Integrability approaches to differential equations

Transcription

1 XXIV Fall Workshop on Geometry and Physics Zaragoza, CUD, September 2015

2 What is integrability? On a fairly imprecise first approximation, we say integrability is the exact solvability or regular behavior of solutions of a system.

3 What is integrability? On a fairly imprecise first approximation, we say integrability is the exact solvability or regular behavior of solutions of a system....nevertheless, there are various, distinct notions of integrable systems...

4 What is integrability? On a fairly imprecise first approximation, we say integrability is the exact solvability 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.

5 Why integrability? appeared with Classical Mechanics with a quest for exact solutions to Newton s equation of motion. Integrable systems present a number of conserved quantities: angular momentum, linear momentum, energy... Indeed, some systems present an infinite number of conserved quantities. But finding conserved quantities is more of an exception rather than a rule! Hence, the need for characterization and search of criteria for integrability!

6 Notions of integrability Geometrical viewpoint of dynamical systems: Differential are interpreted in terms of Pfaffian systems and the Frobenius theorem.

7 Notions of integrability Geometrical viewpoint of dynamical systems: Differential are interpreted in terms of Pfaffian systems and the Frobenius theorem. In the context of differentiable dynamical systems: The notion of integrability refers to Liouville integrability.

8 Notions of integrability Geometrical viewpoint of dynamical systems: Differential are interpreted in terms of Pfaffian systems and the Frobenius theorem. 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

9 Notions of integrability Geometrical viewpoint of dynamical systems: Differential are interpreted in terms of Pfaffian systems and the Frobenius theorem. 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?: method, existence of, the inverse scattering transform, the Hirota bilinear method...

10 approaches What comes to integrability of, we focus on the approaches method: this is a quasi-algorithmic method to check whether an ODE or PDE is integrable. The existence of Derivation of with the singular manifold method. Solvability through the Inverse scattering method. solutions or the Hirota bilinear method approaches with different geometric approaches.

11 method To discern whether our equation is integrable or not, involves a deep inspection of its geometrical properties and aforementioned methods. Now, a question arises: Is there any algorithmical method to check the integrability of a equation? The answer is affirmative. It focuses on the singularity analysis of the equation, attending to a fundamental property: being a fixed or a movable singularity, or a singularity not depending or depending on the initial conditions, respectively. Sophia Kovalevskaya centered herself in the study of 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.

12 Fixed and movable singularity Consider a manifold N locally diffeomorphic to R TR, with local coordinates {t, u(t), u t}. Consider the equation (t c)u t = bu and c, b R. Its general solution reads u(t) = k 0(t c) b, where k 0 is a constant of integration. Depending on the value of the exponent 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.

13 Fixed and movable singularity Let us now consider an ODE on R T 2 R with local coordinates {t, u, u t, u tt}, which reads buu tt + (1 b)u 2 t = 0, with b R. The general solution to this equation is u(t) = k 0(t t 0) b. If b is a negative integer, the singularity t = t 0 is a singularity that depends on the initial conditions through the constant of integration t 0. In this case, we say that the singulary is movable.

14 Painlevé, Gambier et al oriented their study towards second-order. on R T 2 R with local coordinates {t, u, u t, u tt}, of the type u tt = F (t, u, u t), where F is a rational function in u, u t and analytic in t. He found that there were 50 different of this type whose unique movable singularities were poles. Out of the 50 types, 44 were integrated in terms of known functions as Riccati, elliptic, linear, etc., and the 6 remaining, although having meromorphic solutions, they do not possess algebraic integrals that permit us to reduce them by quadratures. These 6 functions are called Painlevé transcendents (P I P VI ), because they cannot be expressed in terms of elementary or rational functions or solutions expressible in terms of special functions.

15 and Test We say that an ODE has the PP if all the movable singularities of its solution are poles. Test Given a general ODE on R T p R with local coordinates {t, u, u t,..., u t,...,t}, F = F (t, u(t),..., u t,...,t), (1) the PT analyzes local properties by proposing solutions in the form u(t) = a j (t t 0) (j α), (2) j=0 where t 0 is the singularity, a j, 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

16 test 1 We determine the value of α by balance of dominant terms, which will permit us to obtain a 0, simultaneously. The values of α and a 0 are not necessarily unique, and α must be a positive integer. 2 Having introduced (2) into the equation (1), we obtain a relation of recurrence for the rest of coefficients a j (j β 1) (j β n)a j = F j (t,..., u k, (u k ) t,... ), k < j, (3) which arises from setting equal to zero different orders in (t t 0). This gives us a j in terms of a k for k < j. Observe that when j = β l with l = 1,..., n, the left-hand side of the equation is null and the associated a βl is arbitrary. Those values of j, are called resonances and the equation (3) turns into a relation for a k for k < β l which is known as the resonance condition. 3 If resonance conditions are satisfied identically, F j = 0 for every j = β l, we say that the ODE posesses the PP. The resonances have to be positive except j = 1, which is associated with the arbitrariness of t 0. In the case of PDEs, Weiss, Tabor and Carnevale carried out the generalization of the Painleve method, the so called WTC method.

17 Painleve test for PDEs The Ablowitz-Ramani-Segur conjecture (ARS) says that a PDE is integrable in the Painlevé sense, if all of its reductions have the Painlevé property. We can extend the Painlevé test to PDEs by substituting the function (t t 0) by an arbitrary function φ(x i ) for all i = 1,..., n, which receives the name of movable singularity manifold. We propose a Laurent expansion series which incorporates u l (x i ) as functions of the coordinates x i. 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.

18 A Lax pair (LP) or spectral problem is a pair of linear operators L(t) and P(t), acting on a fixed Hilbert space H, that satisfy a corresponding 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 its spectrum of eigenvalues is independent of the evolution variable. We call eigenvalue problem the relation Lψ = λψ, where ψ H, henceforth called a spectral function or eigenfunction, and λ is a spectral value or eigenvalue. are interesting because they guarantee the integrability of certain. 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.

19 The inverse scattering method The IST guarantees the existence of analytical solutions of the PDE (when it can be applied). The name inverse transform comes from the idea of recovering the time evolution of the potential u(x, t) from the time evolution of its scattering data, opposed to the direct scattering which finds the scattering matrix from the evolution of the potential. Consider L and P acting on H, where L depends on an unknown function 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 obtain ψ(x, 0). If P is known, we can propagate the eigenfunction with the equation (x, t) = Pψ(x, t) with initial condition ψ(x, 0). ψ t 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.

20 The inverse scattering method scattering data Initial potential, u(x, t = 0) time difference Spectrum L(0), ψ(x, t = 0) scattering data t>0 Inverse scattering data Potential at time t, u(x, t) dψ/dt = Pψ, i.c. ψ(x, t = 0) Reconstruct L(t),t>0

21 The singular manifold method The singular manifold method (SMM) focuses on solutions which arise from truncated series of the generalized PP method. We require the solutions of the PDE written in the form of a Laurent expansion to select the truncated terms u l (x i ) u (l) 0 (x i)φ(x i ) α + u (l) 1 (x i)φ(x i ) 1 α + + u (l) α (x i ), (4) for every l. In the case of several branches of expansion, this truncation needs to be formulated for every value of α. Here, the function φ(x i ) is no longer arbitrary, but a singular manifold equation whose expression arises from the truncation. An expression of the type F = F (φ, φ xi, φ xi,x j,... ), arises. The SMM is interesting because it contributes substantially in the derivation of a Lax pair.

22 The SMM and The singular manifold method has shown its efficiency in the derivation of. Through the singular manifold, with a general expression F = F (φ, φ xi, φ xi x j,... ), we introduce the quantities ω = φ t/φ x, v = φ xx/φ x, s = v x v 2 /2. From the compatibility condition φ xt = φ tx, we achieve v t = (ω x + ωv) x, s t = ω xxx + 2sω x + ωs x. We obtain a final expression of the type F = F (ω, s) which is linearizable from which a Lax pair can be retrieved.

23 The Another method guaranteeing the integrability of a PDE is the Hirota s bilinear method. The major advantage of the HBM over the IST is the obtainance of possible multi-soliton solutions by imposing Ansätze. Hirota noticed that the best dependent variables for constructing soliton solutions are those in which the soliton appears as a finite number of exponentials. To apply this method it is necessary that the equation is quadratic and that the derivatives can be expressed using Hirota s D-operator defined by Dx n f g = ( x1 x2 ) n f (x 1)g(x 2) x2 =x 1 =x. Unfortunately, the process of bilinearization is far from being algorithmic, and it is hard to know how many variables are needed for bilinearization.

24 The notion of has been widely considered since the 19th century as a way to find solutions of dynamical systems......out of symmetry, we achieve conserved quantities that imply the reduction or possible integration of a system. Reduced versions of an unidentified system can occur as well known in the scientific literature.

25 consist on the mixing of the role of the dependent and independent variables to achieve simpler versions or even linearized versions of the initial, nonlinear PDE. Two different, seemingly unrelated, happen to be equivalent versions of a same equation after a reciprocal transformation. In this way, the big number of integrable in the literature, could be greatly diminished by establishing a method to discern which are disguised versions of a common problem. Then, the next question comes out: Is there a way to identify different versions of a common nonlinear problem?

26 consist on the mixing of the role of the dependent and independent variables to achieve simpler versions or even linearized versions of the initial, nonlinear PDE. Two different, seemingly unrelated, happen to be equivalent versions of a same equation after a reciprocal transformation. In this way, the big number of integrable in the literature, could be greatly diminished by establishing a method to discern which are disguised versions of a common problem. Then, the next question comes out: Is there a way to identify different versions of a common nonlinear problem? In principle, the only way to ascertain is by proposing different 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.

27 Example CHH(2 + 1) mchh(2 + 1) Miura-reciprocal transf. reciprocal transf. reciprocal transf. CBS equation Miura transf. mcbs equation Figure: Miura-reciprocal transformation.

28 A Lie system is system of ODEs that admits a superposition rule, i.e., a map allowing us to express the general solution of the system of ODEs in terms of a family of particular a set of constants related to initial conditions. Φ : N m N N of the form x = Φ(x (1),..., x (m) ; k) allowing us to write the 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 k N. These superposition principles are, in general, nonlinear.

29 Main theorem Theorem A system X t is a Lie system and consequently admits a superposition rule if and only if X t = rα=1 bα(t)xα spans an r-dimensional Lie algebra of vector fields, X 1,..., X r the so-called Vessiot Guldberg Lie algebra associated with X t, for certain functions b 1(t),..., b r (t). Consider the first-order Riccati equation on the real line ẋ = a 0(t) + a 1(t)x + z 2(t)x 2 This equation admits the decomposition in terms of a time dependent vector field X (x, t) = a 0(t)X 1 + a 1(t)X 2 + a 2(t)X 3 where X 1 = / x, X 2 = x / x, X 3 = x 2 / x that span a V.G. Lie algebra V isomorphic to sl(2, R).

30 Examples of play a relevant role in Cosmology, quantum mechanical problems, Financial Mathematics, Control theory, Biology... Many can be studied through the theory of, even though they are not. This is the case of Riccati and generalized versions (matrix Riccati...) Kummer Schwarz, Milne Pinney, Ermakov system, Winternitz Smorodinsky oscillators, Buchdahl, Second-order Riccati Riccati on different types of composition algebras, as the complex, quaternions, Study numbers. Viral models Reductions of Yang Mills Complex Bernoulli

31 Other Lie Hamilton systems are that admit Vessiot Guldberg Lie algebras of Hamiltonian vector fields with respect to a Poisson structure. Lie Hamilton systems posses a time-dependent Hamiltonian given by a curve in a finite-dimensional Lie algebra of functions with respect to a Poisson bracket related with the Poisson structure, a Lie Hamilton algebra. 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 of motion in the case of Lie Hamilton systems

32 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 symmetries and reciprocal Available at: ArXiv: