v 8 + 20u2ux2
8
− 12u5vx + 60uv 4vx − 20v 2vx2)
β2 6
(u + v 6)
+
4
19
PART II:
Differential-difference Equations
• Conservation Laws for DDEs
Consider a system of DDEs, continuous in time, discretized in space
u̇n = F(..., un−1, un, un+1, ...)
un and F are vector dynamical variables
Conservation law :
ρ̇n = Jn − Jn+1
ρn is the density, Jn is the flux
Both are polynomials in un and its shifts
d X
X
X
( ρn) = ρ̇n = (Jn − Jn+1)
n
n
dt n
If Jn is bounded for all n, with suitable boundary conditions
X
n
ρn = constant
20
• Definitions
Define: D shift-down operator, U shift-up operator
Dm = m|n→n−1,
U m = m|n→n+1
For example,
Dun+2vn = un+1vn−1,
U un−2vn−1 = un−1vn
Compositions of D and U define an equivalence relation
All shifted monomials are equivalent, e.g.
un−1vn+1 ≡ un+2vn+4 ≡ un−3vn−1
Use equivalence criterion:
If two monomials, m1 and m2, are equivalent, m1 ≡ m2, then
m1 = m2 + [Mn − Mn+1]
for some polynomial Mn
For example, un−2un ≡ un−1un+1 since
un−2un = un−1un+1 + [un−2un − un−1un+1] = un−1un+1 + [Mn − Mn+1]
with Mn = un−2un
21
Main representative of an equivalence class; the monomial with label
n on u (or v)
For example, unun+2 is the main representative of the class with elements un−1un+1, un+1un+3, etc.
Use lexicographical ordering to resolve conflicts
For example, unvn+2 (not un−2vn) is the main representative of the
class with elements un−3vn−1, un+2vn+4, etc.
• Algorithm: Toda Lattice
u̇n = vn−1 − vn,
v̇n = vn(un − un+1)
Simplest conservation law (by hand):
u̇n = ρ̇n = vn−1 − vn = Jn − Jn+1 with Jn = vn−1
First pair:
ρ(1)
n = un ,
Jn(1) = vn−1
Second pair:
2
1
ρ(2)
n = 2 un + vn ,
22
Jn(2) = unvn−1
Key observation: The DDE
u̇n = vn−1 − vn,
v̇n = vn(un − un+1)
and the two pairs
ρ(1)
n = un ,
Jn(1) = vn−1
2
1
ρ(2)
n = 2 un + vn ,
Jn(2) = unvn−1
are invariant under the scaling symmetry
(t, un, vn) → (λt, λ−1un, λ−2vn)
Dimensional analysis:
un corresponds to one derivative with respect to t
For short, un ∼
d
dt ,
and similarly, vn ∼
d2
dt2
Our algorithm exploits this symmetry to find conserved densities:
1. Determining the weights
2. Constructing the form of density
3. Determining the unknown coefficients
23
• Step 1: Determine the weights
The weight, w, of a variable is equal to the number of derivatives with
respect to t the variable carries
Weights are positive, rational, and independent of n
d
)=1
Set w( dt
For the Toda lattice: w(un) = 1, and w(vn) = 2
The rank of a monomial is the total weight of the monomial, in terms
of derivatives with respect to t
In each equation of the Toda lattice, all the terms are uniform in rank
Requiring uniformity in rank for each equation, allows one to compute
the weights of the dependent variables
Indeed
w(un) + 1 = w(vn),
w(vn) + 1 = w(un) + w(vn)
yields
w(un) = 1,
w(vn) = 2
which is consistent with the scaling symmetry
24
• Step 2: Construct the form of the density
For example, compute the form of the density of rank 3
List all monomials in un and vn of rank 3 or less:
G = {un3, un2, unvn, un, vn}
Next, for each monomial in G, introduce enough t-derivatives, so that
each term exactly has weight 3. Using the equations of Toda lattice,
d0
d0
3
3
(un ) = un ,
(unvn) = unvn,
dt0
dt0
d
d
(un2) = 2unvn−1 − 2unvn,
(vn) = unvn − un+1vn,
dt
dt
d2
(un) = un−1vn−1 − unvn−1 − unvn + un+1vn
dt2
Gather the resulting terms in a set
H = {un3, unvn−1, unvn, un−1vn−1, un+1vn}
Identify members that belong to the same equivalence classes and replace them by the main representatives.
For example, since unvn−1 ≡ un+1vn both are replaced by unvn−1.
H is replaced by
I = {un3, unvn−1, unvn}
containing the building blocks of the density.
Form a linear combination of the monomials in I
ρn = c1 un3 + c2 unvn−1 + c3 unvn
with constant coefficients ci
25
• Step 3: Determine the unknown coefficients
Require that the conservation law holds
Compute ρ̇n
Use the equations to remove u̇n, v̇n, etc.
Group the terms
ρ̇n = (3c1 − c2)un2vn−1 + (c3 − 3c1)un2vn + (c3 − c2)vn−1vn
+c2un−1unvn−1 + c2vn−12 − c3unun+1vn − c3vn2
Use the equivalence criterion to modify ρ̇n.
Replace un−1unvn−1 by unun+1vn + [un−1unvn−1 − unun+1vn].
The goal is to introduce the main representatives. Therefore,
ρ̇n = (3c1 − c2)un2vn−1 + (c3 − 3c1)un2vn
+(c3 − c2)vnvn+1 + [(c3 − c2)vn−1vn − (c3 − c2)vnvn+1]
+c2unun+1vn + [c2un−1unvn−1 − c2unun+1vn]
+c2vn2 + [c2vn−12 − c2vn2] − c3unun+1vn − c3vn2
Group the terms outside of the square brackets and move the pairs
inside the square brackets to the bottom. Rearrange the latter terms
so that they match the pattern [Jn − Jn+1]. Hence,
ρ̇n = (3c1 − c2)un2vn−1 + (c3 − 3c1)un2vn
+(c3 − c2)vnvn+1 + (c2 − c3)unun+1vn + (c2 − c3)vn2
+[{(c3 − c2)vn−1vn + c2un−1unvn−1 + c2vn−12}
−{(c3 − c2)vnvn+1 + c2unun+1vn + c2vn2}]
26
The terms inside the square brackets determine:
Jn = (c3 − c2)vn−1vn + c2un−1unvn−1 + c2vn−12
The terms outside the square brackets must all vanish, thus
S = {3c1 − c2 = 0, c3 − 3c1 = 0, c2 − c3 = 0}
The solution is 3c1 = c2 = c3. Choose c1 = 13 , thus c2 = c3 = 1
ρn = 13 un3 + un(vn−1 + vn),
Jn = un−1unvn−1 + vn−12
Analogously, conserved densities of rank ≤ 5:
= un ,
ρ(1)
n
2
1
ρ(2)
n = 2 un + vn
ρ(3)
=
n
3
1
3 un
+ un(vn−1 + vn)
=
ρ(4)
n
4
1
4 un
+ un2(vn−1 + vn) + unun+1vn + 21 vn2 + vnvn+1
ρ(5)
=
n
5
1
5 un
+ un3(vn−1 + vn) + unun+1vn(un + un+1)
+unvn−1(vn−2 + vn−1 + vn) + unvn(vn−1 + vn + vn+1)
27
• Application: A parameterized Toda lattice
u̇n = α vn−1 − vn,
v̇n = vn (β un − un+1)
α and β are nonzero parameters. The system is integrable if α = β = 1
Compute the compatibility conditions for α and β, so that there is a
conserved densities of, say, rank 3.
In this case, we have S:
{3αc1 −c2 = 0, βc3 −3c1 = 0, αc3 −c2 = 0, βc2 −c3 = 0, αc2 −c3 = 0}
A non-trivial solution 3c1 = c2 = c3 will exist iff α = β = 1
Analogously, the parameterized Toda lattice has density
ρ(1)
n = un of rank 1 if α = 1
and density
β
2
ρ(2)
n = 2 un + vn of rank 2 if α β = 1
Only when α = β = 1 will the parameterized system have conserved
densities of rank ≥ 3
28
• Example: Nonlinear Schrödinger (NLS) equation
Ablowitz and Ladik discretization of the NLS equation:
i u̇n = un+1 − 2un + un−1 + u∗nun(un+1 + un−1)
where u∗n is the complex conjugate of un.
Treat un and vn = u∗n as independent variables, add the complex conjugate equation, and absorb i in the scale on t
u̇n = un+1 − 2un + un−1 + unvn(un+1 + un−1)
v̇n = −(vn+1 − 2vn + vn−1) − unvn(vn+1 + vn−1)
Since vn = u∗n, w(vn) = w(un).
No uniformity in rank! Circumvent this problem by introducing an
auxiliary parameter α with weight,
u̇n = α(un+1 − 2un + un−1) + unvn(un+1 + un−1)
v̇n = −α(vn+1 − 2vn + vn−1) − unvn(vn+1 + vn−1).
Uniformity in rank requires that
w(un) + 1 = w(α) + w(un) = 2w(un) + w(vn) = 3w(un)
w(vn) + 1 = w(α) + w(vn) = 2w(vn) + w(un) = 3w(vn)
which yields
1
w(un) = w(vn) = ,
2
29
w(α) = 1
Uniformity in rank is essential for the first two steps of the algorithm.
After Step 2, you can already set α = 1.
The computations now proceed as in the previous examples
Conserved densities:
ρ(1)
= c1unvn−1 + c2unvn+1
n
ρ(2)
= c1( 12 un2vn−12 + unun+1vn−1vn + unvn−2)
n
+ c2( 12 un2vn+12 + unun+1vn+1vn+2 + unvn+2)
ρ(3)
= c1[ 31 un3vn−13
n
+unun+1vn−1vn(unvn−1 + un+1vn + un+2vn+1)
+unvn−1(unvn−2 + un+1vn−1)
+unvn(un+1vn−2 + un+2vn−1) + unvn−3]
+ c2[ 31 un3vn+13
+unun+1vn+1vn+2(unvn+1 + un+1vn+2 + un+2vn+3)
+unvn+2(unvn+1 + un+1vn+2)
+unvn+3(un+1vn+1 + un+2vn+2) + unvn+3]
30
• Scope and Limitations of Algorithm & Software
– Systems must be polynomial in dependent variables
– Only two independent variables (x and t) are allowed
– No terms should explicitly depend on x and t
– Program only computes polynomial conserved densities;
only polynomials in the dependent variables and their derivatives;
no explicit dependencies on x and t
– No limit on the number of evolution equations and DDEs.
In practice: time and memory constraints
– Input systems may have (nonzero) parameters.
Program computes the compatibility conditions for parameters such
that densities (of a given rank) exist
– Systems can also have parameters with (unknown) weight.
Allows one to test systems with equations of non-uniform rank
– For systems where one or more of the weights are free,
the program prompts the user to enter values for the free weights
– Negative weights are not allowed
– Fractional weights and ranks are permitted
– Form of ρ can be given in the data file (testing purposes)
31
• Conserved Densities Software
– Conserved densities programs CONSD and SYMCD
by Ito and Kako (Reduce, 1985, 1994 & 1996).
– Conserved densities in DELiA by Bocharov
(Pascal, 1990)
– Conserved densities and formal symmetries FS
by Gerdt and Zharkov (Reduce, 1993), Mikhailov and Yamilov
(MuMath, 1990)
– Conserved densities by Roelofs, Sanders and Wang
(Reduce 1994, Maple 1995, Form 1995-present)
– Conserved densities condens.m by Hereman and Göktaş
(Mathematica, 1996)
– Conservation laws, based on CRACK by Wolf (Reduce, 1995)
– Conservation laws by Hickman (Maple, 1996)
– Conserved densities by Ahner et al.
(Mathematica, 1995). Project halted.
– Conserved densities diffdens.m by Göktaş and Hereman
(Mathematica, 1997)
32
• Conclusions and Further Research
– Two Mathematica programs are available:
condens.m for evolution equations (PDEs)
diffdens.m for differential-difference equations (DDEs)
– Usefulness
∗ Testing models for integrability
∗ Study of classes of nonlinear PDEs or DDEs
– Comparison with other programs
∗ Parameter analysis is possible
∗ Not restricted to uniform rank equations
∗ Not restricted to evolution equations provided that
one can write the equation(s) as a system of evolution equations
– Future work
∗ Generalization towards broader classes of equations (e.g. uxt)
∗ Generalization towards more space variables (e.g. KP equation)
∗ Conservation laws with time and space dependent coefficients
33
∗ Exploit other symmetries in the hope to find conserved densities
of non-polynomial form
∗ Constants of motion for dynamical systems
(e.g. Lorenz and Hénon-Heiles systems)
– Research supported in part by NSF under Grant CCR-9625421
– In collaboration with Ünal Göktaş and Grant Erdmann
– Papers submitted to: J. Symb. Comp., and Phys. Lett. A
– Software: available via FTP, ftp site mines.edu
in subdirectory pub/papers/math cs dept/software/condens
or via the Internet
URL: http://www.mines.edu/fs home/whereman/
34
Table 1:
Conserved Densities for the Sawada-Kotera and Lax 5th-order equations
Density
Sawada-Kotera equation
Lax equation
ρ1
u
u
ρ2
----
1 2
2u
ρ3
1 3
3u
− ux 2
1 3
3u
− 16 ux 2
ρ4
1 4
4u
− 94 uux 2 + 43 u2x 2
1 4
4u
− 12 uux 2 +
ρ5
----
1 5
5u
− u2 ux 2 + 15 uu2x 2 −
ρ6
1 6
6u
1 6
6u
− 53 u3 ux 2 −
−
25 3
2
4 u ux
+2u2x 3 −
ρ7
ρ8
1 7
7u
−
17
4
8 ux
21
2
8 uu3x
− 9u4 ux 2 −
+ 6u2 u2x 2
+ 38 u4x 2
54
4
5 uux
+
2
2
+ 648
35 ux u2x +
489
3
35 uu2x
2
− 288
35 u2x u3x +
81
2
35 uu4x
----
5
+ 63
u2x 3 −
57 3
2
5 u u2x
−
−
261 2
2
35 u u3x
9
2
35 u5x
1 7
7u
5
4
36 ux
1
2
14 uu3x
+
1
2
70 u3x
+ 12 u2 u2x 2
1
2
252 u4x
− 52 u4 ux 2 − 56 uux 4 + u3 u2x 2
+ 12 ux 2 u2x 2 +
10
3
21 uu2x
5
u2x u3x 2 +
− 42
1 8
8u
1
2
20 u2x
−
1
2
42 uu4x
− 72 u5 ux 2 −
3 2
2
14 u u3x
1
2
924 u5x
−
35 2
4
12 u ux
+ 74 u4 u2x 2
+ 72 uux 2 u2x 2 + 35 u2 u2x 3 +
7
4
24 u2x
+ 12 u3 u3x 2
− 14 ux 2 u3x 2 − 56 uu2x u3x 2 +
1 2
2
12 u u4x
7
+ 132
u2x u4x 2 −
1
2
3432 u6x
35
1
2
132 uu5x
+
Table 2:
Conserved Densities for the Kaup-Kuperschmidt and Ito 5th-order equations
Density
Kaup-Kuperschmidt equation
Ito equation
ρ1
u
u
ρ2
----
u2
2
ρ3
u3
3
− 18 ux 2
----
ρ4
u4
4
−
ρ5
----
ρ6
u6
6
−
9
2
16 uux
u7
7
−
35 3
2
16 u ux
31
4
256 ux
−
15
2
128 uu3x
27 4
2
8 u ux
2
− 171
640 u2x u3x +
+
+
− 49 uux 2 + 34 u2x 2
51 2
2
64 u u2x
----
3
2
512 u4x
369
4
320 uux
−
2
2
+ 2619
4480 ux u2x +
ρ8
u4
4
3
2
64 u2x
----
37
+ 256
u2x 3 −
ρ7
+
+
2211
3
2240 uu2x
27
2
560 uu4x
−
69 3
2
40 u u2x
−
----
477 2
2
1120 u u3x
9
2
4480 u5x
----
----
36
Table 3:
Conserved Densities for the Sawada-Kotera-Ito and Lax 7th-order equations
Density
Sawada-Kotera-Ito equation
Lax equation
ρ1
u
u
ρ2
----
u2
ρ3
−u3 + ux 2
−2u3 + ux 2
ρ4
3u4 − 9uux 2 + u2x 2
5u4 − 10uux 2 + u2x 2
ρ5
----
−14u5 + 70u2 ux 2 − 14uu2x 2 + u3x 2
ρ6
6
− 12
7 u +
150 3
2
7 u ux
+
3
2
− 16
21 u2x + uu3x −
ρ7
17
4
7 ux
48 2
2
7 u u2x
− 37 u6 +
163
3
9 uu2x
2
2
− 32
9 u2x u3x + uu4x −
−
70 3
2
3 u ux
+
3
2
− 10
9 u2x + uu3x −
1
2
21 u4x
5u7 − 105u4 ux 2 − 42uux 4 +
+24ux 2 u2x 2 +
ρ8
−
133 3
2
3 u u2x
− 32 u7 +
29 2
2
3 u u3x
35 4
2
3 u ux
− 37 ux 2 u2x 2 −
1
2
27 u5x
5
2
9 u2x u3x
3 8
2u
----
+
35
4
18 ux
− 7u2 u2x 2
1
2
18 u4x
35
4
9 uux
20
3
9 uu2x
− 19 uu4x 2 +
−
14 3
2
3 u u2x
+ u2 u3x 2
1
2
198 u5x
− 42u5 ux 2 − 35u2 ux 4 + 21u4 u2x 2
+42uux 2 u2x 2 + 20u2 u2x 3 + 72 u2x 4 − 6u3 u3x 2
−3ux 2 u3x 2 − 10uu2x u3x 2 + u2 u4x 2
7
+ 11
u2x u4x 2 −
37
1
2
11 uu5x
+
1
2
286 u6x