.
SYMBOLIC SOFTWARE FOR SOLITON THEORY:
INTEGRABILITY, SYMMETRIES
CONSERVATION LAWS
AND EXACT SOLUTIONS
Willy Hereman
Dept. of Mathematical and Computer Sciences
Colorado School of Mines
Golden, Colorado
Kruskal Fest
Symposium in Applied Mathematics:
Nonlinear Waves, Dynamics, Asymtotic Analysis
and Physical Applications
Boulder, Colorado
August 3-6, 1995
I. INTRODUCTION
Symbolic Software
• Painlevé test for systems of ODEs and PDEs
(Macsyma & Mathematica)
• Conservation laws of systems of evolution equations
(Mathematica)
• Solitons via Hirota’s method (Macsyma & Mathematica)
• Lie symmetries for systems of 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, Chris Elmer, Wuning Zhuang
(MS students)
• Ameina Nuseir (Ph.D student)
• Mark Coffey (CU-Boulder)
• Tony Miller & Tracy Otto (BS students)
II. SYMBOLIC SOFTWARE
Program 1 – Macsyma
Painlevé Integrability Test
for Systems of ODEs and PDEs
Integrability of (a systems of) ODEs or PDEs requires that the
only movable singularities in its solution are poles
Definition: A single equation or system has the Painlevé
Property if its solution in the complex plane has no worse
singularities than movable poles
Aim: Verify whether or not the system of equations
satisfies the necessary criteria to have the Painlevé Property
For simplicity, consider the case of a single PDE
The solution f expressed as a Laurent series
f = gα
∞
X
k=0
should only have movable poles
uk g k
Steps of the Painlevé Test
• Step 1:
1. Substitute the leading order term
f ∝ u0 g α
into the given equation
2. Determine the integer α < 0 by balancing
the most singular terms in g
3. Calculate u0
• Step 2:
1. Substitute the generic terms
f ∝ u0 g α + ur g α+r
into the equation, retaining its most singular terms
2. Require that ur is arbitrary
3. Determine the corresponding values of r > 0
called resonances
• Step 3:
1. Substitute the truncated expansion
f =g
R
α X
k=0
uk g k
into the complete equation
(R represents the largest resonance)
2. Determine uk unambiguously at the non-resonance levels
3. Check whether or not the compatibility
condition is satisfied at resonance levels
• An equation or system has the Painlevé Property and
is conjectured to be integrable if:
1. Step 1 thru 3 can be carried out consistently with α < 0
and with positive resonances
2. The compatibility conditions are identically
satisfied for all resonances
• The above algorithm does not detect essential
singularities
Painlevé Integrability Test
• Painlevé test for 3rd order equations by Hajee
(Reduce, 1982)
• Painlevé program (parts) by Hlavatý (Reduce, 1986)
• ODE Painlevé by Winternitz & Rand
(Macsyma, 1986)
• PDE Painlevé by Hereman & Van den Bulck
(Macsyma, 1987)
• Painlevé test by Conte & Musette (AMP, 1988)
• Painlevé analysis by Renner (Reduce, 1992)
• Painlevé test for systems of ODEs and PDEs
by Hereman, Elmer and Göktaş
(Macsyma, 1994-96, under development)
• Painlevé test for single ODEs and PDEs
by Hereman, Miller and Otto
(Mathematica, 1995, under development)
• Painlevé test for systems of ODEs and PDEs
by Hereman and Miller
(Mathematica, 1995-96, planned)
Painlevé Test of Systems
• System of ODEs due to Akhiezer
uxx + 2u3 − 2au + 2bv = 0
vxx + 2v 3 − 2av + 2bu = 0
∞
X
α1
u(x) = g (x)
v(x) = g α2 (x)
k=0
∞
X
k=0
uk g k (x)
vk g k (x)
Leading orders: α1 = α2 = −1
Resonances: r = −1, r = 4
Coefficients:
u0
u1
u2
u3
=
=
=
=
v0
v1
v2
v3
=i
=0
= i(b − a)/3
=0
At level 4: compatibility condition is satisfied
Coefficients u4 and v4 are arbitrary and independent
The system passes the Painlevé test
• Carleman system (simplified version)
ut + ux − c uv = 0
vt − vx + c uv = 0
α1
u(x, t) = g (x, t)
v(x, t) = g α2 (x, t)
∞
X
k=0
∞
X
k=0
uk (x, t) g k (x, t)
vk (x, t) g k (x, t)
Leading orders: α1 = α2 = −1
Resonances: r = −1, r = 1
Coefficients:
1
1
u0 = (gt − gx), v0 = − (gt + gx)
c
c
At level 1: compatibility condition is satisfied
However u1 and v1 dependent on each other
gxx − gtt + c v1(gt − gx)
u1 =
c (gt + gx)
The system passes the Painlevé test
• Coupled KdV Equations (Hirota-Satsuma system)
1
ut − 3uux + 6vvx − uxxx = 0
2
vt + vxxx + 3uvx = 0
Use Kruskal simplification, g(x, t) = x − h(t), in
α1
u(x, t) = g (x, t)
v(x, t) = g α2 (x, t)
∞
X
k=0
∞
X
k=0
uk (x, t) g k (x, t)
vk (x, t) g k (x, t)
Leading orders: First Branch α1 = α2 = −2
Resonances: r = −2, −1, 3, 4, 6, 8
Coefficients:
u0 = −4, v0 = −2
u1 = v 1 = 0
u2 = ht/3, v2 = 2ht/3
htt + 30v3ht
htt − 12v3ht
u5 = −
, v5 =
63
63
v4t + 12v3v4
4hthtt + v3(8h2t − 84v4) + 21v4t
u7 = −
, v7 =
12
504
At level 3: compatibility condition is satisfied
Here u3 = v3 , one function is arbitary
At level 4: compatibility condition is satisfied
Here u4 = 2v4, one function is arbitrary
At level 6: compatibility condition is satisfied
−v3t + 24v6 − 3v32
,
Here u6 =
12
one function is arbitrary
At level 8: compatibility condition is satisfied
Here u8 =
−httt + 6v3htt + (12v3t + 198v32)ht − 3024v8 − 756v42
,
756
one function is arbitrary
Leading orders: Second Branch α1 = −2, α2 = −1
Resonances: r = −1, 0, 1, 4, 5, 6
Coefficients:
u0 = −2, v0 free
u1 = 0, v1 free
u2 = −(2ht + 3v02)/6, v2 = (4v0ht + 3v03)/12
v0t + 3v02v1
u3 = −v0v1, v3 =
12
At level 0: compatibility condition is satisfied
Coefficient v0 is arbitary
At level 1: compatibility condition is satisfied
Coefficient v1 is arbitary
At level 4: compatibility condition is satisfied
16v0h2t + 24v03ht − 24v1t + 9v05 + 72u4v0
Here u4 = −
,
288
one function is arbitrary
At level 5: compatibility condition is satisfied
Here
2htt − 12v0v1ht + 3v0v0t − 9v03v1
u5 =
18
one function is arbitrary
At level 6: compatibility condition is satisfied
Here
1
u6 =
(−64v0h3t − 144v03h2t +(96v1t − 108v05 − 480u4v0)ht
5760
+24v02v1t − 16v0tt + 288v03v12 − 27v07 − 360u4v03 + 576u6v0)
one function is arbitrary
The system passes the Painlevé test
Program 2 – Mathematica
Conserved Densities
• Purpose
Compute polynomial-type conservation laws of
single evolution equations and systems of evolution equations
For simplicity, consider a single evolution equation
ut = F(u, ux, uxx, ..., unx)
Conservation law is of the form
ρt + J x = 0
both ρ(u, ux, u2x, . . . , unx) and J(u, ux, u2x, . . . , unx)
are polynomials in their arguments
Consequently
P =
+∞
−∞ ρ
Z
dx = constant
provided J vanishes at infinity
– Conservation laws describe the conservation of
fundamental physical quantities such as mass,
linear momentum, total energy (compare with constants
of motion in mechanics)
– For nonlinear PDEs, the existence of a sufficiently large
(in principal infinite) number of conservation laws assures complete integrability
– Tool to test numerical integrators for PDEs
• Example
Consider the KdV equation, ut + uux + u3x = 0
Conserved densities:
ρ1
=
u
ρ2
=
u2
ρ3
..
ρ6
=
u3 − 3u2x
..
=
u6 − 60u3u2x − 30u4x + 108u2u22x
648 2
216 2
720 3
u −
uu3x +
u
+
7 2x
7
7 4x
Integrable equations have ∞ many 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
(ii) Compute
= 3u2ut + 2c1uxuxt
∂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
3
∂ρ3
= −[ 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 4
J3 = u − 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
Conserved Densities
• Conserved densities by Ito & Kako
(Reduce, 1985, 1994)
• Conserved densities in DELiA by Bocharov
(Pascal, 1990)
• Conserved densities by Gerdt (Reduce, 1993)
• Conserved densities by Roelofs, Sanders and Wang
(Reduce 1994, Maple 1995)
• Conserved densities by Hereman and Göktaş
(Mathematica, 1993-1995)
• Conservation laws by Wolf (Reduce, 1995)
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
Under what conditions for the parameters α, β, and γ does
this equation admit ∞ many conservation laws?
3 2
γ and β = 2γ then there is a sequence
• If α = 10
(without gaps!) of conserved densities (Lax case)
• If α = 15 γ 2 and β = γ then there is a sequence
(with gaps!) of conserved densities (SK case)
• If α = 15 γ 2 and β = 25 γ then there is a sequence
(with gaps!) of conserved densities (KK case)
2
2
• If α = −2β +7βγ−4γ
or β = 2γ then there is a conserved
45
density of degree 4
2
Combining both conditions gives α = 2γ9 and β = 2γ (Ito
case)
Conserved Densities of Systems of Evolution Eqs.
• Coupled KdV Equations (Hirota-Satsuma system)
ut − a(uxxx + 6uux) − 2bvvx = 0
vt + vxxx + 3uvx = 0
∂2
u ∼ 2,
∂x
∂2
v∼ 2
∂x
ρ1 = u
2
ρ2 = u2 + bv 2
3
1
ρ3 = (1 + a)(u3 − u2x) + b(uv 2 − vx2 )
2
and e.g.
1
ρ4 = u4 − 2uu2x + u2xx
5
4
2
8
2 2
4
+ b(u2v 2 + uvvxx + uvx2 − vxx
) + b2 v 4
5
3
3
3
15
provided a =
1
2
There are infinitely many more conservation laws
• The Ito system
ut − uxxx − 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 − u2x
ρ4 = 5u4 + 6u2v 2 + v 4 − 10uu2x + 2v 2u2x + u22x
and there are infinitely many more conservation laws
• The Drinfel’d-Sokolov system
ut + 3vvx = 0
vt + vux + 2uvx + 2v3x = 0
∂2
u ∼ 2,
∂x
∂2
v free, choose v ∼ 2
∂x
ρ1 = u
ρ2 = v 2
−2 3
1 2 3 2
2
ρ3 =
u + uv + ux − vx
9
6
2
1 4 1 2 2 1 4 1 2
ρ4 = u − u v − v − uux + uvx2
2
6
8
6
−
5 2
1
3 2
v u2x + u22x − v2x
12
36
4
and there are presumably infinitely many more
conservation laws
• The dispersiveless long-wave system
ut + vux + uvx = 0
vt + ux + vvx = 0
u free, v free, but u ∼ 2v
∂
∂
choose u ∼
and 2v ∼
∂x
∂x
ρ1
ρ2
ρ3
ρ4
ρ5
=
=
=
=
=
ρ6 =
v
u
uv
u2 + uv 2
3u2v + uv 3
1 3
1
u + u2v 2 + uv 4
3
6
1 5
ρ7 = u v + u v + uv
10
3
2 3
1 4
1 6
3 2
2 4
ρ8 = u + 2u v + u v + uv
3
15
always the same set irrespective the choice of weights
Further Examples
• A generalized Schamel equation
2
n
2
n ut + (n + 1)(n + 2)u ux + uxxx = 0
n positive integer
ρ1 = u,
ρ 2 = u2
1 2 n2 2+ 2
ρ3 = u x − u n
2
2
no further conservation laws
• The Boussinesq equation
utt − buxx + 3uuxx + 3u2x + au4x = 0
a and b are real constants
Write as system of evolution equations
ut + v x = 0
vt + bux − 3uux − au3x = 0
∂2
u ∼ 2,
∂x
∂2
b ∼ 2,
∂x
∂3
v free, take v ∼ 3
∂x
ρ1 = u
ρ2 = u
ρ3 = bu
ρ4 = bc1u + c2uv
ρ5 = b2c1u + c2(bu2 − u3 + v 2 + au2x)
and there are infinitely many more conservation laws
• A modified vector derivative NLS 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
∂
∂
∂
,
2v ∼
,
β∼
2u ∼
∂x
∂x
∂x
The first 6 conserved densities are:
ρ1 = c 1 u + c 2 v
ρ2 = u 2 + v 2
1 2
ρ3 = (u + v 2)2 − uvx + uxv + βu2
2
ρ4 =
1 2
1
β
(u + v 2)3 + (u2x + vx2 ) − u3vx + v 3ux + (u4 − v 4)
4
2
4
ρ5 =
2
4
1 2
(u + v 2)4 − (uxvxx − uxxvx) + (uux + vvx)2
4
5
5
6
+ (u2 + 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
7 2
2 5 1 2
2
ρ6 =
(u + v ) + (uxx + 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 ) − (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
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
----
+ 83 u24x
5 3
u2x −
+ 63
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
u2x u24x −
+ 132
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
− 640
u2x u23x +
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
----
Example 3 – Macsyma/Mathematica
Solitons – Hirota’s Method
• Hirota’s Direct Method
allows to construct soliton solutions of
– nonlinear evolution equations
– wave equations
– coupled systems
• Test conditions for existence of soliton solutions
• Examples:
– Korteweg-de Vries equation (KdV)
ut + 6uux + u3x = 0
– Kadomtsev-Petviashvili equation (KP)
(ut + 6uux + u3x)x + 3u2y = 0
– Sawada-Kotera equation (SK)
ut + 45u2ux + 15uxu2x + 15uu3x + u5x = 0
Hirota’s Method
Korteweg-de Vries equation
ut + 6uux + u3x = 0
Substitute
∂ 2 ln f (x, t)
u(x, t) = 2
∂x2
Integrate with respect to x
2
f fxt − fxft + f f4x − 4fxf3x + 3f2x
=0
Bilinear form
def
B(f ·f ) = DxDt +
Dx4
(f ·f ) = 0
Introduce the bilinear operator
m
n
Dxm Dtn(f ·g) = (∂x − ∂x0) (∂t − ∂t0) f (x, t) g(x0, t0)|x0=x,t0=t
Use the expansion
f =1+
∞
X
n=1
n fn
Substitute f into the bilinear equation
Collect powers in (book keeping parameter)
O(0)
O(1)
O(2)
O(3)
O(4)
:
:
:
:
:
B(1·1) = 0
B(1·f1 + f1·1) = 0
B(1·f2 + f1·f1 + f2·1) = 0
B(1·f3 + f1·f2 + f2·f1 + f3·1) = 0
B(1·f4 + f1·f3 + f2·f2 + f3·f1 + f4·1) = 0
n
X
n
O( ) : B(
j=0
fj ·fn−j ) = 0
with f0 = 1
Start with
f1 =
N
X
i=1
exp(θi) =
N
X
i=1
exp (ki x − ωi t + δi)
ki, ωi and δi are constants
Dispersion law
ωi = ki3
(i = 1, 2, ..., N )
If the original PDE admits a N-soliton solution
then the expansion will truncate at level n = N
Consider the case N=3
Terms generated by B(f1, f1) determine
f2 =
=
+
+
a12
a12
a13
a23
exp(θ1 + θ2) + a13 exp(θ1 + θ3) + a23 exp(θ2 + θ3)
exp [(k1 + k2) x − (ω1 + ω2) t + (δ1 + δ2)]
exp [(k1 + k3) x − (ω1 + ω3) t + (δ1 + δ3)]
exp [(k2 + k3) x − (ω2 + ω3) t + (δ2 + δ3)]
Calculate the constants a12, a13 and a23
(ki − kj )2
i, j = 1, 2, 3
aij =
2
(ki + kj )
Terms from B(f1·f2 + f2·f1) determine
f3 = b123 exp(θ1 + θ2 + θ3)
= b123 exp [(k1 +k2 +k3)x−(ω1 +ω2 +ω3)t+(δ1 +δ2 +δ3)]
with
(k1 − k2)2 (k1 − k3)2 (k2 − k3)2
b123 = a12 a13 a23 =
(k1 + k2)2 (k1 + k3)2 (k2 + k3)2
Subsequently, fi = 0 for i > 3
Set = 1
f = 1 + exp θ1 + exp θ2 + exp θ3
+ a12 exp(θ1 + θ2) + a13 exp(θ1 + θ3) + a23 exp(θ2 + θ3)
+ b123 exp(θ1 + θ2 + θ3)
Return to the original u(x, t)
∂ 2 ln f (x, t)
u(x, t) = 2
∂x2
Single soliton solution
f = 1 + eθ ,
θ = kx − ωt + δ
k, ω and δ are constants and ω = k 3
Substituting f into
∂ 2 ln f (x, t)
u(x, t) = 2
∂x2
fxxf − fx2
= 2(
)
f2
Take k = 2K
u = 2K 2sech2K(x − 4K 2t + δ)
Two-soliton solution
f = 1 + eθ1 + eθ2 + a12eθ1+θ2
θi = kix − ωit + δi
with ωi =
ki3
, (i = 1, 2) and a12 =
(k1 −k2 )2
(k1 +k2 )2
Select
δi
e
c2i kix−ωit+∆i
= e
ki
1 − 1 (θ̃1+θ̃2)
˜
f = fe 2
4
θ̃i = kix − ωit + ∆i
k + k1
 2

c2i =


k2 − k1 ki
Return to u(x, t)
∂ 2 ln f˜(x, t)
u(x, t) = ũ(x, t) = 2
∂x2
2
 k2


=

−
2


k22cosech2 θ̃22 + k12sech2 θ̃21 



θ̃2
θ̃1 2 
(k2 coth 2 − k1 tanh 2 )
k12  

Program 4 – Macsyma
Lie-point Symmetries
• System of m differential equations of order k
∆i(x, u(k)) = 0,
i = 1, 2, ..., m
with p independent and q dependent variables
x = (x1, x2, ..., xp) ∈ IRp
u = (u1, u2, ..., uq ) ∈ IRq
• The group transformations have the form
x̃ = Λgroup (x, u),
ũ = Ωgroup (x, u)
where the functions Λgroup and Ωgroup are to be determined
• Look for the Lie algebra L realized by the vector field
p
q
∂
∂
X
X
i
α=
η (x, u)
+ ϕl (x, u) l
i=1
∂xi l=1
∂u
Procedure for finding the coefficients
• Construct the k th prolongation pr(k)α of the vector field α
• Apply it to the system of equations
• Request that the resulting expression vanishes
on the solution set of the given system
pr(k)α∆i |∆j =0
i, j = 1, ..., m
• This results in a system of linear homogeneous PDEs
for η i and ϕl , with independent variables x and u
( determining equations)
• Procedure thus consists of two major steps:
deriving the determining equations
solving the determining equations
Procedure for Computing the Determining Equations
• Use multi-index notation J = (j1, j2, ..., jp) ∈ INp,
to denote partial derivatives of ul
ulJ
∂ |J|ul
≡
,
j
j
j
p
1
2
∂x1 ∂x2 ...∂xp
where |J| = j1 + j2 + ... + jp
• u(k) denotes a vector whose components are all the partial
derivatives of order 0 up to k of all the ul
• Steps:
(1) Construct the k th prolongation of the vector field
q X
X
(k)
J
(k) ∂
pr α = α +
ψl (x, u ) l , 1 ≤ |J| ≤ k
∂uJ
l=1 J
The coefficients ψlJ of the first prolongation are:
ψlJi
= Diϕl (x, u) −
p
X
j=1
ulJj Diη j (x, u),
where Ji is a p−tuple with 1 on the ith position and zeros
elsewhere
Di is the total derivative operator
q X
∂
∂
X
l
Di =
+
uJ+Ji l , 0 ≤ |J| ≤ k
∂xi l=1 J
∂uJ
Higher order prolongations are defined recursively:
ψlJ+Ji
=
DiψlJ
−
p
X
j=1
ulJ+Jj Diη j (x, u),
|J| ≥ 1
(2) Apply the prolonged operator pr(k)α to each
equation ∆i(x, u(k)) = 0
Require that pr(k)α vanishes on the solution set of the system
pr(k)α ∆i |∆j =0 = 0 i, j = 1, ..., m
(3) Choose m components of the vector u(k),
say v 1, ..., v m, such that:
(a) Each v i is equal to a derivative of a ul (l = 1, ..., q)
with respect to at least one variable xi (i = 1, ..., p).
(b) None of the v i is the derivative of another one in the
set.
(c) The system can be solved algebraically for the v i in
terms of the remaining components of u(k), which we denoted by w:
v i = S i(x, w),
i = 1, ..., m.
(d) The derivatives of v i,
vJi = DJ S i(x, w),
where DJ ≡ D1j1 D2j2 ...Dpjp , can all be expressed in terms
of the components of w and their derivatives, without ever
reintroducing the v i or their derivatives.
For instance, for a system of evolution equations
uit(x1, ..., xp−1, t) = F i(x1, ..., xp−1, t, u(k)),
i = 1, ..., m,
where u(k) involves derivatives with respect to the variables
xi but not t, choose v i = uit.
(4) Eliminate all v i and their derivatives from the expression prolonged vector field, so that all the remaining
variables are independent
(5) Obtain the determining equations for η i(x, u) and
ϕl (x, u) by equating to zero the coefficients of the remaining independent derivatives ulJ .
Example: The Korteweg-de Vries Equation
ut + auux + uxxx = 0
• one equation (parameter a)
• two independent variables t and x
• one dependent variable u
∂
∂
∂
+ η t ∂t
+ ϕu ∂u
• vector field α = η x ∂x
Format for SYMMGRP.MAX
• variables x[1] = x,
x[2] = t,
u[1] = u
• equation e1 : u[1, [0, 1]] + a ∗ u[1] ∗ u[1, [1, 0]] + u[1, [3, 0]]
• variable to be eliminated v1 : u[1, [0, 1]]
• coefficients of vectorfield in SYMMGRP.MAX:
eta[1] = η x, eta[2] = η t and phi[1] = ϕu
There are only eight determining equations
∂eta[2]
= 0
∂u[1]
∂eta[2]
= 0
∂x[1]
∂eta[1]
= 0
∂u[1]
∂ 2phi[1]
= 0
∂u[1]2
∂ 2phi[1]
∂ 2eta[1]
−
= 0
2
∂u[1]∂x[1]
∂x[1]
∂phi[1]
∂phi[1] ∂ 3phi[1]
+
+
u[1]
= 0
3
∂x[2]
∂x[1]
∂x[1]
∂eta[1] ∂ 3eta[1]
∂ 3phi[1]
−
3
−
∂u[1]∂x[1]2
∂x[2]
∂x[1]3
eta[1]
+ 2 u[1]
+ phi[1] = 0
∂x[1]
∂eta[2]
∂ 3phi[1]
∂eta[1] ∂ 3eta[1]
u[1]
+3
−
−
∂x[2]
∂u[1]∂x[1]2
∂x[2]
∂x[1]3
eta[1]
− u[1]
+ phi[1] = 0
∂x[1]
The solution in the original variables
η x = k1 + k3 t − k4 x
η t = k2 − 3k4 t
ϕ u = k3 + 2 k4 u
The four infinitesimal generators are
G1
G2
G3
G4
=
=
=
=
∂x
∂t
t ∂x + ∂u
x ∂x + 3 t ∂t − 2 u ∂u
Equation is invariant under:
• translations G1 and G2
• Galilean boost G3
• scaling G4
Computation of the flows corresponding to G1 thru G4 shows that
for any solution u = f (x, t) of the KdV equation the transformed
solutions
ũ = f (x − , t)
ũ = f (x, t − )
ũ = f (x − , t) + ũ = e−2f (e−x, e−3t)
will solve the KdV equation
Note that is the parameter of the transformation group
III. PLANS FOR THE FUTURE
Extension of Symbolic Software Packages
(Macsyma/Mathematica)
• Lie symmetries of differential-difference equations
• Solver for systems of linear, homogeneous PDEs
(Hereman)
• Painlevé test for systems of PDEs
(Elmer, Göktaş & Coffey)
• Solitons via Hirota’s method for bilinear equations (Zhuang)
• Simplification of Hirota’s method (Hereman & Nuseir)
• Conservation laws of PDEs with variable coefficients (Göktaş)
• Lax pairs, special solutions, ...
New Software
• Wavelets (prototype/educational tool)
• Other methods for Differential Equations