Geometric Asymptotics

Transcription

1 Geometric Asymptotics Nalini Odlyzko 011 Supported by the Australian Research Council

2 Motion Gridlock in India

3 Motion follows curves In initial value space But continuation of the flow fails at singularities of curves and at base points where families of curves all intersect.

4 An example y = x 3 The curve has a singularity at 0,0), which can be resolved.

5 Resolution 1 x 1 y 1 = x = y x, x = x 1 y = x 1 y 1

6 Resolution 1 x 1 y 1 = x = y x, x = x 1 y = x 1 y 1

7 Resolution 1 x 1 y 1 = x = y x, x = x 1 y = x 1 y 1 f = y x 3 = x 1 y 1 = x 1 y 1 x 3 1 x 1

8 Resolution 1 x 1 y 1 = x = y x, x = x 1 y = x 1 y 1 f = y x 3 = x 1 y 1 = x 1 y 1 x 3 1 x 1 e 0 = {x 1 =0}, f 1) = y 1 x 1

9 Resolution 1 e 0 x 1 y 1 = x = y x, x = x 1 y = x 1 y 1 C 1) f = y x 3 = x 1 y 1 = x 1 y 1 x 3 1 x 1 e 0 = {x 1 =0}, f 1) = y 1 x 1

10 Resolution x = x 1 y 1 y = y 1, x 1 = x y y 1 = y

11 Resolution x = x 1 y 1 y = y 1, x 1 = x y y 1 = y

12 Resolution x = x 1 y 1 y = y 1, x 1 = x y y 1 = y f 1) x 1,y 1 )=y x y = y y x

13 Resolution x = x 1 y 1 y = y 1, x 1 = x y y 1 = y f 1) x 1,y 1 )=y x y = y y x e 1 = {y =0}, f ) = y x

14 Resolution e 1) 0 C ) x = x 1 y 1 y = y 1, x 1 = x y y 1 = y e 1 f 1) x 1,y 1 )=y x y = y y x e 1 = {y =0}, f ) = y x

15 Resolution 3 x 3 = x y 3 = y, x x = x 3 y = x 3 y 3

16 Resolution 3 x 3 = x y 3 = y, x x = x 3 y = x 3 y 3

17 Resolution 3 x 3 = x y 3 = y, x x = x 3 y = x 3 y 3 f ) x,y )=x 3 y 3 x 3 = x 3 y 3 1

18 Resolution 3 x 3 = x y 3 = y, x x = x 3 y = x 3 y 3 f ) x,y )=x 3 y 3 x 3 = x 3 y 3 1 e = {x 3 =0}, f 3) = y 3 1

19 Resolution 3 x 3 = x y 3 = y, x x = x 3 y = x 3 y 3 e ) 0 e C 3) f ) x,y )=x 3 y 3 x 3 = x 3 y 3 1 e 1) 1 e = {x 3 =0}, f 3) = y 3 1

20 Intersection theory - e ) 0 e 1) 1 e - -1 C 3) Each line has a selfintersection number. Exceptional lines have self-intersection -1. Each blow up reduces the self-intersection number by 1. The lines of selfintersection - play a special role.

21 Intersection theory - e ) 0 e 1) 1 e - -1 C 3) Each line has a selfintersection number. Exceptional lines have self-intersection -1. Each blow up reduces the self-intersection number by 1. The lines of selfintersection - play a special role.

22 DuVal Correspondence The curves with selfintersection - correspond to nodes of a Dynkin diagram. This is DuVal or Mackay) correspondence. e ) 0 e 1) 1 e C 3)

23 DuVal Correspondence The curves with selfintersection - correspond to nodes of a Dynkin diagram. This is DuVal or Mackay) correspondence. e ) 0 e 1) 1 e C 3)

24 DuVal Correspondence The curves with selfintersection - correspond to nodes of a Dynkin diagram. e ) 0 This is DuVal or Mackay) correspondence. + e 1) 1 e C 3) A

25 Three Painlevé Equations P I : w tt =6w t P II : w tt =w 3 + tw + P IV : w tt = w t w + 3 w3 +4tw + t + )w + w What are the behaviours of solutions the limit t!1? Duistermaat & J 011); Howes & J 014); J & Radnovic 015, 016, 017)

26 Properties Solutions possess movable poles General solutions are highly transcendental functions. Asymptotic behaviours needed for applications. Fornberg & Weideman 009 PI

27 System form PI: w tt =6w in system form d dt w1 w = t w 6 w1 t has t-dependent Hamiltonian H = w w tw 1

28 Perturbed Form In Boutroux s coordinates: w 1 = t 1/ u 1 z), w = t 3/4 u z), z = 4 5 t5/4 u1 u 1 u1 = u 6u 1 1 5z 3u a perturbation of an elliptic curve as z!1 E = u u u 1 ) de dz = 1 5z 6E +4u 1)

29 Perturbed Form In Boutroux s coordinates: w 1 = t 1/ u 1 z), w = t 3/4 u z), z = 4 5 t5/4 u1 u 1 u1 = u 6u 1 1 5z 3u a perturbation of an elliptic curve as z!1 E = u u u 1 ) de dz = 1 5z 6E +4u 1)

30 Pencil of cubic curves Analogous to Weierstrass cubic curves y =4x 3 g x g 3 where g = 1/ and g 3 is free. In homogeneous coordinates in CP the curves are wv =4u 3 g uw g 3 w 3 Base point: 0, 1, 0) at infinity.

31 Charts at infinity u 031,u 03 ) CP u 011,u 01 ) u 01,u 0 ) [u 1 : u : 1] = [1 : u 1 1 u : u 1 1 ]=[1:u 0 : u 01 ] [u 1 : u : 1] = [u 1 u 1 :1:u 1 ]=[u 031 :1:u 03 ]

32 Resolution of PI There are nine base points: b 0 : u 031 =0,u 03 =0 b 1 : u 111 =0,u 11 =0 b : u 11 =0,u 1 =0 b 3 : u 311 =4,u 31 =0 b 4 : u 411 =4,u 41 =0 b 5 : u 511 =0,u 51 =0 b 6 : u 611 =0,u 61 =0 b 7 : u 711 = 3,u 71 =0 b 8 : u 811 = 8 5 z),u 81 =0 Only the last one differs from the elliptic case.

33 PI L9 L8 1) L7 ) L6 3) L5 4) L4 5) L3 6) L0 9) L1 9) L 8) Duistermaat & Joshi, 011

34 PI L9 L8 1) L7 ) L6 3) L5 4) L4 5) L3 6) L0 9) L1 9) L 8) Duistermaat & Joshi, 011

35 PI L9 E8 1) L8 1) L7 ) L6 3) L5 4) L4 5) L3 6) L0 9) L1 9) L 8) Duistermaat & Joshi, 011

36 PI L9 autonomous case E8 1) L8 1) L7 ) L6 3) L5 4) L4 5) L3 6) L0 9) L1 9) L 8) Duistermaat & Joshi, 011

37 PII Howes & Joshi, 014

38 PII Howes & Joshi, 014

39 PII E7 1) Howes & Joshi, 014

40 PII E7 1) autonomous case Howes & Joshi, 014

41 PIV L 0 L 4 L 5 L 6 L 7 z) L 1 L L L 8 z) L 9 z) 3 Joshi & Radnovic, 015

42 PIV L 0 L 4 L 5 L 6 L 7 z) L 1 L L L 8 z) L 9 z) 3 Joshi & Radnovic, 015

43 PIV E6 1) L 0 L 4 L 5 L 6 L 7 z) L 1 L L L 8 z) L 9 z) 3 Joshi & Radnovic, 015

44 PIV E6 1) L 0 L 4 L 5 L 6 L 7 z) L 1 L L L 8 z) L 9 z) 3 autonomous case Joshi & Radnovic, 015

45 Initial-Value Space For each t, let S9t) be the resolved space. The union S = [ tc S 9 t) is Okamoto s space. The union It) = 8[ i=0 L 9 i) i t) is the infinity set. For each solution wt) in S9t)\It), define the limit set w ={s S 9 1)\I1)s.t. 9 t j!1, wt j )! s, as j!1}

46 Global results for PI, PII, PIV The infinity set is a repeller for the flow. The complex limit set is non-empty, connected and compact. Every solution of PI, every solution of PII whose limit set is not {0}, and every non-rational solution of PIV intersects the last exceptional lines) infinitely many times infinite number of movable poles and movable zeroes. Duistermaat & J 011); Howes & J 014); J & Radnovic 015, 016, 017)

47 Ingredients of proofs Use the energy and Jacobians of each chart as a measure of the distance between the flow and the exceptional lines. Estimate the domain corresponding to each chart in which the solution is analytic. Use compactness to deduce results about the limit set. E = u u u 1 J ij ij ij1 1

48 Summary Global dynamics of solutions of nonlinear equations, whether they are differential or discrete, can be found through geometry. Geometry provides the only analytic approach available in C for discrete equations. Tantalising questions about finite properties of solutions remain open.