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Disguised diﬀerential equations towards canonicity Cristina Sard´ on ICMAT-CSIC A workshop to honor Professor Orlando Ragnisco Burgos 2016
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### Transcript of Disguised differential equations towards canonicity · Burgos 2016. Disguised di erential equations...

• Disguised differential equationstowards canonicity

Cristina SardónICMAT-CSIC

A workshop to honorProfessor Orlando Ragnisco

Burgos 2016

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

1. Integrability of differential equations

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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 quasi-algorithmic 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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Painlevé, Gambier et al oriented their study towards second-orderdifferential 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

ICMAT-CSIC

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

ICMAT-CSIC

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 left-hand 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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Painleve test for PDEs

The Ablowitz-Ramani-Segur 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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

1. Canonical differential equations

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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 Degasperis-Procesi equation

ut − uxxt + 4uux − 3uxuxx − uuxxx = 0 (5)

Vakhnenko equation

uuxxt − uxuxt + u2ut = γu (6)

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

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 shallow-water 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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Reciprocal transformation for CH

The n-component Camassa-Holm 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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Reciprocal transformation for CH

The n-component Camassa-Holm 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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Reciprocal transformation for CH

The n-component Camassa-Holm 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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Reciprocal transformation for Qiao

The n-component 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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Miura-reciprocal 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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Miura-reciprocal 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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Example

CHH(1 + 1)reciprocal transf. //

KSMiura-reciprocal transf.

��

CBS equation

Miura transf.

��

mCHH(1 + 1) mCBS equationreciprocal transf.oo

Figure: Miura-reciprocal transformation.

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Setting

Let us consider a general manifold NRn ' Rk × Rn, where the first k-tuplerefers to the dependent variables u = (u1, . . . , uk) ∈ Rk and the n-tupledenotes the independent variables x = (x1, . . . , xn) ∈ Rn. We denote byJp(Rn,Rk) the space of jets of order p on NRn .

The space of p-jets 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 higher-order 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 higher-order derivatives.

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

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

ICMAT-CSIC

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

−n∑

i 6=î=1

XziXz

dxi , (14)

where we have used that dzi = dxi for all i 6= î according to (12).

• Disguiseddifferentialequationstowards

canonicity

CristinaSardón

ICMAT-CSIC

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

.

(15)

Now we perform the extension of the transformation to higher-orderderivatives. In the case of first-order derivatives, it is

uxî

=∂u

∂zî

∂zî∂xî

+n∑

i 6=î=1

∂u

∂zi

∂zi∂xî

=uz

Xzî

,

uxi =∂u

∂zî

∂zî∂xi

+n∑

i 6=î=1

∂u

∂zi

∂zi∂xi

= −XziXz

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

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

ICMAT-CSIC

Integrability

Notions ofintegrability

The PainleveProperty

Canonicity

The SMM

Reciprocaltransformations

Bibliography

Hodograph Transformations for a Camassa-Holm hierarchy in 2+1dimensionsP. G. Estvez, J. Prada.J. Phys. A: Math and Gen, 38, 1-11, (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