Integrability of quad equations: CAC vs. BT
Jarmo Hietarinta
Department of Physics and Astronomy
University of Turku, FIN-20014 Turku, Finland

SIDE13, Fukuoka, 11-17 Nov, 2018

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

In general terms: Integrable systems have some beautiful
underlying mathematical structure, which makes the
dynamics regular but not trivial.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

In general terms: Integrable systems have some beautiful
underlying mathematical structure, which makes the
dynamics regular but not trivial.
The definition of integrability is elusive, in fact there isno universal definition.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

In general terms: Integrable systems have some beautiful
underlying mathematical structure, which makes the
dynamics regular but not trivial.
The definition of integrability is elusive, in fact there isno universal definition.
Rather there are certain well defined properties associatedwith integrability. Over the years it has also been observedthat if a given equation has one good property it will most likely
have several other special properties.
Example: equations that have three-soliton solutions
also pass the Painlevé test.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

In general terms: Integrable systems have some beautiful
underlying mathematical structure, which makes the
dynamics regular but not trivial.
The definition of integrability is elusive, in fact there isno universal definition.
Rather there are certain well defined properties associatedwith integrability. Over the years it has also been observedthat if a given equation has one good property it will most likely
have several other special properties.
Example: equations that have three-soliton solutions
also pass the Painlevé test.
Therefore for many particular equations we have reached aconsensus that they should be called integrable.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

In general terms: Integrable systems have some beautiful
underlying mathematical structure, which makes the
dynamics regular but not trivial.
The definition of integrability is elusive, in fact there isno universal definition.
Rather there are certain well defined properties associatedwith integrability. Over the years it has also been observedthat if a given equation has one good property it will most likely
have several other special properties.
Example: equations that have three-soliton solutions
also pass the Painlevé test.
Therefore for many particular equations we have reached aconsensus that they should be called integrable.
The concept of “partial integrability” is also interesting.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Integrability concepts for lattice equations:
For lattice equations the following properties have been
considered:1

Algebraic entropy: The analysis is based of growth ofcomplexity as a function of the number of iterations n:• if growth is linear in n ⇒ equation is linearizable.
• if growth is polynomial in n ⇒ equation is integrable.
• if growth is exponential in n ⇒ equation is chaotic.

This may be taken as a rigorous test for integrability.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Integrability concepts for lattice equations:
For lattice equations the following properties have been
considered:1

Algebraic entropy: The analysis is based of growth ofcomplexity as a function of the number of iterations n:• if growth is linear in n ⇒ equation is linearizable.
• if growth is polynomial in n ⇒ equation is integrable.
• if growth is exponential in n ⇒ equation is chaotic.

This may be taken as a rigorous test for integrability.
2

Singularity confinement: Very efficient method, necessary
but not sufficient in its basic form.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Integrability concepts for lattice equations:
For lattice equations the following properties have been
considered:1

Algebraic entropy: The analysis is based of growth ofcomplexity as a function of the number of iterations n:• if growth is linear in n ⇒ equation is linearizable.
• if growth is polynomial in n ⇒ equation is integrable.
• if growth is exponential in n ⇒ equation is chaotic.

This may be taken as a rigorous test for integrability.
2

Singularity confinement: Very efficient method, necessary
but not sufficient in its basic form.

3

Symmetry analysis, somewhat complicated.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

There is still another property and it is used here:
4

Multidimensional consistency (MDC):
For an equation defined on a 2D lattice it should bepossible to extend it into 3D consistently.

MDC is related to the existence of hierarchies of soliton
equations.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

There is still another property and it is used here:
4

Multidimensional consistency (MDC):
For an equation defined on a 2D lattice it should bepossible to extend it into 3D consistently.

MDC is related to the existence of hierarchies of soliton
equations.
It is typically tested by “Consistency-Around-a-Cube” (CAC).

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

There is still another property and it is used here:
4

Multidimensional consistency (MDC):
For an equation defined on a 2D lattice it should bepossible to extend it into 3D consistently.

MDC is related to the existence of hierarchies of soliton
equations.
It is typically tested by “Consistency-Around-a-Cube” (CAC).
In this talk I will discuss the strength of this definition:
MDC is necessary in some form, but how strongly is it
associated with integrability?

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

The lattice setting
The basic elementary square or quadrilateral:xn,m = x00 = xxn+1,m = x10 = x[1] = xe

xb s

e
sx

x s

e
sx

b

xn,m+1 = x01 = x[2] = xbexn+1,m+1 = x11 = x[12] = xb

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

The lattice setting
The basic elementary square or quadrilateral:xn,m = x00 = xxn+1,m = x10 = x[1] = xe

xb s

e
sx

x s

e
sx

b

xn,m+1 = x01 = x[2] = xbexn+1,m+1 = x11 = x[12] = xb

The four corner values are related by a multi-linear equation:e; p, q) = 0
Q(x, xe, xb, xb
This allows propagation from initial data given on a staircase oron a corner.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
b xb

x r
e bd
xb
a

xe b
x r
xe r

Jarmo Hietarinta

r xb
b xb
e

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
b xb

x r
e bd
xb
a

xe b
x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

back:

Q23 (x, xb, x, xb ; q, r ) = 0,

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

back:

Q23 (x, xb, x, xb ; q, r ) = 0,

left: Q31 (x, x, xe, xe ; r , p) = 0,
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

back:

Q23 (x, xb, x, xb ; q, r ) = 0,e ; p, q) = 0,
top: Q12 (x, xe, xb, xb

left: Q31 (x, x, xe, xe ; r , p) = 0,
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

e ; p, q) = 0,
top: Q12 (x, xe, xb, xb

back:

Q23 (x, xb, x, xb ; q, r ) = 0,e, xe, xb
e ; q, r ) = 0,
front: Q23 (xe, xb

left: Q31 (x, x, xe, xe ; r , p) = 0,
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

e ; p, q) = 0,
top: Q12 (x, xe, xb, xb

back:

Q23 (x, xb, x, xb ; q, r ) = 0,e, xe, xb
e ; q, r ) = 0,
front: Q23 (xe, xb

left: Q31 (x, x, xe, xe ; r , p) = 0,e, xbe ; r , p) = 0.
right: Q31 (xb, xb, xb

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

In general
Discrete equations in general
2D lattice equations and MDC

Consistency-Around-a-Cube
e bd
xb
a

xe b

•
•

b xb

x r

•

•

x r
xe r

r xb
b xb
e

The equations on the faces of the consistency cube are:bottom:e ; p, q) = 0,
Q12 (x, xe, xb, xb

e ; p, q) = 0,
top: Q12 (x, xe, xb, xb

back:

Q23 (x, xb, x, xb ; q, r ) = 0,e, xe, xb
e ; q, r ) = 0,
front: Q23 (xe, xb

left: Q31 (x, x, xe, xe ; r , p) = 0,e, xbe ; r , p) = 0.
right: Q31 (xb, xb, xb

e must be the same.
CAC: the three computed values for xb
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Search: The class of equations
The following assumptions are made about the equations:1

The equations depend on all corner variables of thequadrilateral.

2

The equations are affine linear in each corner variable.

3

The equations are irreducible.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Search: The class of equations
The following assumptions are made about the equations:1

The equations depend on all corner variables of thequadrilateral.

2

The equations are affine linear in each corner variable.

3

The equations are irreducible.

4

Equations Q12 , Q23 and Q31 may be different.

5

On parallel planes all quad equations are identical.

6

The equations are homogeneous quadratic.

Assumptions 1,2,3 are standard, 6 is specific to this work.
Assumption 4 means we are considering triplets of equations(cf. Boll’s work, but he does not have assumption 5)
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

The typical equation is
ec6 = 0,e c4 +x xbc5 +xexb
e c3 +x xb
e; p, q) := x xe c1 +xexbc2 +xbxb
Q12 (x, xe, xb, xb
cj may depend on the lattice variables p, q.
Similarly for Q23 (x, xb, x, xb; q, r ) and Q31 (x, x, xe, xe; r , p) by cyclic
permutation p → q → r , xe → xb → x.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

The typical equation is
ec6 = 0,e c4 +x xbc5 +xexb
e c3 +x xb
e; p, q) := x xe c1 +xexbc2 +xbxb
Q12 (x, xe, xb, xb
cj may depend on the lattice variables p, q.
Similarly for Q23 (x, xb, x, xb; q, r ) and Q31 (x, x, xe, xe; r , p) by cyclic
permutation p → q → r , xe → xb → x.
At the beginning cj (p, q), cj (q, r ) and cj (r , p) are independentand not obtainable from some cj (x, y ) by substitution.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

The typical equation is
ec6 = 0,e c4 +x xbc5 +xexb
e c3 +x xb
e; p, q) := x xe c1 +xexbc2 +xbxb
Q12 (x, xe, xb, xb
cj may depend on the lattice variables p, q.
Similarly for Q23 (x, xb, x, xb; q, r ) and Q31 (x, x, xe, xe; r , p) by cyclic
permutation p → q → r , xe → xb → x.
At the beginning cj (p, q), cj (q, r ) and cj (r , p) are independentand not obtainable from some cj (x, y ) by substitution.
The triplets of equations are named by numbers indicating
which ci are nonzero. If |ci | = 1 if ci 6= 0 then
NQ = |c1 | + |c2 | 2 + |c3 | 4 + |c4 | 8 + |c5 | 16 + |c6 | 32

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

The typical equation is
ec6 = 0,e c4 +x xbc5 +xexb
e c3 +x xb
e; p, q) := x xe c1 +xexbc2 +xbxb
Q12 (x, xe, xb, xb
cj may depend on the lattice variables p, q.
Similarly for Q23 (x, xb, x, xb; q, r ) and Q31 (x, x, xe, xe; r , p) by cyclic
permutation p → q → r , xe → xb → x.
At the beginning cj (p, q), cj (q, r ) and cj (r , p) are independentand not obtainable from some cj (x, y ) by substitution.
The triplets of equations are named by numbers indicating
which ci are nonzero. If |ci | = 1 if ci 6= 0 then
NQ = |c1 | + |c2 | 2 + |c3 | 4 + |c4 | 8 + |c5 | 16 + |c6 | 32
A detailed description of the search has been given in preprint
arXiv:1806.08511.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Some result
The result include Q3(δ = 0), Q1(δ = 0), H3(δ = 0) along with
their asymmetric reductions.
There are also more exotic triplets with CAC property, including
e + c1 (p, q)x xe + c5 (p, q)x xb = 0,
exe + c3 (p, q)xbxb
c6 (p, q)xb
r (x xb + x xb) + x xb + x xb = 0,r (x xe + x xe) + x xe + x xe = 0.
with code name {53,15,58}. The triply shifted variable ise = −xxb

c1 (p, q)xe + c5 (p, q)xb,
c6 (p, q)xe + c3 (p, q)xb

which clearly has the tetrahedron property.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Two more examples: {63, 10, 10}e c3 (p, q) + x xb
e c4 (p, q) + x xbc5 (p, q) + xebex xe c1 (p, q) + xexbc2 (p, q) + xbxb
c 6 (p, q)=0,xbx − x xb=

0,x xe − x xe=0.

c (p, q)xx xb + c1 (p, q)x xex + c2 (p, q)xexbx
e=− 5
xb
.
c4 (p, q)x 2 + c6 (p, q)x xe + c3 (p, q)x xb{10, 58, 15}e = 0,
xbxe − x xb
xxc5 (q, r ) + x xbc4 (q, r ) + x xbc2 (q, r ) + xbxbc6 (q, r ) = 0,
xxc1 (r , p) + x xec4 (r , p) + x xec2 (r , p) + xexec3 (r , p) = 0.
e=xxb(c1 (r , p)x + c2 (r , p)xe)(c5 (q, r )x + c2 (q, r )xb).
(c4 (r , p)x + c3 (r , p)xe)(c4 (q, r )x + c6 (q, r )xb)

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Gauge invariance, rotations and reflections

The equations can be changed by a gauge transformation
provided that the transformation does not break the uniformityof the 3D lattice. That is, it should not introduce
non-autonomous terms, except possibly as overall multipliers.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Gauge invariance, rotations and reflections

The equations can be changed by a gauge transformation
provided that the transformation does not break the uniformityof the 3D lattice. That is, it should not introduce
non-autonomous terms, except possibly as overall multipliers.

For a given triplet other related triplets can be obtained byrotations and reflections, they form the orbit of the triplet.
It is only necessary to consider one representative of an orbit.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Reductions, part 1
The results can be arranged into groups whose entries are
related by reductions. For example:{63,63,63}-2 is Q3(δ = 0), {63,63,63}-1 is Q1(δ = 0) and{53,53,53} is H3(δ = 0). All others are asymmetric reductions.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Reductions, part 1, an example
Starting with {63,63,63}-2, which is Q3

Scale x → x/r and then take the leading term as r → ∞.
Also change signs of p, q. Then get

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

The class of equations for searchingSome results

Reductions, part 2
Two further chains of reductions, here the entries contain somearbitrary functions.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failDefinition of Bäcklund transformation
We can arrange the 6 equations of the cube as follows (y = x):e; p, q) = 0,
Q12 (y , ye, yb,ybe; q, r ) = 0e, ye, yb
Q23 (x, xb, y , yb; q, r ) = 0, Q23 (xe, xb
e; r , p) = 0,e, yb
Q (x, y , xe, ye; r , p) = 0, Q (xb, yb, xb(2a)e; p, q) = 0,
Q12 (x, xe, xb,xb

(2d)

31

31

Jarmo Hietarinta

Integrability: CAC vs. BT

(2b)
(2c)

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failDefinition of Bäcklund transformation
We can arrange the 6 equations of the cube as follows (y = x):e; p, q) = 0,
Q12 (y , ye, yb,ybe; q, r ) = 0e, ye, yb
Q23 (x, xb, y , yb; q, r ) = 0, Q23 (xe, xb
e; r , p) = 0,e, yb
Q (x, y , xe, ye; r , p) = 0, Q (xb, yb, xb(2a)e; p, q) = 0,
Q12 (x, xe, xb,xb

(2d)

31

31

This provides a Bäcklund transformation (BT) if
e from three of the equations in• After eliminating ye, yb, yb(2b),(2c), the fourth equation factorizes as
e; p, q) = 0(something depending on y) × Q12 (x, xe, xb, xb

Jarmo Hietarinta

Integrability: CAC vs. BT

(2b)
(2c)

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failDefinition of Bäcklund transformation
We can arrange the 6 equations of the cube as follows (y = x):e; p, q) = 0,
Q12 (y , ye, yb,ybe; q, r ) = 0e, ye, yb
Q23 (x, xb, y , yb; q, r ) = 0, Q23 (xe, xb
e; r , p) = 0,e, yb
Q (x, y , xe, ye; r , p) = 0, Q (xb, yb, xb(2a)e; p, q) = 0,
Q12 (x, xe, xb,xb

(2d)

31

31

(2b)
(2c)

This provides a Bäcklund transformation (BT) if
e from three of the equations in• After eliminating ye, yb, yb(2b),(2c), the fourth equation factorizes as
e; p, q) = 0(something depending on y) × Q12 (x, xe, xb, xb
e we get Q12 (y , ye, yb, yb
e; p, q).
• Similarly after eliminating xe, xb, xb
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failThree ways to generate equations by BT:• top and bottom from left-back-right-front• front and back from bottom-left-top-right• left and right from bottom-back-top-front

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may fail

Recall:e from three of the equations in (2b),(2c), the fourth• After eliminating ye, yb, yb
equation factorizes as
e; p, q) = 0(something depending on y) × Q12 (x, xe, xb, xb

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may fail

Recall:e from three of the equations in (2b),(2c), the fourth• After eliminating ye, yb, yb
equation factorizes as
e; p, q) = 0(something depending on y) × Q12 (x, xe, xb, xb

Since we have the CAC property, everything should work as
described.
Problem: The factor “(something depending on y)” may vanish
identically and therefore no equation Q12 is produced.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may fail

Example {X , 10, 10} (recall y = x):xby − x yb = 0,
xey − x ye = 0,

Jarmo Hietarinta

e = 0,eye − xeyb
xbbb
xeyb − xbye = 0.

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may fail

Example {X , 10, 10} (recall y = x):xby − x yb = 0,
xey − x ye = 0,e = 0,eye − xeyb
xbbb
xeyb − xbye = 0.

We find that

y
y
y
e ,
e = xbyb = xb , ye = xe , ybxxx
and this is enough to satisfy all 4 equations.
Thus no equation for x is generated and we can say BT fails(equivalently: the Lax pair is fake).

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may fail

Example {X , 10, 10} (recall y = x):xby − x yb = 0,
xey − x ye = 0,e = 0,eye − xeyb
xbbb
xeyb − xbye = 0.

We find that

y
y
y
e ,
e = xbyb = xb , ye = xe , ybxxx
and this is enough to satisfy all 4 equations.
Thus no equation for x is generated and we can say BT fails(equivalently: the Lax pair is fake).

This can be predicted: With these side equations CAC ispossible with bottom-top equations containing free functions.
There is no way the simple side equations can generate them.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may fail

Example {X , 10, 10} (recall y = x):xby − x yb = 0,
xey − x ye = 0,e = 0,eye − xeyb
xbbb
xeyb − xbye = 0.

We find that

y
y
y
e ,
e = xbyb = xb , ye = xe , ybxxx
and this is enough to satisfy all 4 equations.
Thus no equation for x is generated and we can say BT fails(equivalently: the Lax pair is fake).

This can be predicted: With these side equations CAC ispossible with bottom-top equations containing free functions.
There is no way the simple side equations can generate them.
The mere possibility of free functions implies BT fails.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failBT results

This table gives the number of equations produced by BTs.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Jarmo Hietarinta

Definition
BT in all directions
BT may failIntegrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failBT with back-left-front-right equations yield:e gete − xbxe).
• if eliminate ye, yb, yb(bottom equation)·y · (x xbbb• if eliminate xe, xb, xe get(top equation)·x · (y ye − ybye).
The extra factor corresponds to {10, 58, 15}.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Jarmo Hietarinta

Definition
BT in all directions
BT may failIntegrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failBT with back-left-front-right equations yield:
e get• if eliminate ye, yb, yb
e + x xbxe + x xb
exe + xbxb
exe)(bottom equation) ·(x xbxbb2e).
• if eliminate xe, xb, xe get (top equation)·x · (y + ye + yb + yb
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Jarmo Hietarinta

Definition
BT in all directions
BT may failIntegrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failResults for some more trivial triplets with CAC.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Jarmo Hietarinta

Definition
BT in all directions
BT may failIntegrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failSet σ1 = σ2 = 1. Try to generate left and right equations.
BT with bottom-back-top-front equations yields (y = xb) :• eliminate y : (xx − xexe) · [c4 (p, q)(x xe + x xe) + c5 (p, q)(xx + xexe)]• eliminate x: (y y − yeye) · [c4 (p, q)(y ye + y ye) + c5 (p, q)(y y + yeye)]

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failSet σ1 = σ2 = 1. Try to generate left and right equations.
BT with bottom-back-top-front equations yields (y = xb) :• eliminate y : (xx − xexe) · [c4 (p, q)(x xe + x xe) + c5 (p, q)(xx + xexe)]• eliminate x: (y y − yeye) · [c4 (p, q)(y ye + y ye) + c5 (p, q)(y y + yeye)]
The extra factor corresponds to {58, 48, 15} but has wrong
parameter dependence.
Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failConclusions

1

The CAC condition is only weakly associated withintegrability, it is definitely not sufficient.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failConclusions

1

The CAC condition is only weakly associated withintegrability, it is definitely not sufficient.

2

Multidimensional consistency is necessary in some form.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failConclusions

1

The CAC condition is only weakly associated withintegrability, it is definitely not sufficient.

2

Multidimensional consistency is necessary in some form.

3

Our results suggest that the existence of a unique BT in all
directions of the cube is rather strongly associated withintegrability, while non-uniqueness may be a signal ofpartial integrability.

Jarmo Hietarinta

Integrability: CAC vs. BT

What is integrability
Gathering evidence
Bäcklund transformations

Definition
BT in all directions
BT may failConclusions

1

The CAC condition is only weakly associated withintegrability, it is definitely not sufficient.

2

Multidimensional consistency is necessary in some form.

3

Our results suggest that the existence of a unique BT in all
directions of the cube is rather strongly associated withintegrability, while non-uniqueness may be a signal ofpartial integrability.

4

Integrability may be lost during reductions.

More in preprint

arXiv:1806.08511.

Jarmo Hietarinta

Integrability: CAC vs. BT