Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Construction of Lax Pairs in Matrix Form and the
Drinfel’d-Sokolov Method for Conservation Laws
Jacob Rezac
Colorado School of Mines
06/15/2012
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
1
Introduction
2
Construction of Lax pairs
3
Conservation Laws
4
Conclusions
Collaborators: Willy Hereman, Sara Clifton, Oscar Aguilar (CSM)
Funded by NSF research award no. CCF-0830783
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
What is a matrix Lax pair?
For a PDE,
ut = f (u, ux , uxx , ...),
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
What is a matrix Lax pair?
For a PDE,
ut = f (u, ux , uxx , ...),
matrices X and T which satisfy
Φx = X Φ
and
Φt = T Φ,
for an eigenfunction Φ are called a Lax pair.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
What is a matrix Lax pair?
For a PDE,
ut = f (u, ux , uxx , ...),
matrices X and T which satisfy
Φx = X Φ
and
Φt = T Φ,
for an eigenfunction Φ are called a Lax pair.
This is equivalent to the equation
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
What is a matrix Lax pair?
For a PDE,
ut = f (u, ux , uxx , ...),
matrices X and T which satisfy
Φx = X Φ
and
Φt = T Φ,
for an eigenfunction Φ are called a Lax pair.
This is equivalent to the equation
Xt − Tx + [X , T ] =
˙ 0
for [X , T ] = XT − TX .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
What is a matrix Lax pair?
For a PDE,
ut = f (u, ux , uxx , ...),
matrices X and T which satisfy
Φx = X Φ
and
Φt = T Φ,
for an eigenfunction Φ are called a Lax pair.
This is equivalent to the equation
Xt − Tx + [X , T ] =
˙ 0
for [X , T ] = XT − TX .
As before, =
˙ means equality on solutions of the PDE.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the Korteweg-de Vries (KdV) equation,
ut + αuux + uxxx = 0.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the Korteweg-de Vries (KdV) equation,
ut + αuux + uxxx = 0.
A known Lax pairs is
XKdV =
TKdV =
−4λ2 +
0
1
λ − 61 αu 0
1
6 αux
1 2 2
1
αλu
+ 18
α u
3
Jacob Rezac
,
+ 16 αuxx
−4λ − 13 αu
− 61 αux
Computation of Lax Pairs
.
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the Korteweg-de Vries (KdV) equation,
ut + αuux + uxxx = 0.
A known Lax pairs is
XKdV =
TKdV =
−4λ2 +
0
1
λ − 61 αu 0
1
6 αux
1 2 2
1
αλu
+ 18
α u
3
,
+ 16 αuxx
−4λ − 13 αu
− 61 αux
.
The isospectral parameter λ (λt = 0) is necessary for non-triviality.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Substituting these into the Lax equation,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Substituting these into the Lax equation,
0
0
(XKdV )t − (TKdV )x + [XKdV , TKdV ] =
− 1 α (ut + αuux + uxxx ) 0
6 0 0
=
˙
.
0 0
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, ν).
Jacob Rezac
Computation of Lax Pairs
(1)
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, ν).
Then,
Jacob Rezac
Computation of Lax Pairs
(1)
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, ν).
Then,
ut + uux + uxxx = 0 → κ−5 (ντ + ννξ + νξξξ ) .
Jacob Rezac
Computation of Lax Pairs
(1)
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, ν).
(1)
Then,
ut + uux + uxxx = 0 → κ−5 (ντ + ννξ + νξξξ ) .
Call each exponent on κ in (1) the weight of a variable (denoted
W (·)). That is,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Scaling Invariance and Weight
Rescale the variables in the KdV equation to
(x, t, u) → (κ−1 x, κ−3 t, κ2 u) = (ξ, τ, ν).
(1)
Then,
ut + uux + uxxx = 0 → κ−5 (ντ + ννξ + νξξξ ) .
Call each exponent on κ in (1) the weight of a variable (denoted
W (·)). That is,
∂
∂
=1
W
=3
W (u) = 2.
W
∂x
∂t
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
Jacob Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
Jacob Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
Jacob Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
Jacob Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
Jacob Rezac
∂
∂t
=3
W (u) = 2.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
∂
∂t
=3
W (u) = 2.
Why is W (λ) = 2?
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Consider again the KdV equation X matrix,
 weight 0
weight 0
z}|{
z}|{

0
1

1
XKdV = 
 λ − αu
0
|{z}

| {z6 } weight 0






weight 2
We have
W
∂
∂x
=1
W
∂
∂t
=3
W (u) = 2.
Why is W (λ) = 2? Set
λ = iκ2 .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
Jacob Rezac
∂
∂t
=3
W (u) = 1
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
Jacob Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
{z2iλu}
|4iλ +




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
Jacob Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
{z2iλu}
|4iλ +




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
Jacob Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
{z2iλu}
|4iλ +




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
Jacob Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
{z2iλu}
|4iλ +




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
More interestingly, the modified KdV (mKdV) equation,
ut + αu 2 ux + uxxx = 0,
with weights
W
∂
∂x
=1
W
∂
∂t
=3
has Lax pair with T -component

weight 3
z
}|
{


−4iλ3 − 2iλu 2
TmKdV = 
 4λ2 u − 2iλux + 2u 3 − uxx
|
{z
}
weight 3
Jacob Rezac
W (u) = 1
weight 3

z
}|
{
4λ2 u + 2iλux + 2u 3 − uxx
3
2
{z2iλu}
|4iλ +




Computation of Lax Pairs
weight 3
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Construction
Assume Lax matrices have weight invariant elements, e.g.,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Construction
Assume Lax matrices have weight invariant elements, e.g.,
W (X11 ) W (X12 )
W (T11 ) W (T12 )
W (X ) =
W (T ) =
.
W (X21 ) W (X22 )
W (T21 ) W (T22 )
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Construction
Assume Lax matrices have weight invariant elements, e.g.,
W (X11 ) W (X12 )
W (T11 ) W (T12 )
W (X ) =
W (T ) =
.
W (X21 ) W (X22 )
W (T21 ) W (T22 )
We substitute these weights into the Lax equation and force scale
invariance,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Construction
Assume Lax matrices have weight invariant elements, e.g.,
W (X11 ) W (X12 )
W (T11 ) W (T12 )
W (X ) =
W (T ) =
.
W (X21 ) W (X22 )
W (T21 ) W (T22 )
We substitute these weights into the Lax equation and force scale
invariance,
W (Xt ) = W (Tx ) = W ([X , T ]) ,
or,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Construction
Assume Lax matrices have weight invariant elements, e.g.,
W (X11 ) W (X12 )
W (T11 ) W (T12 )
W (X ) =
W (T ) =
.
W (X21 ) W (X22 )
W (T21 ) W (T22 )
We substitute these weights into the Lax equation and force scale
invariance,
W (Xt ) = W (Tx ) = W ([X , T ]) ,
or,
W (X ) + W
∂
∂x
Jacob Rezac
∂
= W (T ) + W
∂t
= W (X ) + W (T ) .
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving the system of equations produced from this yields
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving the system of equations produced from this yields
∂
W ∂x
W (X12)
W (X ) =
∂
∂
2W ∂x
− W (X12 ) W ∂x
and
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving the system of equations produced from this yields
∂
W ∂x
W (X12)
W (X ) =
∂
∂
2W ∂x
− W (X12 ) W ∂x
and
W (T ) =
W
∂
∂t
∂
W ∂t
− W (X12 ) + W
Jacob Rezac
∂
∂x
W
∂
∂t
+ W (X12) − W
∂
W ∂t
Computation of Lax Pairs
∂
∂x
.
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Step 3: Substitute matrices into Lax equation.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Step 3: Substitute matrices into Lax equation.
Step 4: Solve for undetermined coefficients.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Method of Construction
Step 1: Pick W (X12 )
Step 2: Based on the weight matrices of X and T , generate elements
of X and T with undetermined coefficients.
Step 3: Substitute matrices into Lax equation.
Step 4: Solve for undetermined coefficients.
Step 5: Substitute back into X and T .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the KdV equation (α = 6) and assume (Step 1)
W (X12 ) = 0. Then, our candidate Lax pair is (Step 2)
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the KdV equation (α = 6) and assume (Step 1)
W (X12 ) = 0. Then, our candidate Lax pair is (Step 2)
c1 k
c3
X =
c4 k 2 + c5 u −c1 k
and
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the KdV equation (α = 6) and assume (Step 1)
W (X12 ) = 0. Then, our candidate Lax pair is (Step 2)
c1 k
c3
X =
c4 k 2 + c5 u −c1 k
and
T =
T11 T12
T21 −T11
,
where
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider the KdV equation (α = 6) and assume (Step 1)
W (X12 ) = 0. Then, our candidate Lax pair is (Step 2)
c1 k
c3
X =
c4 k 2 + c5 u −c1 k
and
T =
T11 T12
T21 −T11
,
where
T11 = c6 k 3 + c7 ku + c8 ux ,
T12 = c9 k 2 + c10 u,
T21 = c11 k 4 + c12 k 2 u + c13 kux + c14 u 2 + c15 uxx .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
We substitute this into the Lax equation (Step 3) and get, for the
(1, 1)-element,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
We substitute this into the Lax equation (Step 3) and get, for the
(1, 1)-element,
(c11 c3 − c4 c9 ) k 4 + (c12 c3 − c10 c4 − c5 c9 ) k 2 u + (c14 c3 − c10 c5 ) u 2
+ (c13 c3 − c7 ) kux + (c15 c3 − c8 ) uxx = 0.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
We substitute this into the Lax equation (Step 3) and get, for the
(1, 1)-element,
(c11 c3 − c4 c9 ) k 4 + (c12 c3 − c10 c4 − c5 c9 ) k 2 u + (c14 c3 − c10 c5 ) u 2
+ (c13 c3 − c7 ) kux + (c15 c3 − c8 ) uxx = 0.
We break this up in powers of k and u to get the system of
equations
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
We substitute this into the Lax equation (Step 3) and get, for the
(1, 1)-element,
(c11 c3 − c4 c9 ) k 4 + (c12 c3 − c10 c4 − c5 c9 ) k 2 u + (c14 c3 − c10 c5 ) u 2
+ (c13 c3 − c7 ) kux + (c15 c3 − c8 ) uxx = 0.
We break this up in powers of k and u to get the system of
equations
c11 c3 − c4 c9 = 0,
c12 c3 − c10 c4 − c5 c9 = 0,
c14 c3 − c10 c5 = 0,
c13 c3 − c7 = 0,
c15 c3 − c8 = 0.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving with the equations from the other three matrix elements
(Step 4) gives many solutions. Choose (Step 5) a non-trivial one –
more on this later:
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving with the equations from the other three matrix elements
(Step 4) gives many solutions. Choose (Step 5) a non-trivial one –
more on this later:
#
"
0
− cc10
2
11
X =
c11
2
0
2c10 c10 k + 2c11 u
and
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving with the equations from the other three matrix elements
(Step 4) gives many solutions. Choose (Step 5) a non-trivial one –
more on this later:
#
"
0
− cc10
2
11
X =
c11
2
0
2c10 c10 k + 2c11 u
and
T =
T11 T12
T21 −T11
,
where
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving with the equations from the other three matrix elements
(Step 4) gives many solutions. Choose (Step 5) a non-trivial one –
more on this later:
#
"
0
− cc10
2
11
X =
c11
2
0
2c10 c10 k + 2c11 u
and
T =
T11 T12
T21 −T11
,
where
T11 = ux ,
2c10
T12 = 3 −c10 k 2 + c11 u ,
c11
T21 = c10 k 4 + c11 k 2 u −
Jacob Rezac
2
c11
2u 2 + uxx .
c10
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Solving with the equations from the other three matrix elements
(Step 4) gives many solutions. Choose (Step 5) a non-trivial one –
more on this later:
#
"
0
− cc10
2
11
X =
c11
2
0
2c10 c10 k + 2c11 u
and
T =
T11 T12
T21 −T11
,
where
T11 = ux ,
2c10
T12 = 3 −c10 k 2 + c11 u ,
c11
T21 = c10 k 4 + c11 k 2 u −
2
c11
2u 2 + uxx .
c10
Setting c10 = 4, c11 = 2i, and κ2 = −iλ gives the pair from before.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
Gauge freedom
X̂ = GXG −1 + Gx G −1
Jacob Rezac
and
T̂ = GTG −1 + Gt G −1
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
Gauge freedom
X̂ = GXG −1 + Gx G −1
and
T̂ = GTG −1 + Gt G −1
Use is limited to scaling invariant PDEs
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
Gauge freedom
X̂ = GXG −1 + Gx G −1
and
T̂ = GTG −1 + Gt G −1
Use is limited to scaling invariant PDEs
Computational complexity
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients.
Gauge freedom
X̂ = GXG −1 + Gx G −1
and
T̂ = GTG −1 + Gt G −1
Use is limited to scaling invariant PDEs
Computational complexity
How do we choose our undetermind coefficients?
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Problems with Method
Large number of possible values for undetermined coefficients
Gauge freedom
X̂ = GXG −1 + Gx G −1
and
T̂ = GTG −1 + Gt G −1
Use is limited to scaling invariant PDEs
Computational complexity
How do we choose our undetermind coefficients?
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Definition
A Conservation Law in (1+1)-dimensions with density ρ and flux J
is an equation of the form
ρt + Jx =
˙ 0,
when evaluated on a PDE.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Definition
A Conservation Law in (1+1)-dimensions with density ρ and flux J
is an equation of the form
ρt + Jx =
˙ 0,
when evaluated on a PDE.
The first few conservation laws of the KdV equation are easy to
compute:
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Definition
A Conservation Law in (1+1)-dimensions with density ρ and flux J
is an equation of the form
ρt + Jx =
˙ 0,
when evaluated on a PDE.
The first few conservation laws of the KdV equation are easy to
compute:
1 2
(u)t +
u + uxx =
˙ 0,
2
x
2 3
2
2
u t+
u − ux + 2uuxx =
˙ 0,
3
x
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Definition
A Conservation Law in (1+1)-dimensions with density ρ and flux J
is an equation of the form
ρt + Jx =
˙ 0,
when evaluated on a PDE.
The first few conservation laws of the KdV equation are easy to
compute:
1 2
(u)t +
u + uxx =
˙ 0,
2
x
2 3
2
2
u t+
u − ux + 2uuxx =
˙ 0,
3
x
and have been known for some time.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Continuing this pattern, we might try ρ = u 3 .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Continuing this pattern, we might try ρ = u 3 .
u 3 t = 3u 2 ut
= −3(u 3 ux + u 2 uxxx )
= Jx .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Continuing this pattern, we might try ρ = u 3 .
u 3 t = 3u 2 ut
= −3(u 3 ux + u 2 uxxx )
= Jx .
Thus,
Z
J = −3
(u 3 ux + u 2 uxxx )dx.
However, this integral can’t be written as a polynomial in
(u, ux , uxx , . . .).
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Some experimentation yields ρ = u 3 − 3ux2 and
3 4
2
2
2
3
2
u − 6uux + 3u uxx + 3uxx − 6ux uxxx =
˙ 0.
u − 3ux t +
4
x
These are well-known and have been for some time (e.g., M.
Kruskal’s “speach” during A. Newell’s talk).
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Some experimentation yields ρ = u 3 − 3ux2 and
3 4
2
2
2
3
2
u − 6uux + 3u uxx + 3uxx − 6ux uxxx =
˙ 0.
u − 3ux t +
4
x
These are well-known and have been for some time (e.g., M.
Kruskal’s “speach” during A. Newell’s talk).
It has been conjectured that any integrable PDE will have an
infinite number of Lax pairs.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Relation to Lax Pairs
Compare the zero-curvature equation and defining conservation
law equation
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Relation to Lax Pairs
Compare the zero-curvature equation and defining conservation
law equation
Xt − Tx + [X , T ] =
˙ 0
Jacob Rezac
and
ρt + Jx =
˙ 0.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Relation to Lax Pairs
Compare the zero-curvature equation and defining conservation
law equation
Xt − Tx + [X , T ] =
˙ 0
and
ρt + Jx =
˙ 0.
These are the same if we can make the commutator vanish from
the zero-curvature equation,
Xt − Tx = 0.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Relation to Lax Pairs
Compare the zero-curvature equation and defining conservation
law equation
Xt − Tx + [X , T ] =
˙ 0
and
ρt + Jx =
˙ 0.
These are the same if we can make the commutator vanish from
the zero-curvature equation,
Xt − Tx = 0.
How do we make the commutator vanish?
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
X = X0 − X1 λ,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
X = X0 − X1 λ,
for X0 off-diagonal and X1 diagonal.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
X = X0 − X1 λ,
for X0 off-diagonal and X1 diagonal.
Let S be a power series in λ,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
X = X0 − X1 λ,
for X0 off-diagonal and X1 diagonal.
Let S be a power series in λ,
S =I+
∞
X
Γi λ−i ,
i=1
for unknown (off-diagonal) matrices Γi . Define X̃ = SXS −1 as
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
X = X0 − X1 λ,
for X0 off-diagonal and X1 diagonal.
Let S be a power series in λ,
S =I+
∞
X
Γi λ−i ,
i=1
for unknown (off-diagonal) matrices Γi . Define X̃ = SXS −1 as
X̃ = Dx − X1 λ +
∞
X
Pi λ−i ,
i=0
for undetermined (diagonal) matrices, Pi and Dx =
Jacob Rezac
Computation of Lax Pairs
∂
∂x I .
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Drinfel’d-Sokolov Method
If we know a Lax pair, expand X as
X = X0 − X1 λ,
for X0 off-diagonal and X1 diagonal.
Let S be a power series in λ,
S =I+
∞
X
Γi λ−i ,
i=1
for unknown (off-diagonal) matrices Γi . Define X̃ = SXS −1 as
X̃ = Dx − X1 λ +
∞
X
Pi λ−i ,
i=0
for undetermined (diagonal) matrices, Pi and Dx =
Finally, set T̃ = STS −1 .
Jacob Rezac
Computation of Lax Pairs
∂
∂x I .
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
This set-up will do two things:
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
This set-up will do two things:
[X̃ , T̃ ] = 0, so X̃ and T̃ are the (matrix) density and flux of a
conservation law.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
This set-up will do two things:
[X̃ , T̃ ] = 0, so X̃ and T̃ are the (matrix) density and flux of a
conservation law.
Equating powers of λ in the equation X̃ = SXS −1 gives a
recurrence relation between Pn and Γn+1 :
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
This set-up will do two things:
[X̃ , T̃ ] = 0, so X̃ and T̃ are the (matrix) density and flux of a
conservation law.
Equating powers of λ in the equation X̃ = SXS −1 gives a
recurrence relation between Pn and Γn+1 :
Pn + [Γn+1 , X1 ] = Γn X0 − Γ0n −
n−1
X
Pi Γn−i ,
i=0
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
This set-up will do two things:
[X̃ , T̃ ] = 0, so X̃ and T̃ are the (matrix) density and flux of a
conservation law.
Equating powers of λ in the equation X̃ = SXS −1 gives a
recurrence relation between Pn and Γn+1 :
Pn + [Γn+1 , X1 ] = Γn X0 − Γ0n −
n−1
X
Pi Γn−i ,
i=0
which will allow us to algorithmically solve for each Pn .
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider again the KdV equation, with Lax pair X = X0 − X1 λ,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider again the KdV equation, with Lax pair X = X0 − X1 λ,
0 u
i 0
X0 =
and
X1 =
.
−1 0
0 −i
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Consider again the KdV equation, with Lax pair X = X0 − X1 λ,
0 u
i 0
X0 =
and
X1 =
.
−1 0
0 −i
Set P0 as
P0 =
Jacob Rezac
0 0
0 0
.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Using the recurrence relationship, we find
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Using the recurrence relationship, we find
1
1
0 u
u 0
and
P1 = i
.
Γ1 = i
1 0
0 u
2
2
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Using the recurrence relationship, we find
1
1
0 u
u 0
and
P1 = i
.
Γ1 = i
1 0
0 u
2
2
Similarly,
1
Γ2 = − i
8
0 u 2 − uxx
u
0
Jacob Rezac
and
1
P2 = −
4
Computation of Lax Pairs
ux
0
0
0
.
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Using the recurrence relationship, we find
1
1
0 u
u 0
and
P1 = i
.
Γ1 = i
1 0
0 u
2
2
Similarly,
1
Γ2 = − i
8
0 u 2 − uxx
u
0
and
1
P2 = −
4
So, the first few densities are
Jacob Rezac
Computation of Lax Pairs
ux
0
0
0
.
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Using the recurrence relationship, we find
1
1
0 u
u 0
and
P1 = i
.
Γ1 = i
1 0
0 u
2
2
Similarly,
1
Γ2 = − i
8
0 u 2 − uxx
u
0
and
1
P2 = −
4
So, the first few densities are
ρ1 = u,
Jacob Rezac
ρ2 = ux .
Computation of Lax Pairs
ux
0
0
0
.
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Using the recurrence relationship, we find
1
1
0 u
u 0
and
P1 = i
.
Γ1 = i
1 0
0 u
2
2
Similarly,
1
Γ2 = − i
8
0 u 2 − uxx
u
0
and
1
P2 = −
4
So, the first few densities are
ρ1 = u,
ρ2 = ux .
Continuing, we find ρ3 = u 2 , ρ5 = u 3 − 3ux2 , . . ..
Jacob Rezac
Computation of Lax Pairs
ux
0
0
0
.
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Finding the Flux
We could do the same process to find J - but it’s tricky. We use
simple integration.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Finding the Flux
We could do the same process to find J - but it’s tricky. We use
simple integration.
Integrating our conservation law, we have
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Finding the Flux
We could do the same process to find J - but it’s tricky. We use
simple integration.
Integrating our conservation law, we have
Z
Z
ρt dx + Jx dx =
˙ 0.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Finding the Flux
We could do the same process to find J - but it’s tricky. We use
simple integration.
Integrating our conservation law, we have
Z
Z
ρt dx + Jx dx =
˙ 0.
Rearranging,
Z
J=−
Jacob Rezac
ρt dx.
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Continuing with the KdV equation, consider
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Continuing with the KdV equation, consider
ρ1 = u.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Continuing with the KdV equation, consider
ρ1 = u.
Then,
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Continuing with the KdV equation, consider
ρ1 = u.
Then,
Z
J1 = −
Z
ut dx =
−αuux + uxxx dx.
Jacob Rezac
Computation of Lax Pairs
So,
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example
Continuing with the KdV equation, consider
ρ1 = u.
Then,
Z
J1 = −
So,
Z
ut dx =
−αuux + uxxx dx.
1
J1 = − αu 2 + uxx .
2
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Software Demonstration
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Issues
Only works when X = X0 − X1 λ
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Issues
Only works when X = X0 − X1 λ
Produces non-standard conservation laws
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Issues
Only works when X = X0 − X1 λ
Produces non-standard conservation laws
Every other conservation law is trivial
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
with a number of possible forms
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
with a number of possible forms
Build conservation laws given Lax pairs
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
with a number of possible forms
Build conservation laws given Lax pairs
of a specific form
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
with a number of possible forms
Build conservation laws given Lax pairs
of a specific form
which help to narrow down possible Lax pairs
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
with a number of possible forms
Build conservation laws given Lax pairs
of a specific form
which help to narrow down possible Lax pairs
Despite the constraints, our methods are reliable and work on
a large class of physically important equations.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Our goal was to generate Lax pairs and conservation laws easily
and algorithmically - this has been done to some degree. We can:
Construct Lax pairs
for scaling invariant equations
with a number of possible forms
Build conservation laws given Lax pairs
of a specific form
which help to narrow down possible Lax pairs
Despite the constraints, our methods are reliable and work on
a large class of physically important equations.
In the future, we hope to expand these methods to a larger class of
integrable equations.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Conclusions
Thanks!
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example: Short Pulse Equation
The Short Pulse Equation (SPE),
uxt + αu + β(u 3 )xx = 0,
with weights
∂
=1
W
∂x
W
∂
∂t
=3
W (u) = 1
has a Lax pair with X -component
λ
XSPE = 1
λu
x
α
Jacob Rezac
λux
λ
Computation of Lax Pairs
W (α) = 4
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example: Short Pulse Equation
The Short Pulse Equation (SPE),
uxt + αu + β(u 3 )xx = 0,
with weights
∂
=1
W
∂x
W
∂
∂t
=3
W (u) = 1
has a Lax pair with X -component
λ
XSPE = 1
λu
x
α
λux
λ
W (α) = 4
This can be constructed with the method proposed in this
talk.
Jacob Rezac
Computation of Lax Pairs
Introduction
Construction of Lax pairs
Conservation Laws
Conclusions
Example: Short Pulse Equation
The Short Pulse Equation (SPE),
uxt + αu + β(u 3 )xx = 0,
with weights
∂
=1
W
∂x
W
∂
∂t
=3
W (u) = 1
has a Lax pair with X -component
λ
XSPE = 1
λu
x
α
λux
λ
W (α) = 4
This can be constructed with the method proposed in this
talk.
We cannot build conservation laws from it (it is in WKI-form
rather than AKNS-form).
Jacob Rezac
Computation of Lax Pairs