Padé approximations for Painlevé I and II transcendents

Padé approximations for
Painlevé I and II transcendents
Victor Novokshenov
Institute of Mathematics
Ufa Scientific Center
Russian Academy of Sciences
The Fifth International Workshop
«SOLITONS, COLLAPSES AND TURBULENCE:
Achievements, Developments and Perspectives»
CHERNOGOLOVKA, Moscow region, RUSSIA
August 2-7, 2009

Outline
• Motivation: focusing NLS equation and PI
• Poles of PI tritronqueé solution
• Fast algorithm for Padé approximation
• Tronqueé vs. tritronqueé
• Exponentially vanishing terms
• Dubrovin’s conjecture
• Padé approximations for PII
V.Novokshenov, TMPh, 159(3): 853-862 (2009).

Focusing NLS equation
A model for incipient modulation instability
with initial profile
2
2
it
xx
| | 0, 1
2
2
( x,0) A( x)exp iS( x) / .
x
x
Take u | | , v ,
2i
ut
( uv) x
0,
2
2
uxx
u
x
vt vvx ux
2
4 u 2u
x
Elliptic umbilic catastrophe [1] :
A( x) A sech x, S( x) tanh x
0
2
1 2 2 2
H uv u ux
,
2 8u
u ( H ) 0, v ( H ) 0.
t v x t u x
[1]
B.Dubrovin, T.Grava and C.Klein, J.Nonlin.Sci.19 (2009) 57-94

Universality of critical behavior: PI special solution
2/5
4/5
( x, t, ) ( X , T ) ( z),
z z( X , T ) / , where X,T – slow space-time,
2
zz
6 z - Painlevé I equation
tritronquée solution ( z)
Pierre
Boutroux
1880 - 1922
• All solutions are meromorphic
• The poles accumulate along the rays
2
i
arg z n, n 0, 1, 2.
5
• Tronquée solution: no poles in two
consecutive sectors (say, along z )
• Trironquée solution: no poles along three
consecutive rays [2]
• Asymptotics at infinity:
z
z O z z
6
2
( ) 1 ( ) , .
[2]
'P.Boutroux,. Ann.Ecole Norm., 30 (1913) 265 - 375.

PI and isomonodromic deformations
2 2
z
2
2
z 2
2( )
z
,
0 2
z
1 0
14 1 4
1
( z)
3
k
( z)
e
2
14 1
4
4
1 0
5
0 1
52 12
( z)
z
3
( z) ( z)
S
k1 k k k k 1
S
The Lax pair
normalization
k
1
s 1 0
S
2k1
2k1 2k
0 1 s2k
1
1 1 1
13 z
3
z
2 1 1
12
( z) lim 4 ( ) e
( )
1
11
Inversion formula
Tritronquée solution: s1 s2 s3 s4 i, s5
0

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)
0
1z
1
z
1 1
2
The idea of algorithm: an invariance of Painlevé equations under Möbius
transformation.
2 2 3
A Bu u C Du u B u
E Fu Gu Hu
2 0,
zz z z
where A…H are polynomials of z vanishing at zero, and
retains the form of equation.
u
u
u

Let
u
n
n
, then
1 zu
n1
The Fair-Luke algorithm [3]
2 2 3
A B u u C D u u B u
E F u G u H u
2 0,
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 1 An nBn , Bn 1
zAn , Cn
1
2 Cn nDn
,
z
C
D E
D 2 A zC , F 3 2 F
G ,
n n n n
n1 n n n1
n n n
z n
E z E F G H , H z A ,
1 1 2 1 2
n1 n n n n n n n n1
n n
G z A C z E F
1 1
n1
2
n n
3
n n n
,
n1
un
1
0
E
F
n1
n
1
, then
(0) .
(0)
0
Pn
( z)
u( z) u0( z)
1z 1
Qn
( z)
2z
1 1
[3]
W.Fair, Y.Luke, Math.Comp. 20 (1968), p.602-605

Padé approximation for PI
zz
2
6 z,
(0) ,
0
(0)
1
u( z)
Put
2
0 1
2 2 3 4 2
0 1 0 1
0
Pn
( z)
1z
Qn
( z)
n-th order Pade fraction
1 1
1
( z) z z u( z),
then
2
z u 4zu 2 1 6 z 6 z u 6z u z 6 z 0
generic tronquée tritronquée
0.1
0
0.3
1
0.16
0
0.2193934573994778462
1
0.18755430834118852651722253248835
0
0.30490556026020212024373994560756
1

Why the precision of
Exponentially vanishing terms
,
0 1
is crucial?
z 1
( z)
2
6 48z
a
8 3
4
5/ 4
4/5
exp z
5/8 1 O( z ) | z |
z
5 2
18
3
a arg z
( s s ), z
7 /8
2
1 4 1 5
( z)
( z)
a 0
z
6
z
6
a 0 - tronquée, a 0
- tritronquée
Exponential terms are hard to control numerically.
Initial conditions
while a is changing.
,
0 1
are almost the same
But not the distribution of the poles!
a 0

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Evolution of poles towards tritronqueé solution
2
uzz
6u z

Elliptic function asymptotics at infinity
i 5/ 4
( z) ( e z g2( ) g3( )) | z |
where is Weierstrass function with modules
g ( ) 2 i
2
o
e
g
( ) A( ) 4
3 1 2 3
a
( ) b
( )
ln( is22k
)
2
i
b
( ) ln(
1
is5 2k
) O (
z )
2
i
4
6
arg z [4]
5 5
. - poles of elliptic function
- Padé approximation
The Boutroux problem
2 3 1 i 1
w ( ) e A( )
2 4
( )( )( )
1 2 3
1 2 3
0
1 i
1 2
23 31
e
2
Re w( ) d
0
a,
b
1 4
1 3
as
6
5
Find deformation of the
elliptic curve
such that
1 6
6
5
i
/5
2 3
e
as
[4]
A.A.Kapaev, J. Phys.A:Math.Gen.37 (2004)11149–11167

Dubrovin’s conjecture
zz
2
6 z,
z 1
( z) | z |
2
6 48z
( z)
is tritronquée
solution
4
arg z
5
There are no poles of at infinity as
Proof: Kapaev’s theorem, [4]
Conjecture. [1]
A way to prove: Let
( z)
Reduce the Lax pair equation
to a scalar ODE as 0
( ) 0,
Y V z Y
0
1/ 4 4
5/ 2 1/ 2
Y s1 exp i z0 , ,
5
4
arg z
5
has no poles in the sector
z 0 2
0 a pole, put
2
4
arg z
5
1 z
( z) , z z0
0
10
3 2 2 4
2 2 2 0
2
2 3
2( ) 2
1/ 4 4 5/ 2 1/ 2
Y s4 | | exp i | | z0
| | , .
5
Prove that spectrum z
4
0 does not contain arg z0
5

Painlevé II equation. Hastings-McLeod solution [5]
u zu 2u
zz
3
u 1 1/ 4 2 3/ 2
( z ) exp , ,
2
z
3
z
z
z 1 73
u( z) 1
, z .
3 6
2 8z
128z
[5] C.Tracy and H.Widom, Comm.Math.Phys. 177 (1996) 727–754..

Painlevé II equation. Ablowitz-Segur solution [6]
u zu 2u
zz
a 2
u z z z
z
2 3
1/ 4 3/ 2
( ) exp , ,
2 3
u z b z z b z z
3 4
1/ 4 3/ 2 2
( ) ( ) sin ( ) log( )
, .
3
[6] M.J.Ablowitz and H.Segur, Stud.Appl.Math.57 (1977) 13–44.

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 | 10
• Dubrovin’s conjecture justified
• Applications to focusing NLS equation
2
1.5
10
1
0.5
5
0
-10
0
-5
0
5
10 -10 -5