Disguised differential equations towards canonicity · Burgos 2016. Disguised di erential equations...
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Disguised differential equationstowards canonicity
Cristina SardónICMATCSIC
A workshop to honorProfessor Orlando Ragnisco
Burgos 2016

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Outline
Two main discussions
What is integrability. How do we understand it in the case of
differential equations.
Notions of integrability.An example: the Painleve test.
Classification and “canonicity” PDES.
The singular manifold method.The usefulness of reciprocal transformations.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
What is integrability?
On a fairly imprecise first approximation, we say integrability is the exactsolvability or regular behavior of solutions of a system.
...Nevertheless, there are various, distinct notions of an integrable system.
1 The characterization and unified definition of integrability is anontrivial matter.
2 The classification of integrable systems and their canonicalpresentation are not straightforward.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
What is integrability?
On a fairly imprecise first approximation, we say integrability is the exactsolvability or regular behavior of solutions of a system.
...Nevertheless, there are various, distinct notions of an integrable system.
1 The characterization and unified definition of integrability is anontrivial matter.
2 The classification of integrable systems and their canonicalpresentation are not straightforward.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
What is integrability?
On a fairly imprecise first approximation, we say integrability is the exactsolvability or regular behavior of solutions of a system.
...Nevertheless, there are various, distinct notions of an integrable system.
1 The characterization and unified definition of integrability is anontrivial matter.
2 The classification of integrable systems and their canonicalpresentation are not straightforward.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
1. Integrability of differential equations

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Why integrability?
Integrability appeared with Classical Mechanics with a quest for exactsolutions to Newton’s equation of motion.
Integrable systems present a number of conserved quantities: angularmomentum, linear momentum, energy... Indeed, some systems present aninfinite number of conserved quantities. But finding conserved quantities ismore of an exception rather than a rule.
Hence, the need for characterization and search of criteria for
integrability.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Notions of integrability
Geometrical viewpoint of dynamical systems: Differential equationsare interpreted in terms of Pfaffian systems and the Frobeniustheorem.
In the context of differentiable dynamical systems: The notion ofintegrability refers to Liouville integrability.
In Hamiltonian systems: Existence of maximal set of commutinginvariants with the Hamiltonian.
{I ,H} = 0
Any (quasi) algorithmic methods?: The Painleve method, existenceof Lax pairs, the inverse scattering transform, the Hirota bilinearmethod...

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Notions of integrability
Geometrical viewpoint of dynamical systems: Differential equationsare interpreted in terms of Pfaffian systems and the Frobeniustheorem.
In the context of differentiable dynamical systems: The notion ofintegrability refers to Liouville integrability.
In Hamiltonian systems: Existence of maximal set of commutinginvariants with the Hamiltonian.
{I ,H} = 0
Any (quasi) algorithmic methods?: The Painleve method, existenceof Lax pairs, the inverse scattering transform, the Hirota bilinearmethod...

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Notions of integrability
Geometrical viewpoint of dynamical systems: Differential equationsare interpreted in terms of Pfaffian systems and the Frobeniustheorem.
In the context of differentiable dynamical systems: The notion ofintegrability refers to Liouville integrability.
In Hamiltonian systems: Existence of maximal set of commutinginvariants with the Hamiltonian.
{I ,H} = 0
Any (quasi) algorithmic methods?: The Painleve method, existenceof Lax pairs, the inverse scattering transform, the Hirota bilinearmethod...

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Notions of integrability
Geometrical viewpoint of dynamical systems: Differential equationsare interpreted in terms of Pfaffian systems and the Frobeniustheorem.
In the context of differentiable dynamical systems: The notion ofintegrability refers to Liouville integrability.
In Hamiltonian systems: Existence of maximal set of commutinginvariants with the Hamiltonian.
{I ,H} = 0
Any (quasi) algorithmic methods?: The Painleve method, existenceof Lax pairs, the inverse scattering transform, the Hirota bilinearmethod...

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Integrability approaches
What comes to integrability of differential equations, there are different“integrability criteria”.
The Painleve method: this is a quasialgorithmic method to checkwhether an ODE or PDE is integrable.
The existence of Lax pairs
Solitonic solutions or the Hirota bilinear method
Reciprocal transformations
Lie symmetry approaches and reduction

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Fixed and movable singularity
Consider a manifold N locally diffeomorphic to R× TR, with localcoordinates {t, u(t), ut}. Consider the differential equation on R× TNthat reads
(t − c)ut = buand c, b ∈ R. Its general solution reads
u(t) = k0(t − c)b,
where k0 is a constant of integration. Depending on the value of theexponent b, we have different types of singularities
If b is a positive integer, then, u(t) is a holomorphic function.
If b is a negative integer, then c is a pole singularity.
In case of b rational, c is a branch point.
Nevertheless, the singularity t = c does not depend on initial conditions.
We say that the singularity is fixed.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Fixed and movable singularity
Let us now consider an ODE on R× T2R with local coordinates{t, u, ut , utt}, which reads
buutt + (1− b)u2t = 0,
with b ∈ R. The general solution to this equation is
u(t) = k0(t − t0)b.
If b is a negative integer, the singularity t = t0 is a singularity that depends
on the initial conditions through the constant of integration t0. In this
case, we say that the singulary is movable.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Painlevé, Gambier et al oriented their study towards secondorderdifferential equations on R× T2R with local coordinates {t, u, ut , utt}, ofthe type
utt = F (t, u, ut),
where F is a rational function in u, ut and analytic in t.
He found that there were 50 different equations of this type whose uniquemovable singularities were poles. Out of the 50 types, 44 were integratedin terms of known functions as Riccati, elliptic, linear, etc., and the 6remaining, although having meromorphic solutions, they do not possessalgebraic integrals that permit us to reduce them by quadratures.
These 6 functions are called Painlevé transcendents (PI − PVI ), becausethey cannot be expressed in terms of elementary or rational functions or
solutions expressible in terms of special functions.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
The Painleve Property and Test
We say that an ODE has the PP if all the movable singularities of itssolution are poles. The Painleve Test Given a general ODE on R× TpR
with local coordinates {t, u, ut , . . . , ut,...,t},
F = F (t, u(t), . . . , ut,...,t), (1)
the PT analyzes local properties by proposing solutions in the form
u(t) =∞∑j=0
aj(t − t0)(j−α), (2)
where t0 is the singularity, aj ,∀j are constants and α is necessarily apositive integer. If (2) is a solution of an ODE, then, the ODE is
conjectured integrable. To prove this, we have to follow a number of steps

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
The Painleve test
1 We determine the value of α by balance of dominant terms, whichwill permit us to obtain a0, simultaneously. The values of α and a0are not necessarily unique, and α must be a positive integer. Havingintroduced (2) into the differential equation (1),
2 We obtain a relation of recurrence for the rest of coefficients aj
(j − β1) · · · · · (j − βn)aj = Fj(t, . . . , uk , (uk)t , . . . ), k < j , (3)which arises from setting equal to zero different orders in (t − t0).This gives us aj in terms of ak for k < j . Observe that when j = βlwith l = 1, . . . , n, the lefthand side of the equation is null and theassociated aβl is arbitrary. Equation (3) turns into a relation for akfor k < βl which is known as the resonance condition. Consideringthat the number of arbitrary constants of motion that an ODEadmits is equal to its order, we must find the order of the equationminus one, as t0 is already one of the arbitrary constants.
3 If resonance conditions are satisfied identically, Fj = 0 for everyj = βl , we say that the ODE posesses the PP. The resonances haveto be positive except j = −1, which is associated with thearbitrariness of t0.
In the case of PDEs, Weiss, Tabor and Carnevale carried out the
generalization of the Painleve method, the so called WTC method.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Painleve test for PDEs
The AblowitzRamaniSegur conjecture (ARS) says that a PDE isintegrable in the Painlevé sense, if all of its reductions have the Painlevéproperty.
We can extend the Painlevé test to PDEs by substituting the function(t − t0) by an arbitrary function φ(xi ) for all i = 1, . . . , n, which receivesthe name of movable singularity manifold. We propose a Laurent expansionseries which incorporates ul(xi ) as functions of the coordinates xi .
It is important to mention that the PT is not invariant under changes of
coordinates. This means that an equation can be integrable in the Painlevé
sense in certain variables, but not when expressed in others, i.e.,the PT is
not intrisecally a geometrical property.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Example: The KdV equation
For example, the well known Korteweg de Vries equation (KdV) includesdispersion and nonlinearity for the propagation of long wave regime. It ischaracteristic of solitary waves in shallow water canals.
ut + uux + µuxxx = 0
with x , t, u representing spatial coordinate, time and amplitude of the waverespectively. Its solutions can be solitons: unperturbable solitary wavesthat do not change their profile even under dispersion and nonlinearbackgrounds.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Painleve and KdV
We propose an expansion of the type
u =U−2φ2
+U−1φ
+ U0 +∑k=3
Ukφk
φ2, k ≥ 3
and we obtain
U−2(x , t) = −12(∂φ
∂x
)2, U−1(x , t) = 12
∂2φ
∂x2,
U0 = −−2φxx + 4φxφxx + φxφt
φ2x
The resonances appear in k = −1, 4, 6. It can be checked that they areidentically satisfied and that U4(x , t) and U6(x , t) are arbitrary.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
1. Canonical differential equations

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Canonical?
What do we mean by canonicity?
By canonicity we shall refer to a unique way of representing severaldifferential equations which seem different but are equivalent descriptionsof a same problem expressed in different variables, for example.
How can we detect this phenomena?
Reciprocal transformations help us identify two different equations,although seemingly unrelated, happen to be equivalent versions of a sameequation after a reciprocal transformation. In this way, the big number ofintegrable equations could be greatly diminished by establishing a methodto discern which equations are disguised versions of a common underlyingproblem.The singular manifold method provides an equation or a system ofequations that can be understood as a canonical representation of the PDEunder study.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Canonical?
What do we mean by canonicity?
By canonicity we shall refer to a unique way of representing severaldifferential equations which seem different but are equivalent descriptionsof a same problem expressed in different variables, for example.
How can we detect this phenomena?
Reciprocal transformations help us identify two different equations,although seemingly unrelated, happen to be equivalent versions of a sameequation after a reciprocal transformation. In this way, the big number ofintegrable equations could be greatly diminished by establishing a methodto discern which equations are disguised versions of a common underlyingproblem.The singular manifold method provides an equation or a system ofequations that can be understood as a canonical representation of the PDEunder study.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
The singular manifold method
The singular manifold method (SMM) focuses on solutions which arisefrom truncated series of the generalized PP method. We require thesolutions of the PDE written in the form of a Laurent expansion to selectthe truncated terms
ul(xi ) ' u(l)0 (xi )φ(xi )−α + u
(l)1 (xi )φ(xi )
1−α + · · ·+ u(l)α (xi ),
for every l . In the case of several branches of expansion, this truncationneeds to be formulated for every value of α. Here, the function φ(xi ) is nolonger arbitrary, but a singular manifold equation whose expression arisesfrom the truncation.
An expression of the type F = F (φ, φxi , φxi ,xj , . . . ), arises.
The SMM is interesting because it contributes substantially in the
derivation of a Lax pair.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
At this point it is very useful to introduce the following quantities
ω =φtφx, v =
φxxφx
, s = vx −v 2
2
From the obvious condition φxt = φtx we can easily write the followingrelationships
vt = (ωx + ωv)x , st = ωxxx + 2sωx + ωsx
To see the importance of these quantities, note that the PP is invariantunder homographic transformations of φ. It is easy to see that ω and s arehomographic invariants, but not v . Since u0 is a solution of the equation,the expansion is an autoBacklund transformation.
We want to express the solutions u in terms of singular manifold quantities
u = u(v , ω, s)
where ω and s must satisfy the singular manifold equations (SME).

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
SMM and KdV
Reconsider the KdV equation in the form
ut − 6uux + uxxx = 0
If we propose the truncated solution, we obtain the singular manifoldequations
6u = ω + 4vx + v2,
ω + s + 6λ = 0
Furthermore, the second equation can be linearized with the followingchange
φx = ψ2 → v = 2ψx
ψ
such that the corresponding linear system is
ψxx = (u0 + λ)ψ,
ψt = −uxψ + 2(u0 − 2λ)ψx
whose compatibility condition demands that u0 must be a solution of the
KdV equation and λ is the spectral parameter, independent of time.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Reciprocal transformations
Reciprocal transformations consist on the mixing of the role of thedependent and independent variables to achieve simpler versions or evenlinearized versions of the initial, nonlinear PDE.
Then, the next question comes out: Is there a way to identify differentversions of a common nonlinear problem?
In principle, the only way to ascertain is by proposing differenttransformations and obtain results by recurrent trial and error.
It is desirable to derive a canonical form by using the explained SMM, but
it is still a conjecture to be proven.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Reciprocal transformations
Reciprocal transformations consist on the mixing of the role of thedependent and independent variables to achieve simpler versions or evenlinearized versions of the initial, nonlinear PDE.
Then, the next question comes out: Is there a way to identify differentversions of a common nonlinear problem?
In principle, the only way to ascertain is by proposing differenttransformations and obtain results by recurrent trial and error.
It is desirable to derive a canonical form by using the explained SMM, but
it is still a conjecture to be proven.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Example
Given the equation(Hx1x1x2 + 3Hx2Hx1 + n0
H2x1x2Hx2
)x1
= Hx2x3 (4)
is integrable for n0 = 0, 3/4.
After a reciprocal transformation and a reduction of order, this equationretrieves
The DegasperisProcesi equation
ut − uxxt + 4uux − 3uxuxx − uuxxx = 0 (5)
Vakhnenko equation
uuxxt − uxuxt + u2ut = γu (6)

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
The necessity of reciprocal transformations
The Camassa Holm equation is a completely integrable dispersive shallowwater equation, namely,
ut + 2kux − uxxt + 3uux − 2uxuxx + uuxxx = 0
where u is the fluid velocity in the x direction and k is a constant relatedto the critical shallowwater wave speed. The limit k = 0 is given specialattention for the search of solutions of the form U(x − vt). It was foundthat:
U = Ce−x−ct + O(k log k)
Nonetheless, this equation is not integrable from the point of view of thePainleve integrability.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
The problem
Finding a proper reciprocal transformation is usually a very complicatedtask. Notwithstanding, in the cases that concern us: as it can be the caseof equations in fluid mechanics, a change of this type is usually reliable.
For systems of hydrodynamic type with time evolution
(uj)t =k∑
l=1
v jl (u)(ul)xi , ∀1 ≤ i ≤ n, j = 1, . . . , k (7)
and v jl (u) are infinitely differentiable functions. These evolution equations
appear in gas dynamics, hydrodynamics, chemical kinetics, differential
geometry and topological field theory.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Reciprocal transformation for CH
The ncomponent CamassaHolm hierarchy in 1 + 1 dimensions can bewritten in a compact form in terms of a recursion operator as follows:
UT = R−nUX .
with K , J defined as K = ∂XXX − ∂X and J = − 12 (∂XU + U∂X ).
If we use
that R = KJ−1 and include auxiliary fields Ω(i) with i = 1, . . . , n when theinverse of an operator appears, and the change U = P2,
PT = −1
2
(PΩ(1)
)X,
Ω(i)XXX − Ω
(i)X = −P
(PΩ(i+1)
)X, i = 1, . . . , n − 1
P2 = Ω(n)XX − Ω
(n).
Given the conservative form of one of the equations, the followingtransformation arises naturally:
dT0 = PdX −1
2PΩ(1)dT , dT1 = dT , d
2T0 = 0

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Reciprocal transformation for CH
The ncomponent CamassaHolm hierarchy in 1 + 1 dimensions can bewritten in a compact form in terms of a recursion operator as follows:
UT = R−nUX .
with K , J defined as K = ∂XXX − ∂X and J = − 12 (∂XU + U∂X ). If we usethat R = KJ−1 and include auxiliary fields Ω(i) with i = 1, . . . , n when theinverse of an operator appears, and the change U = P2,
PT = −1
2
(PΩ(1)
)X,
Ω(i)XXX − Ω
(i)X = −P
(PΩ(i+1)
)X, i = 1, . . . , n − 1
P2 = Ω(n)XX − Ω
(n).
Given the conservative form of one of the equations, the followingtransformation arises naturally:
dT0 = PdX −1
2PΩ(1)dT , dT1 = dT , d
2T0 = 0

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Reciprocal transformation for CH
The ncomponent CamassaHolm hierarchy in 1 + 1 dimensions can bewritten in a compact form in terms of a recursion operator as follows:
UT = R−nUX .
with K , J defined as K = ∂XXX − ∂X and J = − 12 (∂XU + U∂X ). If we usethat R = KJ−1 and include auxiliary fields Ω(i) with i = 1, . . . , n when theinverse of an operator appears, and the change U = P2,
PT = −1
2
(PΩ(1)
)X,
Ω(i)XXX − Ω
(i)X = −P
(PΩ(i+1)
)X, i = 1, . . . , n − 1
P2 = Ω(n)XX − Ω
(n).
Given the conservative form of one of the equations, the followingtransformation arises naturally:
dT0 = PdX −1
2PΩ(1)dT , dT1 = dT , d
2T0 = 0

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
By considering the former independent variable X as a dependent field ofthe new pair of independent variables X = X (T0,T1), and therefore,dX = X0 dT0 + X1 dT1 where the subscripts zero and one refer to partialderivative of the field X with respect to T0 and T1, correspondingly. Theinverse transformation takes the form:
dX =dT0P
+1
2Ω(1)dT1, dT = dT1
which, by direct comparison with the total derivative of the field X , weobtain:
X0 =1
P, X1 =
Ω(1)
2.
We can now extend the transformation by introducing n − 1 independentvariables T2, . . . ,Tn which account for the transformation of the auxiliary
fields Ω(i) = 2Xi , with i = 2, . . . , n and Xi =∂X∂Ti
. Then, X is a function
X = X (T0,T1,T2, . . . ,Tn) of n + 1 variables.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
−(Xi+1X0
)0
=
{(X00X0
+ X0
)0
− 12
(X00X0
+ X0
)2}i
, i = 1, . . . , n − 1.
which constitutes n− 1 copies of the same system, each of which is writtenin three variables T0,Ti ,Ti+1. We introduce the change
Mi = −1
4
(Xi+1X0
),
M0 =1
4
{(X00X0
+ X0
)0
− 12
(X00X0
+ X0
)2}
with M = M(T0,Ti ,Ti+1) and i = 1, . . . , n − 1.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
The compatibility condition of X000 and Xi+1 in this system gives rise to aset of equations written entirely in terms of M:
M0,i+1 + M000i + 4MiM00 + 8M0M0i = 0, i = 1, . . . , n − 1
that are n − 1 CBS equations, each one in three variablesM = M(T0, ..Ti ,Ti+1, ...Tn). These equations have the PP and the SMM
can be applied to derive its LP. Making use of CBS’s LP and with the aid
of the inverse reciprocal transform, we could derive a LP for the initial
CH1 + 1

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Reciprocal transformation for Qiao
The ncomponent Qiao hierarchy in 1 + 1 dimensions can be written in acompact form in terms of a recursion operator:
ut = r−nux
such that r = kj−1 and k = ∂xxx − ∂x and j = −∂xu(∂x)−1u∂x .
ut = −(uω(1)
)x,
v (i)xxx − v (i)x = −(uω(i+1)
)x, i = 1, . . . , n − 1,
u = v (n)xx − v (n).
in which the change ω(i)x = uv
(i)x with other n auxiliary fields ω
(i) hasnecessarily been included to operate with the inverse term present in j .From the conservative form, we propose:
dT0 = u dx − uω(1)dt, dT1 = dt
such that d2T0 = 0.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
We now propose a reciprocal transformation by considering the initialindependent variable x as a dependent field of the new independentvariables such that x = x(T0,T1), and therefore, dx = x0 dT0 + x1 dT1.The inverse transformation adopts the form:
dx =dT0u
+ ω(1)dT1, dt = dT1.
By direct comparison of the inverse transform with the total derivative ofx , we obtain that:
x0 =1
u, x1 = ω
(1).
We shall prolong this transformation in such a way that we introduce newvariables T2, . . . ,Tn such that x = x(T0,T1, ...Tn) according to thefollowing rule ω(i) = xi , xi =
∂x∂Ti
for i = 2, . . . , n.
m0 =x202,
mi =xi+1x0
+xi00x0, i = 1, . . . , n − 1.
which can be considered as modified versions of the CBS equation withm = m (T0, ..Ti ,Ti+1, ...Tn).

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Miurareciprocal transformations
From the literature, we know that CBS and mCBS can be transformed oneinto another. In this manner, the fields present in CBS and mCBS arerelated through the following formula:
4M = x0 −m.
This is the point at which the question of whether Qiao1 + 1 couldpossibly be a modified version of CH1 + 1 arises. Nevertheless, the relationbetween these two hierarchies cannot be a simple Miura transform, sinceeach of them is written in different variables (X ,T ) and (x , t). However,both triples lead to the same final triple (T0,T1)
PdX − 12PΩ(1)dT = udx − uω(1)dt, dt = dT
that yields a relationship between the variables in CH1 + 1 and Qiao1 + 1.
4M0 = x00 −m0 ⇒X00X0
+ X0 = x0,
4Mi = x0i −mi ⇒ −Xi+1X0
= x0i −x00ix0− xi+1
x0, i = 1, . . . , n − 1.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Miurareciprocal transformations
Using the inverse reciprocal transform
1
u=
(1
P
)X
+1
P
PΩ(i+1) = 2(v (i) − v (i)x )⇒ ω(i+1) =Ω
(i+1)X + Ω
(i+1)
2, i = 1, . . . , n − 1.
x = X − lnP.
This equation gives us an important relation between the initial
independent variables X and x in CH1 + 1 and Qiao1 + 1, respectively.
Notice that the rest of independent variables, y , t and Y ,T do not appear
in these expressions.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Example
CHH(1 + 1)reciprocal transf. //
KSMiurareciprocal transf.
��
CBS equation
Miura transf.
��
mCHH(1 + 1) mCBS equationreciprocal transf.oo
Figure: Miurareciprocal transformation.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Setting
Let us consider a general manifold NRn ' Rk × Rn, where the first ktuplerefers to the dependent variables u = (u1, . . . , uk) ∈ Rk and the ntupledenotes the independent variables x = (x1, . . . , xn) ∈ Rn. We denote byJp(Rn,Rk) the space of jets of order p on NRn .
The space of pjets will be locally coordinatized by
xi , uj , (uj)xi1 , (uj)x j1i1 ,xj2i2
, . . . , (uj)x j1i1 ,xj2i2,x
j3i3,...,x
jnin
and such that i = 1, . . . , n, j = 1, . . . , k, j1 + · · ·+ jn ≤ p.
Let us consider a general (possibly nonlinear) system of a number q ofPDEs on Jp(Rn,Rk), with higherorder derivative of order p,
Ψl = Ψl(xi , uj , (uj)xi1 , (uj)x j1i1 ,x
j2i2
, . . . , (uj)x j1i1 ,xj2i2,x
j3i3,...,x
jnin
), (8)
for all l = 1, . . . , q and i = 1, . . . , n, j = 1, . . . , k, j1 + · · ·+ jn ≤ p. Thenotation accords to the usual: (uj)xi1 = ∂uj/∂xi1 , and this definition isextensible to higherorder derivatives.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Suppose that we know a number “n” of conserved quantities that areexpressible in the following form
A(j)xi
(xi1 , uj , (uj)xi1 , (uj)x j1i1 ,x
j2i2
, . . .
)= A(j
′)xi′
(xi , uj , (uj)x j1i1
, (uj)x j1i1 ,xj2i2
, . . .
),
xi 6= xi′ , A(j) 6= A(j′),
1 ≤ i ≤ n, j1 + · · ·+ jn ≤ p, 1 ≤ j , j ′ ≤ 2n.(9)
where A(j)xi = ∂A
(j)/∂xi , A(j′)xi′ = ∂A
(j′)/∂xi′ and A(j),A(j
′) ∈ C∞Jp(Rn,Rk),are different.If the number of equations in (9) is equal to the number of indepentvariables, we propose a transformation for each {x1, . . . , xn} to a new set{z1, . . . , zn} as
dzi = A(j)dxi + A
(j′)dxi′ , ∀1 ≤ i , i ′ ≤ n, ∀1 ≤ j , j ′ ≤ 2n. (10)
such that if the property of closeness is satisfied, d2zi = 0 for all zi ,i = 1, . . . , n, we recover the conserved quantities given in (9).

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Construction
We focus on the case in which only one conserved quantity equation isused. It implies that only one independent variable xî is transformed.We now search for a function X (z1, . . . , zn) such that
xî = X (z1, . . . , zn) (11)
Simultaneously,
dzî = A(j)dxi + A
(j′)dxi′ ,
dzi = dxi ,(12)
∀1 ≤ i 6= i ′ ≤ n and for a fixed value 1 ≤ î ≤ n, where the independentvariables xi , ∀i 6= î = 1, . . . , n are untransformed but renamed as zi .Deriving relation (11),
dxî =n∑
i=1
Xzi dzi , Xzi =∂X
∂zi(13)
and by isolating dzî in (13), we have
dzî =dxîXz
î
−n∑
i 6=î=1
XziXz
î
dxi , (14)
where we have used that dzi = dxi for all i 6= î according to (12).

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Here, by direct comparison of coefficients in (12) and (14), and if weidentify zî with zi1 , we have that
A(j′) =
1
Xzî
,
A(j) = −XziXz
î
.
(15)
Now we perform the extension of the transformation to higherorderderivatives. In the case of firstorder derivatives, it is
uxî
=∂u
∂zî
∂zî∂xî
+n∑
i 6=î=1
∂u
∂zi
∂zi∂xî
=uz
î
Xzî
,
uxi =∂u
∂zî
∂zî∂xi
+n∑
i 6=î=1
∂u
∂zi
∂zi∂xi
= −XziXz
î
uzî
+ uzi , ∀i 6= î .(16)
This process is recursively applied to achieve higher order derivatives. Inthis way, using expressions in (15), (16), etc., we transform a system (8)with initial variables {x1, . . . , xn} into a new system written in variables{z1, . . . , zn} and scalar fields uj(z1, . . . , zn), ∀j = 1, . . . , k.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
From (15), we can extract expressions for
zi , uj , (uj)zi1 , (uj)z j1i1 ,zj2i2
, . . . , (uj)z j1i1 ,zj2i2,...,z
jnin
(17)
for j1 + · · ·+ jn ≤ p, if possible, given the particular form of A(j),A(j
′) ∈ C∞Jp(Rn,Rk), in each case.Bearing in mind expression (11), the transformation of the initial system(8) will then read
Ψl = Ψl(zi ,X ,Xzi ,Xz j1i1 ,z
j2i2
, . . . ,Xzj1i1,z
j2i2,z
j3i3,...,z
jnin
)(18)
for all l = 1, . . . , q such that j1 + · · ·+ jn ≤ p and zi = z1, . . . , zn.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Conclusions
The definition of integrability is not unique.
Reciprocal transformations have proven their suitability for theidentification of alike equations spread in the literature of nonlinearintegrable systems.
They are an useful instrument to derive Lax pairs.
They can be used to simplify nonlinear models into linearizedequations.
They serve as a way to derive solutions: they transform peakonequations into other peakon equations.
Reciprocal transformations turn an initial equation which is notintegrable (in the Painleve sense) to another that is.
The singular manifold method could be a canonical way of classifyingPDES.

Disguiseddifferentialequationstowards
canonicity
CristinaSardón
ICMATCSIC
Integrability
Notions ofintegrability
The PainleveProperty
Canonicity
The SMM
Reciprocaltransformations
Bibliography
Hodograph Transformations for a CamassaHolm hierarchy in 2+1dimensionsP. G. Estvez, J. Prada.J. Phys. A: Math and Gen, 38, 111, (2005).
Miura Reciprocal transformation for two integrable hierarchies in 1+1dimensionsP. G. Estevez, C. Sardón,PROCEEDINGS GADEIS
Lie systems, Lie symmetries and reciprocal transformationsC. Sardón,arXiv:1508.00726
IntegrabilityNotions of integrabilityThe Painleve Property
CanonicityThe SMMReciprocal transformations