Padé approximations of the Painlevé transcendents. Victor Novokshenov Institute of Mathematics Ufa Scientific Center Russian Academy of Sciences

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1 Padé approximations of the Painlevé transcendents Victor Novokshenov Institute of Mathematics Ufa Scientific Center Russian Academy of Sciences

2 Outline Motivation: focusing NLS equation and PI Poles of PI tritronqueé solution Fast algorithm for Padé Tronqueé vs. tritronqueé Exponentially vanishing terms Dubrovin s conjecture Padé approximations for PII

3 Focusing NLS equation A model for incipient modulation instability with initial profile t xx, ( x,) A( x)exp is( x) /. x x Take u, v, i ut ( uv) x, uxx u x vt vvx ux 4 u u x Elliptic umbilic catastrophe [] : A( x) A sech x, S( x) tanh x H uv u ux, 8u u ( H ), v ( H ). t v x t u x [] B.Dubrovin, T.Grava and C.Klein, arxiv:74.v, (7).

4 Universality of critical behavior: PI special solution / 4/ ( x, t, ) ( X, T ) ( z), z z( X, T ) /, where X,T slow space-time, zz 6 z - Painlevé I equation tritronquée solution ( z) Pierre Boutroux 88-9 All solutions are meromorphic The poles accumulate along the rays i arg z n, n,,. Tronquée solution: no poles in two consecutive sectors (say, along z ) Trironquée solution: no poles along three consecutive rays [] Asymptotics at infinity: z z O z z 6 ( ) ( ),. [] 'P.Boutroux,. Ann.Ecole Norm., 3 (93) 6-37.

5 PI and isomonodromic deformations z z ( ) z, z 4 4 ( z) 3 k ( z) e ( z) z 3 ( z) ( z) S k k k k k S The Lax pair normalization k s S k k k sk 3 z 3 z ( z) lim 4 ( ) e ( ) Inversion formula Tritronquée solution: s s s3 s4 i, s

6 Padé approximations for PI solutions Q: How to find poles of PI function? A: For large by Isomonodromy Method, else numerically. z Main diagonal Padé approximation: ( z) z z The idea of algorithm: an invariance of Painlevé equations under Möbius transformation. 3 A Bu u C Du u B u E Fu Gu Hu, zz z z where A H are polynomials of z vanishing at zero, and retains the form of equation. u u u

7 Let u n n, then zu n The Fair-Luke algorithm [3] 3 A B u u C D u u B u E F u G u H u, n n n n n n n n n n n n n n n n n and the recurrence holds and If An nbn An An nbn, Bn zan, Cn Cn ndn, z C D E D A zc, F 3 F G, n n n n n n n n n n n z n E z E F G H, H z A, n n n n n n n n n n n G z A C z E F n n n 3 n n n, n un E F n n, then (). () u( z) u ( z) Pn ( z) z Qn ( z) z [3] W.Fair, Y.Luke, Math.Comp. (968), p.6-6

8 Padé approximation for PI zz 6 z, (), () Put then ( z) z z u( z), z u zu z z u z u z z Pn ( z) u( z) z Qn ( z) n-th order Pade fraction generic tronquée tritronquée

9 Exponentially vanishing terms Why the precision of, is crucial? z ( z) 6 48z a / 4 4/ exp z /8 O( z ) z z 8 3 a arg z ( s s ), z 7 /8 4 a a - tronquée, a - tritronquée Exponential terms are hard to control numerically. Initial conditions while a is changing., are almost the same But not the distribution of the poles! a

10 Elliptic function asymptotics at infinity i / 4 ( z) ( e z g( ) g3( )) z where is Weierstrass function with modules g ( ) i o e g ( ) A( ) a ( ) b ( ) ln( isk ) i b ( ) ln( is k ) O ( z ) i 4 6 arg z [4]. - poles of elliptic function - Padé approximation The Boutroux problem 3 i w ( ) e A( ) 4 ( )( )( ) 3 3 i 3 3 e Re w( ) d a, b 4 3 as 6 Find deformation of the elliptic curve such that 6 6 i / 3 e as [4] A.A.Kapaev, J. Phys.A:Math.Gen.37 (4)49 67

11 Dubrovin s conjecture zz 6 z, z ( z) z 6 48z ( z) is tritronquée solution 4 arg z There are no poles of at infinity as Proof: Kapaev s theorem, [4] Conjecture. [] A way to prove: Let ( z) Reduce the Lax pair equation to a scalar ODE as ( ), Y V z Y / 4 4 / / Y s exp i z,, 4 arg z has no poles in the sector z a pole, put 4 arg z z ( z), z z ( ) / 4 4 / / Y s4 exp i z,. Prove that spectrum z 4 does not contain arg z

12 Painlevé II equation. Tracy-Widom solution [] u zu u zz 3 u / 4 3/ ( z ) exp,, z 3 z z z 73 u( z), z z 8z [] C.Tracy and H.Widom, Comm.Math.Phys. 77 (996)

13 Painlevé II equation. Ablowitz-Segur solution [6] u zu u zz a u z z z z 3 / 4 3/ ( ) exp,, 3 u z b z z b z z 3 4 / 4 3/ ( ) ( ) sin ( ) log( ),. 3 [6] M.J.Ablowitz and H.Segur, Stud.Appl.Math.7 (977) 3 44.

14 Conclusion Padé approximations has been found for PI and PII solutions The Boutroux lines of poles are visualized Distribution of poles at infinity has a good correlation with that of z Dubrovin s conjecture justified Applications to focusing NLS equation