Integrability approaches to differential equationscud.unizar.es/sites/default/files/personal/pers... · 2015-10-05 · approaches to di erential equations Cristina Sard on The Painleve

Integrability approaches to differentialequations

Cristina Sardon

XXIV Fall Workshop on Geometry and PhysicsZaragoza, CUD, September 2015

Integrabilityapproaches to

differentialequations

CristinaSardon

The PainleveProperty

Lax pairs

Solitonicsolutions andHirota method

Lie symmetry

Reciprocaltransforma-tions

Lie systems

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 integrable systems...

The characterization and unified definition of integrability is a nontrivial

matter.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems





matter.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems





matter.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

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!



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Notions of integrability

Geometrical viewpoint of dynamical systems: Differentialequations are interpreted in terms of Pfaffian systems and theFrobenius theorem.

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...



CristinaSardon


Lax pairs


Lie symmetry


Lie systems





{I ,H} = 0




CristinaSardon


Lax pairs


Lie symmetry


Lie systems





{I ,H} = 0




CristinaSardon


Lax pairs


Lie symmetry


Lie systems





{I ,H} = 0




CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Integrability approaches

What comes to integrability of differential equations, we focus on theapproaches

The Painleve method: this is a quasi-algorithmic method to checkwhether an ODE or PDE is integrable.

The existence of Lax pairs

Derivation of Lax pairs with the singular manifold method.Solvability through the Inverse scattering method.

Solitonic solutions or the Hirota bilinear method

Reciprocal transformations

Lie symmetry approaches

Lie systems with different geometric approaches.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

The Painleve method

To discern whether our equation is integrable or not, involves a deepinspection of its geometrical properties and aforementioned methods. Now,a question arises: Is there any algorithmical method to check theintegrability of a differential equation? The answer is affirmative.

It focuses on the singularity analysis of the differential equation, attendingto a fundamental property: being a fixed or a movable singularity, or asingularity not depending or depending on the initial conditions,respectively.

Sophia Kovalevskaya centered herself in the study of equations for solid

rigid dynamics: singularities and properties of single-valuedness of poles of

PDEs on the complex plane, etc. Eventually, she expanded her results to

other physical systems.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Fixed and movable singularity

Consider a manifold N locally diffeomorphic to R× TR, with localcoordinates {t, u(t), ut}. Consider the differential equation

(t − c)ut = bu

and 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.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

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 dependson the initial conditions through the constant of integration t0. In thiscase, we say that the singulary is movable.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Painleve, 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 Painleve transcendents (PI − PVI ), becausethey cannot be expressed in terms of elementary or rational functions orsolutions expressible in terms of special functions.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

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 isconjectured integrable. To prove this, we have to follow a number of steps



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

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 a0

are not necessarily unique, and α must be a positive integer.

2 Having introduced (2) into the differential equation (1), we obtain arelation 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 = βl

with l = 1, . . . , n, the left-hand side of the equation is null and theassociated aβl is arbitrary. Those values of j , are called resonancesand the equation (3) turns into a relation for ak for k < βl which isknown as the resonance condition.

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 thegeneralization of the Painleve method, the so called WTC method.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Painleve test for PDEs

The Ablowitz-Ramani-Segur conjecture (ARS) says that a PDE isintegrable in the Painleve sense, if all of its reductions have the Painleveproperty.

We can extend the Painleve 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 Laurentexpansion series which incorporates ul (xi ) as functions of the coordinatesxi .

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 Painleve

sense in certain variables, but not when expressed in others, i.e.,the PT is

not intrisecally a geometrical property.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Lax pairs

A Lax pair (LP) or spectral problem is a pair of linear operators L(t) andP(t), acting on a fixed Hilbert space H, that satisfy a correspondingdifferential equation, the so called Lax equation

dL

dt= [P, L],

where [P, L] = PL− LP. The operator L(t) is said to be isospectral if itsspectrum of eigenvalues is independent of the evolution variable. We calleigenvalue problem the relation

Lψ = λψ,

where ψ ∈ H, henceforth called a spectral function or eigenfunction, and λis a spectral value or eigenvalue.

Lax pairs are interesting because they guarantee the integrability of certain

differential equations. Some PDEs can equivalently be rewritten as the

compatilibity condition of a spectral problem. Sometimes, it is easier to

solve the associated LP rather than the equation itself. Hence, the inverse

scattering method arose.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

The inverse scattering method

The IST guarantees the existence of analytical solutions of the PDE (whenit can be applied). The name inverse transform comes from the idea ofrecovering the time evolution of the potential u(x , t) from the timeevolution of its scattering data, opposed to the direct scattering whichfinds the scattering matrix from the evolution of the potential.

Consider L and P acting on H, where L depends on an unknownfunction u(x , t) and P is independent of it in the scattering region.

We can compute the spectrum of eigenvalues λ for L(0) and obtainψ(x , 0).

If P is known, we can propagate the eigenfunction with the equation∂ψ∂t

(x , t) = Pψ(x , t) with initial condition ψ(x , 0).

Knowing ψ(x , t) in the scattering region, we construct L(t) andreconstruct u(x , t) by means of the Gelfand–Levitan–Marchenkoequation.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

The inverse scattering method

Initial potential, u(x, t = 0)scattering data //

KS

time difference

��

Spectrum L(0), ψ(x, t = 0)

scattering data t>0

��

Potential at time t, u(x, t) dψ/dt = Pψ, i.c. ψ(x, t = 0)Inverse scattering data

Reconstruct L(t),t>0

oo



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

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 ), (4)

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.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

The SMM and Lax pairs

The singular manifold method has shown its efficiency in the derivation ofLax pairs. Through the singular manifold equations, with a generalexpression F = F (φ, φxi , φxi xj , . . . ), we introduce the quantities

ω = φt/φx , v = φxx/φx , s = vx − v 2/2.

From the compatibility condition φxt = φtx , we achieve

vt = (ωx + ωv)x , st = ωxxx + 2sωx + ωsx .

We obtain a final expression of the type F = F (ω, s) which is linearizable

from which a Lax pair can be retrieved.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

The Hirota method

Another method guaranteeing the integrability of a PDE is the Hirota’sbilinear method. The major advantage of the HBM over the IST is theobtainance of possible multi-soliton solutions by imposing Ansatze.

Hirota noticed that the best dependent variables for constructing solitonsolutions are those in which the soliton appears as a finite number ofexponentials. To apply this method it is necessary that the equation isquadratic and that the derivatives can be expressed using Hirota’sD-operator defined by

Dnx f · g = (∂x1 − ∂x2 )nf (x1)g(x2)

∣∣∣x2=x1=x

.

Unfortunately, the process of bilinearization is far from being algorithmic,and it is hard to know how many variables are needed for bilinearization.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Lie symmetry

The notion of Lie symmetry has been widely considered since the 19thcentury as a way to find solutions of dynamical systems...

...Out of symmetry, we achieve conserved quantities that imply thereduction or possible integration of a system.

Reduced versions of an unidentified system can occur as well known

equations in the scientific literature.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems


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.

Two different equations, seemingly unrelated, happen to be equivalentversions of a same equation after a reciprocal transformation. In this way,the big number of integrable equations in the literature, could be greatlydiminished by establishing a method to discern which equations aredisguised versions of a common problem.

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 different

transformations 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.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems


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.

Two different equations, seemingly unrelated, happen to be equivalentversions of a same equation after a reciprocal transformation. In this way,the big number of integrable equations in the literature, could be greatlydiminished by establishing a method to discern which equations aredisguised versions of a common problem.

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 different

transformations 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.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Example

CHH(2 + 1)reciprocal transf. //

KSMiura-reciprocal transf.

��

CBS equation

Miura transf.

��

mCHH(2 + 1) mCBS equationreciprocal transf.oo

Figure: Miura-reciprocal transformation.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Lie systems

A Lie system is system of ODEs that admits a superposition rule, i.e., amap allowing us to express the general solution of the system of ODEs interms of a family of particular solutions and a set of constants related toinitial conditions.

Φ : Nm × N → N of the form x = Φ(x(1), . . . , x(m); k) allowing us to writethe general solution as

x(t) = Φ(x(1)(t), . . . , x(m)(t); k),

where x(1)(t), . . . , x(m)(t) is a generic family of particular solutions andk ∈ N.

These superposition principles are, in general, nonlinear.



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Main theorem

Theorem

A system Xt is a Lie system and consequently admits a superposition ruleif and only if Xt =

∑rα=1 bα(t)Xα spans an r-dimensional Lie algebra of

vector fields, X1, . . . ,Xr the so-called Vessiot–Guldberg Lie algebraassociated with Xt , for certain functions b1(t), . . . , br (t).

Consider the first-order Riccati equation on the real line

x = a0(t) + a1(t)x + z2(t)x2

This equation admits the decomposition in terms of a time dependentvector field

X (x , t) = a0(t)X1 + a1(t)X2 + a2(t)X3

whereX1 = ∂/∂x , X2 = x∂/∂x , X3 = x2∂/∂x

that span a V.G. Lie algebra V isomorphic to sl(2,R).



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Examples of Lie systems

Lie systems play a relevant role in Cosmology, quantum mechanicalproblems, Financial Mathematics, Control theory, Biology... Manydifferential equations can be studied through the theory of Lie systems,even though they are not Lie systems. This is the case of

Riccati equations and generalized versions (matrix Riccati...)

Kummer–Schwarz equations,

Milne–Pinney equations,

Ermakov system,

Winternitz–Smorodinsky oscillators,

Buchdahl equations,

Second-order Riccati equations

Riccati equations on different types of composition algebras, as thecomplex, quaternions, Study numbers .

Viral models

Reductions of Yang–Mills equations

Complex Bernoulli equations



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Other Lie systems

Lie–Hamilton systems are Lie systems that admit Vessiot–Guldberg Liealgebras of Hamiltonian vector fields with respect to a Poisson structure.

Lie–Hamilton systems posses a time-dependent Hamiltonian given by acurve in a finite-dimensional Lie algebra of functions with respect to aPoisson bracket related with the Poisson structure, a Lie–Hamilton algebra.

Lie systems enjoy a plethora of properties

Superposition rules can be interpreted as zero-curvature connections

Involvement of different geometric structures as Poisson, Dirac,Jacobi, k-symplectic, contact structures and so on

The Poisson coalgebra method can be applied to obtain constants ofmotion in the case of Lie–Hamilton systems



CristinaSardon


Lax pairs


Lie symmetry


Lie systems

Bibliography

My thesis dissertation is based on the aforementioned methods and

includes an exhaustive variety of examples. A total of 330 pages can serve

you as a handbook for further details.

“Lie systems, Lie symmetriesand reciprocal transformations”

Available at: ArXiv:1508.00726

Integrability approaches to differential equationscud.unizar.es/sites/default/files/personal/pers... · 2015-10-05 · approaches to di erential equations Cristina Sard on The Painleve

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