Integrability, Nonintegrability and the Poly-Painlev´eTest Rodica D. Costin Sept. 2005 Nonlinearity 2003, 1997, preprint, Meth.Appl.An. 1997, Inv.Math. 2001 Talk dedicated to my professor, Martin Kruskal. Happy birthday! Historical perspective: Motivated by the idea that new functions can be defined as solutions of linear differential equations (Airy, Bessel, Mathieu, Lam´e,Legendre...), a natural question arose: Can nonlinear equations define new, \transcendental" functions? 1 Fuchs' intuition: A given equation does define functions if movable singularities in C of general solution are no worse than poles (no branch points, no essential singularities). Such an equation has solutions meromorphic on a (common) Riemann surface. 1 u0 = gen sol u = (x − C)1=2 movable branch point 2u 1 u0 = gen sol u = x1=2 + C fixed branch point 2x1=2 The property of a diff.eq. to have general solution without movable singularities (except, perhaps, poles) is now called \the Painlev´eProperty"(PP). 2 Directions: discovery of Painlev´eequations (1893-1910), development of Kowalevski integrability test (1888). • Q: Fuchs 1884: find the nonlinear equations with the Painlev´eproperty. Among first and second order equations class: F (w0; w; z) = 0, algebraic; w00 = F (z; w; w0) rational in w, w0, analytic in z equations with (PP) were classified. Among them 6 define new functions: the Painlev´etranscendents. 3 00 2 00 3 PI w = 6w + zP II w = 2w + zw + a 1 1 1 d P w00 = (w0)2 − w0 + aw2 + b + cw3 + III w z z w 1 3 b P w00 = (w0)2 + w3 + 4zw2 + 2(z2 − a)w + IV 2w 2 w 1 1 1 (w − 1)2 b w w(w + 1) P w00 = + (w0)2 − w0 + aw + + c + d V 2w w − 1 z z2 w z w − 1 1 1 1 1 1 1 1 P w00 = + + (w0)2 − + + w0 VI 2 w w − 1 w − z z z − 1 w − z w(w − 1)(w − z) bz z − 1 z(z − 1) + a + + c + d z2(z − 1)2 w2 (w − 1)2 (w − z)2 Painlev´etranscendents: in many physical applications (critical behavior in stat mech, in random matrices, reductions of integrable PDEs,...). 4 In another direction: Sophie Kowalevski linked the idea of absence of movable singularities (other than poles) to integrability. She developed a method of identifying integrable cases, and applied it to the equations describing the motion of a rigid body (a top) rotating about a fixed point. Kowalevski determines parameters for which there are no movable branch points (in C). She found 3 cases: 1) Euler, 2) Lagrange, 3) Kowalevski. In the new case Kowalevski expressed solutions using theta-functions of two variables. 1888 5 The method: of finding parameters for absence of movable branch points has been refined (Painlev´e-Kowalevski test), generalized to PDEs, used extensively, and successfully to identify integrable systems. Kruskal, Ablowitz, Clarkson, Deift, Flaschka, Grammaticos, Newell, Ramani, Segur, Tabor, Weiss... 6 Many aspects of the Painlev´e-Kowalevski test remain, however, not completely understood. Kruskal: • Is not clear if all the movable singularities of a given eq. have been determined. • Movable essential sing are usually not looked for. In fact, this is a very difficult question. • Painlev´e property is not invariant under coordinate transformations. 0 1 1=2 2 E.g. u = 2u has gen sol u = (x − C) : Set v = u : • A general connection to existence of first integrals is not rigorously known. 7 Classical problem: For a given DE establish or disprove existence of first integrals (smooth functions constant on the integral curves). Important: exact number of first integrals determines the dimension of the set filled by an integral curve. Extensive research (theoretical and applications). Poincar´e series: identify (or introduce) a small parameter in equation, assume first integral analytic, and expand in power series. Ziglin, extended and generalized by Ramis, Morales: expand eq. around special solutions. Study first approximation using differential Galois theory: if it does not have a meromorphic integral, then original DE does not have a meromorphic integral. However: this is a restrictive assumption, generic integrals are not meromorphic. 8 Kruskal- The Poly-Painlev´eTest Q: When does a diff. eq. have single-valued first integrals? Will illustrate the concepts on two equations which can be explicitly solved. 0 a Ex 1: u (x) = x−b has gen sol u(x) = a ln(x − b) + C A single-valued first integral F (x; u) = Φ(C) satisfies Φ(C) = Φ(u−a ln(x−b)) = Φ(u−a ln(x−b)−2nπia) = Φ(C−2nπia) for all n 2 Z We found the monodromy: how constants change upon analytic continuation along closed loops. exp(u=a) Hence we may take Φ(C) = exp(C=a) , so F (x; u) = x−b 9 Ex 2: u0(x) = a1 + a2 + a3 gen. sol. x−b1 x−b2 x−b3 u = a1 ln(x − b1) + a2 ln(x − b2) + a3 ln(x − b3) + C A single-valued first integral F (x; u) = Φ(C) should satisfy Φ(C) = Φ(C − 2n1πia1 − 2n2πia2 − 2n3πia3) ; n1;2;3 2 Z so Φ(C) = Φ(C +s) for all s 2 S = f2n1πia1 + 2n2πia2 + 2n3πia3 ; n1;2;3 2 Zg For generic (a1; a2; a3) the set S is dense in C. Hence Φ ≡const: no single-valued first integrals. Obstructions to integrability are encoded in the monodromy | branching properties of solutions . Dense branching no single-valued first integrals. 10 How can one find branching properties? Study equations in singular regions. Results: non-integrability analysis, and criteria for • systems with one regular singular point, • with several reg sing points, also: obstructions to linearization, classification; • for polynomial systems; new monodromy groups. •Irregular singular point: general structure of movable singularities solutions develop. 11 Region with one regular singular point xu0 = Mu + h(x; u) n 0 00 u; h 2 C , x 2 C, M 2 Mn;n(C), h analytic on D = f(x; u); juj < r ; r < jxj < r000g, having a zero of order 2 at u = 0. Theorem (RDC, Nonlin.97) For generic matrices M a complete description of the number of single-valued independent integrals on D is given. Generically integrals exist on D, and are not meromorphic. 12 Illustrate approach on 1-d: xu0 − µu = h(x; u). Set u = U. Eq. becomes xU 0 − µU = U 2h~(x; U) 2 Solutions U(x) = U0(x) + U1(x) + U2(x) + ::: 0 µ Then xU0 − µU0 = 0 so gen. sol. U0(x) = Cx It is relatively easy to see if this first approximation is integrable. (i) µ 2 C n R is the generic case. Then F0(x; U0) = P(ln U0 − µ ln x) is single-valued if P is doubly periodic, with periods 2πi, 2πiµ. Note: First integral is not meromorphic (singularities accumulate at U0 = 0). There is no meromorphic integral. 13 q −p p (ii) If µ 2 Q then F0(x; U0) = U0 x for µ = q . (iii) No single-valued integrals if µ 2 R n Q: µ µ 2nπiµ 2nπiµ Φ(C) = F0(x; Cx ) = F0(x; Cx e ) = Φ Ce 8n 2 Z. The set fe2nπiµ ; n 2 Zg is dense in S1; hence F (x; u) ≡const. No single-valued integrals. If µ 2 R n Q no single-valued first integrals. Conjecture: It is natural to expect that the original, nonlinear equation xu0 − µu = h(x; u) is nonintegrable as well. Q: How to prove? 14 A1: If one addresses the question of existence of regular integrals, they can be easily ruled out. Indeed, if there was one, then 2 F (x; u) = F (x; U) = F (x; U0 + U1 + :::) = F0(x) + F1(x)U0 + ::: j Then F0=const and the first j with Fj 6= 0 would give a first integral Fj(x)U0 for the reduced equation. A similar argument works for F meromorphic. However: in our case we can not expect meromorphic first integrals (we saw that first approximation had, generically, nonmeromorphic integrals). 15 A2 : Show that the original, nonlinear equation is analytically equivalent to its linear part. Normal Form of diff.eq. in an annulus around a regular singular point (Nonlinearity, 1997) Theorem Let xu0 = Mu + h(x; u) n 0 00 000 u; h 2 C , x 2 C, M 2 Mn;n(C), h holomorphic for juj < r , r < jxj < r , having a zero of order 2 at u = 0. Assume eigenvalues µ1; :::; µn of M satisfy the Diophantine condition: n n X X −ν n j kjµj +l−µsj > C( kj +jlj) for all s 2 f1; :::; ng; l 2 Z; k 2 N ; jkj ≥ 2 j=1 j=1 Then the system is biholomorphically equivalent to its linear part in D0 ⊂ D. 16 The proof has some similarity with the case of periodic coefficients. Steps: 1) Find equivalence map as a formal power series. 2) Showing convergence. There is a small denominator problem, overcome by a generalization of Newton's method. Note: The set of all Diophantine M has full measure. Consequences: For such matrices, study of nonlinear case reduces to the study of the linear equation (done in the paper). Generically: equations do have first integrals in D, and they are not meromorphic. Q: What about wider regions, containing several regular singular points? 17 Several regular singular points 2 0 1 1 u f~(x;u) u = Au + f(x; u) A = xµ1 + x−1µ2, f = x(x−1) ; x 2 D ⊃ f0; 1g ; juj < r Valid in more dimensions (at least for special classes of matrices µ1;2). Theorem Obstructions to analytic linearization: (RDC, preprint) Assume <µ1;2 > 0.