Integrability Approaches to Differential Equations

Integrability approaches to diﬀerential equations

Cristina Sard´on

XXIV Fall Workshop on Geometry and Physics Zaragoza, CUD, September 2015 ...Nevertheless, there are various, distinct notions of integrable systems...

The characterization and uniﬁed deﬁnition of integrability is a nontrivial matter.

What is integrability?

Integrability approaches to diﬀerential equations

Cristina Sard´on On a fairly imprecise ﬁrst approximation, we say integrability is the exact The Painleve Property solvability or regular behavior of solutions of a system.

Lax pairs

Solitonic solutions and Hirota method

Lie symmetry

Reciprocal transformations

Lie systems The characterization and uniﬁed deﬁnition of integrability is a nontrivial matter.

What is integrability?

Integrability approaches to diﬀerential equations

Cristina Sard´on On a fairly imprecise ﬁrst approximation, we say integrability is the exact The Painleve Property solvability or regular behavior of solutions of a system.

Lax pairs

Solitonic ...Nevertheless, there are various, distinct notions of integrable systems... solutions and Hirota method

Lie symmetry

Reciprocal transformations

Lie systems What is integrability?

Integrability approaches to diﬀerential equations

Cristina Sard´on On a fairly imprecise ﬁrst approximation, we say integrability is the exact The Painleve Property solvability or regular behavior of solutions of a system.

Lax pairs

Solitonic ...Nevertheless, there are various, distinct notions of integrable systems... solutions and Hirota method

Lie symmetry The characterization and uniﬁed deﬁnition of integrability is a nontrivial

Reciprocal matter. transformations

Lie systems Why integrability?

Integrability approaches to diﬀerential equations

Cristina Sardón Integrability appeared with Classical Mechanics with a quest for exact The Painleve solutions to Newton’s equation of motion. Property Lax pairs Integrable systems present a number of conserved quantities: angular Solitonic momentum, linear momentum, energy... Indeed, some systems present an solutions and Hirota method infinite number of conserved quantities. But finding conserved quantities is more of an exception rather than a rule! Lie symmetry

Reciprocal transforma- Hence, the need for characterization and search of criteria for integrability! tions

Lie systems In the context of diﬀerentiable dynamical systems: The notion of integrability refers to Liouville integrability. In Hamiltonian systems: Existence of maximal set of commuting invariants with the Hamiltonian.

{I , H} = 0

Any (quasi) algorithmic methods?: The Painleve method, existence of Lax pairs, the inverse scattering transform, the Hirota bilinear method...

Notions of integrability

Lax pairs

Solitonic solutions and Hirota method

Lie symmetry

Reciprocal transformations

Lie systems In Hamiltonian systems: Existence of maximal set of commuting invariants with the Hamiltonian.

{I , H} = 0

Any (quasi) algorithmic methods?: The Painleve method, existence of Lax pairs, the inverse scattering transform, the Hirota bilinear method...

Notions of integrability

Integrability approaches to differential equations Geometrical viewpoint of dynamical systems: Differential Cristina equations are interpreted in terms of Pfaffian systems and the Sardón Frobenius theorem. The Painleve Property In the context of differentiable dynamical systems: The notion of integrability refers to Liouville integrability. Lax pairs

Solitonic solutions and Hirota method

Lie symmetry

Reciprocal transformations

Lie systems Any (quasi) algorithmic methods?: The Painleve method, existence of Lax pairs, the inverse scattering transform, the Hirota bilinear method...

Notions of integrability

Solitonic In Hamiltonian systems: Existence of maximal set of commuting solutions and invariants with the Hamiltonian. Hirota method Lie symmetry {I , H} = 0 Reciprocal transformations

Lie systems Notions of integrability

Solitonic In Hamiltonian systems: Existence of maximal set of commuting solutions and invariants with the Hamiltonian. Hirota method Lie symmetry {I , H} = 0 Reciprocal transformations Any (quasi) algorithmic methods?: The Painleve method, existence Lie systems of Lax pairs, the inverse scattering transform, the Hirota bilinear method... Integrability approaches

Integrability approaches to differential equations What comes to integrability of differential equations, we focus on the Cristina approaches Sardón The Painleve method: this is a quasi-algorithmic method to check The Painleve Property whether an ODE or PDE is integrable. Lax pairs The existence of Lax pairs Solitonic solutions and Derivation of Lax pairs with the singular manifold method. Hirota method Solvability through the Inverse scattering method. Lie symmetry Solitonic solutions or the Hirota bilinear method Reciprocal transforma- Reciprocal transformations tions Lie symmetry approaches Lie systems Lie systems with different geometric approaches. The Painleve method

Integrability approaches to diﬀerential equations To discern whether our equation is integrable or not, involves a deep

Cristina inspection of its geometrical properties and aforementioned methods. Now, Sardón a question arises: Is there any algorithmical method to check the integrability of a differential equation? The answer is affirmative. The Painleve Property

Lax pairs It focuses on the singularity analysis of the diﬀerential equation, attending

Solitonic to a fundamental property: being a ﬁxed or a movable singularity, or a solutions and singularity not depending or depending on the initial conditions, Hirota method respectively. Lie symmetry

Reciprocal transforma- Sophia Kovalevskaya centered herself in the study of equations for solid tions rigid dynamics: singularities and properties of single-valuedness of poles of Lie systems PDEs on the complex plane, etc. Eventually, she expanded her results to other physical systems. Fixed and movable singularity

Integrability approaches to differential Consider a manifold N locally diffeomorphic to × T , with local equations R R coordinates {t, u(t), ut }. Consider the differential equation Cristina Sardón (t − c)ut = bu

The Painleve Property and c, b ∈ R. Its general solution reads

Lax pairs b u(t) = k0(t − c) , Solitonic solutions and Hirota method where k0 is a constant of integration. Depending on the value of the exponent b, we have diﬀerent types of singularities Lie symmetry

Reciprocal If b is a positive integer, then, u(t) is a holomorphic function. transformations If b is a negative integer, then c is a pole singularity. Lie systems 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. Fixed and movable singularity

Integrability approaches to diﬀerential equations

Cristina 2 Sard´on Let us now consider an ODE on R × T R with local coordinates {t, u, ut , utt }, which reads The Painleve Property 2 buutt + (1 − b)ut = 0, Lax pairs

Solitonic with b ∈ R. The general solution to this equation is solutions and Hirota method b u(t) = k0(t − t0) . Lie symmetry If b is a negative integer, the singularity t = t is a singularity that depends Reciprocal 0 transforma- on the initial conditions through the constant of integration t0. In this tions case, we say that the singulary is movable. Lie systems Integrability approaches to differential equations Painlevé,Gambier et al oriented their study towards second-order 2 Cristina differential equations. on R × T R with local coordinates {t, u, ut , utt }, of Sardón the type

The Painleve utt = F (t, u, ut ), Property where F is a rational function in u, ut and analytic in t. Lax pairs He found that there were 50 diﬀerent equations of this type whose unique Solitonic solutions and movable singularities were poles. Out of the 50 types, 44 were integrated Hirota method in terms of known functions as Riccati, elliptic, linear, etc., and the 6 Lie symmetry remaining, although having meromorphic solutions, they do not possess Reciprocal algebraic integrals that permit us to reduce them by quadratures. transformations These 6 functions are called Painlev´etranscendents (PI − PVI ), because Lie systems they cannot be expressed in terms of elementary or rational functions or solutions expressible in terms of special functions. The Painleve Property and Test

Integrability approaches to diﬀerential We say that an ODE has the PP if all the movable singularities of its equations solution are poles. Cristina Sard´on The Painleve Test

The Painleve Given a general ODE on × Tp with local coordinates Property R R {t, u, ut ,..., ut,...,t }, Lax pairs

Solitonic F = F (t, u(t),..., ut,...,t ), (1) solutions and Hirota method the PT analyzes local properties by proposing solutions in the form Lie symmetry ∞ Reciprocal X (j−α) transforma- u(t) = aj (t − t0) , (2) tions j=0 Lie systems where t0 is the singularity, aj , ∀j are constants and α is necessarily a positive 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 The Painleve test

Integrability approaches to 1 We determine the value of α by balance of dominant terms, which differential will permit us to obtain a0, simultaneously. The values of α and a0 equations are not necessarily unique, and α must be a positive integer. Cristina Sardón 2 Having introduced (2) into the differential equation (1), we obtain a relation of recurrence for the rest of coefficients aj The Painleve Property (j − β1) ····· (j − βn)aj = Fj (t,..., uk , (uk )t ,... ), k < j, (3) Lax pairs which arises from setting equal to zero different orders in (t − t0). Solitonic This gives us aj in terms of ak for k < j. Observe that when j = βl solutions and Hirota method with l = 1,..., n, the left-hand side of the equation is null and the associated a is arbitrary. Those values of j, are called resonances Lie symmetry βl and the equation (3) turns into a relation for ak for k < βl which is Reciprocal transforma- known as the resonance condition. tions 3 If resonance conditions are satisfied identically, Fj = 0 for every Lie systems j = βl , we say that the ODE posesses the PP. The resonances have to be positive except j = −1, which is associated with the arbitrariness of t0. In the case of PDEs, Weiss, Tabor and Carnevale carried out the generalization of the Painleve method, the so called WTC method. Painleve test for PDEs

Integrability approaches to differential equations The Ablowitz-Ramani-Segur conjecture (ARS) says that a PDE is Cristina integrable in the Painlevésense, if all of its reductions have the Painlevé Sardón property. The Painleve Property We can extend the Painlevétest to PDEs by substituting the function

Lax pairs (t − t0) by an arbitrary function φ(xi ) for all i = 1,..., n, which receives the name of movable singularity manifold. We propose a Laurent Solitonic solutions and expansion series which incorporates ul (xi ) as functions of the coordinates Hirota method xi . Lie symmetry

Reciprocal It is important to mention that the PT is not invariant under changes of transformations coordinates. This means that an equation can be integrable in the Painlev´e

Lie systems sense in certain variables, but not when expressed in others, i.e.,the PT is not intrisecally a geometrical property. Lax pairs

Integrability A Lax pair (LP) or spectral problem is a pair of linear operators L(t) and approaches to differential P(t), acting on a fixed Hilbert space H, that satisfy a corresponding equations differential equation, the so called Lax equation Cristina dL Sardón = [P, L], dt The Painleve Property where [P, L] = PL − LP. The operator L(t) is said to be isospectral if its

Lax pairs spectrum of eigenvalues is independent of the evolution variable. We call eigenvalue problem the relation Solitonic solutions and Hirota method Lψ = λψ, Lie symmetry where ψ ∈ H, henceforth called a spectral function or eigenfunction, and λ Reciprocal transforma- is a spectral value or eigenvalue. tions

Lie systems 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. The inverse scattering method

Integrability approaches to differential The IST guarantees the existence of analytical solutions of the PDE (when equations it can be applied). The name inverse transform comes from the idea of Cristina Sardón recovering the time evolution of the potential u(x, t) from the time evolution of its scattering data, opposed to the direct scattering which The Painleve finds the scattering matrix from the evolution of the potential. Property

Lax pairs Consider L and P acting on H, where L depends on an unknown

Solitonic function u(x, t) and P is independent of it in the scattering region. solutions and Hirota method We can compute the spectrum of eigenvalues λ for L(0) and obtain

Lie symmetry ψ(x, 0).

Reciprocal If P is known, we can propagate the eigenfunction with the equation transforma- ∂ψ (x, t) = Pψ(x, t) with initial condition ψ(x, 0). tions ∂t

Lie systems Knowing ψ(x, t) in the scattering region, we construct L(t) and reconstruct u(x, t) by means of the Gelfand–Levitan–Marchenko equation. The inverse scattering method

Integrability approaches to diﬀerential equations

Cristina Sard´on

scattering data The Painleve Initial potential, u(x, t = 0) / Spectrum L(0), ψ(x, t = 0) Property Lax pairs KS Solitonic time diﬀerence scattering data t>0 solutions and Hirota method Inverse scattering data Lie symmetry Potential at time t, u(x, t) dψ/dt = Pψ, i.c. ψ(x, t = 0) o Reconstruct L(t),t>0 Reciprocal transformations

Lie systems The singular manifold method

Integrability approaches to diﬀerential equations The singular manifold method (SMM) focuses on solutions which arise Cristina from truncated series of the generalized PP method. We require the Sard´on solutions of the PDE written in the form of a Laurent expansion to select the truncated terms The Painleve Property (l) −α (l) 1−α (l) ul (xi ) ' u (xi )φ(xi ) + u (xi )φ(xi ) + ··· + uα (xi ), (4) Lax pairs 0 1 Solitonic for every l. In the case of several branches of expansion, this truncation solutions and Hirota method needs to be formulated for every value of α. Here, the function φ(xi ) is no

Lie symmetry longer arbitrary, but a singular manifold equation whose expression arises from the truncation. Reciprocal transformations An expression of the type F = F (φ, φxi , φxi ,xj ,... ), arises.

Lie systems The SMM is interesting because it contributes substantially in the derivation of a Lax pair. The SMM and Lax pairs

Integrability approaches to diﬀerential equations

Cristina The singular manifold method has shown its eﬃciency in the derivation of Sard´on Lax pairs. Through the singular manifold equations, with a general

The Painleve expression F = F (φ, φxi , φxi xj ,... ), we introduce the quantities Property 2 Lax pairs ω = φt /φx , v = φxx /φx , s = vx − v /2.

Solitonic solutions and From the compatibility condition φxt = φtx , we achieve Hirota method

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

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

Lie systems from which a Lax pair can be retrieved. The Hirota method

Integrability approaches to differential equations Another method guaranteeing the integrability of a PDE is the Hirota’s Cristina Sardón bilinear method. The major advantage of the HBM over the IST is the obtainance of possible multi-soliton solutions by imposing Ansätze. The Painleve Property Hirota noticed that the best dependent variables for constructing soliton Lax pairs solutions are those in which the soliton appears as a finite number of

Solitonic exponentials. To apply this method it is necessary that the equation is solutions and quadratic and that the derivatives can be expressed using Hirota’s Hirota method D-operator deﬁned by Lie symmetry

Reciprocal Dnf · g = (∂ − ∂ )nf (x )g(x ) . x x1 x2 1 2 transforma- x2=x1=x tions

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

Integrability approaches to diﬀerential equations

Cristina Sard´on The notion of Lie symmetry has been widely considered since the 19th The Painleve century as a way to ﬁnd solutions of dynamical systems... Property

Lax pairs ...Out of symmetry, we achieve conserved quantities that imply the Solitonic solutions and reduction or possible integration of a system. Hirota method Lie symmetry Reduced versions of an unidentiﬁed system can occur as well known Reciprocal equations in the scientiﬁc literature. transformations

Lie systems 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.

Reciprocal transformations

Integrability approaches to differential Reciprocal transformations consist on the mixing of the role of the equations dependent and independent variables to achieve simpler versions or even Cristina linearized versions of the initial, nonlinear PDE. Sardón Two different equations, seemingly unrelated, happen to be equivalent The Painleve Property versions of a same equation after a reciprocal transformation. In this way,

Lax pairs the big number of integrable equations in the literature, could be greatly diminished by establishing a method to discern which equations are Solitonic solutions and disguised versions of a common problem. Hirota method

Lie symmetry Then, the next question comes out: Is there a way to identify diﬀerent versions of a common nonlinear problem? Reciprocal transformations

Lie systems Reciprocal transformations

Lie symmetry Then, the next question comes out: Is there a way to identify diﬀerent versions of a common nonlinear problem? Reciprocal transformations In principle, the only way to ascertain is by proposing diﬀerent Lie systems 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. Example

Integrability approaches to diﬀerential equations

Cristina Sard´on reciprocal transf. CHH(2 + 1) / CBS equation The Painleve Property KS Lax pairs Miura-reciprocal transf. Miura transf. Solitonic solutions and Hirota method reciprocal transf. mCHH(2 + 1) mCBS equation Lie symmetry o

Reciprocal transformations Figure: Miura-reciprocal transformation. Lie systems Lie systems

Integrability approaches to diﬀerential equations

Cristina A Lie system is system of ODEs that admits a superposition rule, i.e., a Sard´on map allowing us to express the general solution of the system of ODEs in terms of a family of particular solutions and a set of constants related to The Painleve Property initial conditions.

Lax pairs m Φ: N × N → N of the form x = Φ(x(1),..., x(m); k) allowing us to write Solitonic the general solution as solutions and Hirota method x(t) = Φ(x(1)(t),..., x(m)(t); k), Lie symmetry

Reciprocal where x(1)(t),..., x(m)(t) is a generic family of particular solutions and transformations k ∈ N. Lie systems These superposition principles are, in general, nonlinear. Main theorem

Integrability approaches to diﬀerential equations Theorem

Cristina A system Xt is a Lie system and consequently admits a superposition rule Sardón Pr if and only if Xt = α=1 bα(t)Xα spans an r-dimensional Lie algebra of vector fields, X ,..., X the so-called Vessiot–Guldberg Lie algebra The Painleve 1 r Property associated with Xt , for certain functions b1(t),..., br (t). Lax pairs Consider the first-order Riccati equation on the real line Solitonic solutions and 2 Hirota method x˙ = a0(t) + a1(t)x + z2(t)x

Lie symmetry This equation admits the decomposition in terms of a time dependent Reciprocal transforma- vector ﬁeld tions X (x, t) = a0(t)X1 + a1(t)X2 + a2(t)X3 Lie systems where 2 X1 = ∂/∂x, X2 = x∂/∂x, X3 = x ∂/∂x that span a V.G. Lie algebra V isomorphic to sl(2, R). Examples of Lie systems

Integrability Lie systems play a relevant role in Cosmology, quantum mechanical approaches to differential problems, Financial Mathematics, Control theory, Biology... Many equations differential equations can be studied through the theory of Lie systems, Cristina even though they are not Lie systems. This is the case of Sardón Riccati equations and generalized versions (matrix Riccati...) The Painleve Kummer–Schwarz equations, Property

Lax pairs Milne–Pinney equations, Solitonic Ermakov system, solutions and Hirota method Winternitz–Smorodinsky oscillators, Lie symmetry Buchdahl equations, Reciprocal transforma- Second-order Riccati equations tions Riccati equations on diﬀerent types of composition algebras, as the Lie systems complex, quaternions, Study numbers . Viral models Reductions of Yang–Mills equations Complex Bernoulli equations Other Lie systems

Integrability approaches to differential equations Lie–Hamilton systems are Lie systems that admit Vessiot–Guldberg Lie Cristina Sardón algebras of Hamiltonian vector fields with respect to a Poisson structure.

The Painleve Lie–Hamilton systems posses a time-dependent Hamiltonian given by a Property curve in a ﬁnite-dimensional Lie algebra of functions with respect to a Lax pairs Poisson bracket related with the Poisson structure, a Lie–Hamilton algebra.

Solitonic solutions and Lie systems enjoy a plethora of properties Hirota method Superposition rules can be interpreted as zero-curvature connections Lie symmetry Involvement of diﬀerent geometric structures as Poisson, Dirac, Reciprocal transforma- Jacobi, k-symplectic, contact structures and so on tions The Poisson coalgebra method can be applied to obtain constants of Lie systems motion in the case of Lie–Hamilton systems Bibliography

Integrability approaches to diﬀerential equations

Cristina My thesis dissertation is based on the aforementioned methods and Sard´on includes an exhaustive variety of examples. A total of 330 pages can serve The Painleve you as a handbook for further details. Property

Lax pairs Solitonic “Lie systems, Lie symmetries solutions and Hirota method and reciprocal transformations” Lie symmetry

Reciprocal transformations Available at: ArXiv:1508.00726 Lie systems