.
SYMBOLIC COMPUTATION
OF
CONSERVED DENSITIES
Willy Hereman
Dept. of Mathematical and Computer Sciences
Colorado School of Mines
Golden, Colorado, USA
IMACS Conference
Applications of Computer Algebra
Tuesday, May 16, 1995
15:30
I. INTRODUCTION
Symbolic Software
• Solitons via Hirota’s method (Macsyma & Mathematica)
• Painlevé test for ODEs or PDEs (Macsyma)
• Conservation laws of PDEs (Mathematica)
• Lie symmetries for ODEs and PDEs (Macsyma)
Purpose of the programs
• Study of integrability of nonlinear PDEs
• Exact solutions as bench mark for numerical algorithms
• Classification of nonlinear PDEs
• Lie symmetries −→ solutions via reductions
Collaborators
• Ünal Göktaş (MS student)
• Chris Elmer (MS student)
• Wuning Zhuang (MS student)
• Ameina Nuseir (Ph.D student)
• Mark Coffey (CU-Boulder)
II. MATHEMATICA PROGRAM
FOR CONSERVED DENSITIES
• Purpose
Compute polynomial-type conservation laws
of single PDEs and systems of PDEs
Conservation law:
ρt + J x = 0
both ρ(u, ux, u2x, . . . , unx) and J(u, ux, u2x, . . . , unx)
Consequently
P =
+∞
−∞ ρdx
Z
= constant
provided J vanishes at infinity
Compare with constants of motions in classical
mechanics
• Example
Consider the KdV equation
ut + uux + u3x = 0
Conserved densities:
ρ1
=
u
ρ2
=
u2
ρ3
..
ρ6
=
u3 − 3u2x
..
=
u6 − 60u3u2x − 30u4x + 108u2u22x
720 3
648 2
216 2
+
u2x −
uu3x +
u4x
7
7
7
Integrable equations have ∞ conservation laws
• Algorithm and Implementation
Consider the scaling (weights) of the KdV
∂2
u ∼ 2,
∂x
∂
∂3
∼ 3
∂t ∂x
Compute building blocks of ρ3
(i) Start with building block u3
Divide by u and differentiate twice (u2)2x
Produces the list of terms
[u2x, uu2x] −→ [u2x]
Second list: remove terms that are total derivative
with respect to x or total derivative
up to terms earlier in the list
Divide by u2 and differentiate twice (u)4x
Produces the list: [u4x] −→ [ ]
[ ] is the empty list
Gather the terms:
ρ3 = u3 + c[1]u2x
where the constant c1 must be determined
∂ρ3
= 3u2ut + 2c1uxuxt
(ii) Compute
∂t
Replace ut by −(uux + uxxx) and uxt by −(uux + uxxx)x
(iii) Integrate the result with respect to x
Carry out all integrations by parts
∂ρ3
3
= −[ u4 + (c1 −3)uu2x + 3u2uxx − c1u2xx + 2c1uxuxxx]x
∂t
4
−(c1 + 3)u3x
The last non-integrable term must vanish
Thus, c1 = −3
Result:
ρ3 = u3 − 3u2x
(iv) Expression [. . .] yields
3
J3 = u4 − 6uu2x + 3u2uxx + 3u2xx − 6uxuxxx
4
Computer building blocks of ρ6
(i) Start with u6
Divide by u and differentiate twice
(u5)2x produces the list of terms
[u3u2x, u4u2x] −→ [u3u2x]
Next, divide u6 by u2, and compute (u4)4x
Corresponding list:
[u4x, uu2xu2x, u2u22x, u2uxu3x, u3u4x] −→ [u4x, u2u22x]
6
6
Proceed with ( uu3 )6x = (u3)6x, ( uu4 )8x = (u2)8x
6
and ( uu5 )10x = (u)10x
Obtain the lists:
[u32x, uxu2xu3x, uu23x, u2xu4x, uu2xu4x, uuxu5x, u2u6x] −→
[u32x, uu23x]
[u24x, u3xu5x, u2xu6x, uxu7x, uu8x] −→ [u24x]
and [u10x] −→ [ ]
Gather the terms:
ρ6 = u6 + c1u3u2x + c2u4x + c3u2u22x + c4u32x + c5uu23x + c6u24x
where the constants ci must be determined
(ii) Compute
∂
∂t ρ6
Replace ut, uxt, . . . , unx,t by −(uux + uxxx), . . .
(iii) Integrate the result with respect to x
Carry out all integrations by parts
Require that non-integrabe part vanishes
Set to zero all the coefficients of the independent
combinations involving powers of u and
its derivatives with respect to x
Solve the linear system for unknowns c1, c2, . . . , c6
Result:
ρ6 = u6 − 60u3u2x − 30u4x + 108u2u22x
648 2
216 2
720 3
u2x −
uu3x +
u4x
+
7
7
7
(iv) Flux J6 can be computed by substituting
the constants into the integrable part of ρ6
• Further Examples
* Conservation laws of generalized Schamel equation
2
n2ut + (n + 1)(n + 2)u n ux + uxxx = 0
n positive integer
ρ1 = u
ρ2 = u 2
1 2 n2 2+ 2
ρ3 = u x − u n
2
2
no further conservation laws
* Conserved densities of modified vector derivative
nonlinear Schrödinger equation
∂B⊥
∂
∂B⊥
∂ 2B⊥
2
+ (B⊥B⊥) + αB⊥0B⊥0 ·
+ ex ×
=0
2
∂t
∂x
∂x
∂x
Replace vector equation by
2
2
ut + u(u + v ) + βu − vx x = 0
2
2
vt + v(u + v ) + ux x = 0
u and v denote the components of B⊥ parallel
2
and perpendicular to B⊥0 and β = αB⊥0
The first 5 conserved densities are:
ρ1 = u 2 + v 2
1 2
ρ2 = (u + v 2)2 − uvx + uxv + βu2
2
1 2
β 4
1 2
2 3
2
3
3
ρ3 = (u + v ) + (ux + vx) − u vx + v ux + (u − v 4)
4
2
4
1 2
2
4
2 4
ρ4 = (u + v ) − (uxvxx − uxxvx) + (uux + vvx)2
4
5
5
6 2
+ (u + v 2)(u2x + vx2 ) − (u2 + v 2)2(uvx − uxv)
5
β2 4
β 2
3
6
4 2
6
+ (2ux − 4u vx + 2u + 3u v − v ) + u
5
5
ρ5 =
7 2
1
2
(u + v 2)5 + (u2xx + vxx
)
16
2
5 2
− (u + v 2)(uxvxx −uxxvx) + 5(u2 + v 2)(uux + vvx)2
2
+
15 2
35
(u + v 2)2(u2x + vx2 )2 − (u2 + v 2)3(uvx − uxv)
4
16
+
β 8
(5u + 10u6v 2 − 10u2v 6 − 5v 8 + 20u2u2x
8
− 12u5vx + 60uv 4vx − 20v 2vx2 )
β2 6
+
(u + v 6)
4
• Coupled Systems
* Conserved densities for the Coupled KdV Equations
(Hirota-Satsuma system)
ut − a(uxxx + 6uux) − 2bvvx = 0
vt + vxxx + 3uvx = 0
ρ1 = u
2
ρ2 = u2 + bv 2
3
1 2
ρ3 = (1 + a)(u − ux) + b(uv 2 − vx2 )
2
3
and e.g.
1 2
ρ4 = u −
+ uxx
5
4
2
8
2 2
4
+ b(u2v 2 + uvvxx + uvx2 − vxx
) + b2 v 4
5
3
3
3
15
4
provided a =
2uu2x
1
2
There are infinitely many more conservation laws
* Conserved densities for the Ito system
ut − uxxx − 6uux − 2vvx = 0
vt − 2uxv − 2uvx = 0
ρ1 =
1
u
2
1 2
ρ2 = (u + v 2)
2
1 2
ρ3 = u − ux + uv 2
2
3
and infinitely many more conservation laws
A Class of Fifth-order Evolution Equations
ut + αu2ux + βuxu2x + γuu3x + u5x = 0
Special cases:
α = 30
α = 5
β = 20
β=5
α = 20
α = 2
β = 25
β=6
γ = 10
γ=5
Lax
Sawada Kotera
or Caudry−Dodd−Gibbon
γ = 10
Kaup−Kuperschmidt
γ=3
Ito
Table 1
Density
Conserved Densities for Sawada-Kotera and Lax equations
Sawada-Kotera equation
Lax equation
ρ1
u
u
ρ2
----
1 2
2u
ρ3
1 3
3u
− u2x
1 3
3u
− 61 u2x
ρ4
1 4
4u
− 94 uu2x + 34 u22x
1 4
4u
− 21 uu2x +
ρ6
----
1 5
5u
− u2 u2x + 51 uu22x −
ρ6
1 6
6u
1 6
6u
− 35 u3 u2x −
−
25 3 2
4 u ux
+2u32x −
ρ7
1 7
7u
−
21
2
8 uu3x
----
5 3
+ 63
u2x −
+ 83 u24x
57 3 2
5 u u2x
1 7
7u
489
3
35 uu2x
−
261 2 2
35 u u3x
+ 21 u2x u22x +
−
9 2
35 u5x
5
− 42
u2x u23x +
81
2
35 uu4x
1 8
8u
5 4
36 ux
1
2
14 uu3x
+
2
− 288
35 u2x u3x +
ρ8
+ 6u2 u22x
54
4
5 uux
− 9u4 u2x −
2 2
+ 648
35 ux u2x +
17 4
8 ux
1 2
20 u2x
+
1 2
70 u3x
+ 12 u2 u22x
1 2
252 u4x
− 25 u4 u2x − 65 uu4x + u3 u22x
10
3
21 uu2x
−
1
2
42 uu4x
− 27 u5 u2x −
3 2 2
14 u u3x
1 2
924 u5x
−
35 2 4
12 u ux
+ 27 uu2x u22x + 35 u2 u32x +
+ 74 u4 u22x
7 4
24 u2x
− 41 u2x u23x − 56 uu2x u23x +
7
+ 132
u2x u24x −
1
2
132 uu5x
+ 21 u3 u23x
1 2 2
12 u u4x
+
1
2
3432 u6x
Table 2
Conserved Densities for Kaup-Kuperschmidt and Ito equations
Density
Kaup-Kuperschmidt equation
Ito equation
ρ1
u
u
ρ2
----
u2
2
ρ3
u3
3
− 81 u2x
----
ρ4
u4
4
−
ρ5
----
ρ6
u6
6
−
9
2
16 uux
ρ8
u7
7
−
u4
4
3 2
64 u2x
35 3 2
16 u ux
−
31 4
256 ux
15
2
128 uu3x
27 4 2
8 u ux
−
+
+
51 2 2
64 u u2x
----
3 2
512 u4x
369
4
320 uux
+
2619 2 2
+ 4480
ux u2x +
2211
3
2240 uu2x
171
u2x u23x +
− 640
27
2
560 uu4x
----
− 94 uu2x + 34 u22x
----
37 3
+ 256
u2x −
ρ7
+
69 3 2
40 u u2x
−
−
----
477 2 2
1120 u u3x
9
2
4480 u5x
----