Department of Mathematics\

Singularity Analysis for Ultra-discrete Equations and

Cellular AutomataN. Joshi

S. L.

Department of MathematicsCollege of Charleston

Singularity Analysis: Continuous

Painlevé property for ODEs: no movablebranching points

Painlevé test: necessary condition for theproperty

!

y' '= f (x,y,y ')" y = ai(x # x0)i

i=#p

$

%

Singularity Analysis: Continuous

Practicality: good integrability detector.

Has been successfully extended to PDEs

Several formal results that relates singularity analysis to constant of motion, Hamiltonian structure, Lax pair.

Singularity Confinement

Tool to study singularities of differenceequations

Heuristic connections with: first integrals, LaxPairs, integrable continuous limit

Formal connection with first integral (S.L., Goriely,JMP, 2004)

Formal connection with Backlundtransformation (Cresswell, Joshi, LMP, 2002)

Singularity Confinement

Ref: Ramani, Grammaticos, Hietarinta, Papageorgiou, PRL, 1991

Singularity Confinement

!

xn

= 0

!

xn

= "Singularity if

Actually:

!

xn

= "1/k + #

!

xn+1xn"1 = k +

1

xn

Singularity Confinement

!

xn+1xn"1 = k +

1

xn

Singularity Confinement

!

xn+1xn"1 = #

n+1

xn

Singularity Confinement

Condition

Painleve I equation

Ultra-discrete limit: KdV

KdV

!

cij

= edij

/", # = e

$ /"

!

lim"#0+

" ln eA /" + eB /"( ) =max(A,B)

Ref: Tokihiro, Takahashi, Matsukidaira, Satsuma, PRL, 1996

Ultra-discrete limit: KdV

KdV

Additions become “max” Multiplications become sums

!

d j

t+1" d j

t=max(# + d j"1

t,0) "max(# + d j"1

t+1,0)

Ref: Tokihiro, Takahashi, Matsukidaira, Satsuma, PRL, 1996

Ultra-discrete limit: P eqs

d-P I

!

Xn+1 + X

n+ X

n"1 =max(A + n + Xn,0)

Ref: Ramani, Takahashi, Grammaticos, Ohta, Phys. D, 1998

!

"n

=#$n

!

xn+1xn xn"1 =#$n x

n+1

!

xn

= eXn

/", # = e

A /"

Singularity Analysis: Ultra-D.

First Example: start with a QRT mapping

!

I = xn

+ xn"1 + k

1

xn

+1

xn"1

#

$ %

&

' ( +

1

xnxn"1!

xn+1xn"1 =

1

xn

+ k

Ref: Quispel, Roberts, Thompson, PLA ‘88

Singularity Analysis: Ultra-D.

Ultra-discretize it

!

xn

= eXn/", k = e

K /"

!

xn+1xn"1 =

1

xn

+ k

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ K,0)

Singularity Analysis: Ultra-D.

Ultra-discretize it

!

xn

= eXn/", k = e

K /"

!

I =max(Xn,X

n"1,"Xn+ K,"X

n"1 + K,"Xn"1 " Xn

)!

I = xn

+ xn"1 + k

1

xn

+1

xn"1

#

$ %

&

' ( +

1

xnxn"1

Singularity Analysis: Ultra-D.

Singularity?

Mapping not differentiable at X=-K!

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ K,0)

Singularity Analysis: Ultra-D.

Confinement?

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ K,0)

Singularity Analysis: Ultra-D.

Deautonomization!

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

Singularity Analysis: Ultra-D.

Confinement condition

Ultra-discrete PI equation and its asymmetric version

!

"n+5 #"n+3 #"n+2 + "

n= 0

!

"n

=# + $n + %(&1)n

Singularity Analysis: Ultra-D.

Partial difference case

!

" =1 corresponds to KdV

Singularity Analysis: Ultra-D.

Singularity Pattern

Confinement for σ=1

Singularity Analysis: Ultra-D.

Complete classification of Painleve equations

1+1-D KdV and sine-Gordon

Cellular automata associated with KdV

Bilinearisation P equations andsingularity analysis

Continuous case: Hietarinta and Kruskal, ‘92

Discrete case: Ramani, Grammaticos, and Satsuma, ‘95

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

Introduce an entire function in such a way that the singularityPattern is reproduced

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

Introduce an entire function in such a way that the singularityPattern is reproduced

!

" # function "n

It is entire but goes through zeroes

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

!

xn

= "nD

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

!

xn

="n

"n#1

D2

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

!

xn

="n

"n#1" n#2

D3

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

!

xn

="n"n#3

"n#1" n#2

Bilinearisation of d-PI

Ref:Ramani, Grammaticos, Satsuma, JPA ’95

!

xn+1xn"1 =#$n +

1

xn

!

0,",",0}{

!

xn

="n"n#3

"n#1" n#2

!

"n+1" n#4 =$%n"

n"n#3 + "

n#1" n#2

Bilinearisation of U-PI

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

!

"n

=#n + $

Bilinearisation of U-PI

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

!

"n

=#n + $

!

Xn

= "n( )

++ D

Bilinearisation of U-PI

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

!

"n

=#n + $

!

Xn

= "n( )

+# "

n#1( )+

+ D2

Bilinearisation of U-PI

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

!

"n

=#n + $

!

Xn

= "n( )

+# "

n#1( )+# "

n#2( )+

+ D3

Bilinearisation of U-PI

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

!

"n

=#n + $

!

Xn

= "n( )

+# "

n#1( )+# "

n#2( )+

+ "n#3( )

+

Bilinearisation of U-PI

!

Xn+1 + X

n+ X

n"1 =max(Xn

+ #n,0)

!

"n

=#n + $

!

Xn

= "n( )

+# "

n#1( )+# "

n#2( )+

+ "n#3( )

+

!

"n#4( )

++ "

n+1( )+

= max $n

+ "n( )

++ "

n#3( )+, "

n#1( )+

+ "n#2( )

+( )

Conclusions

Integrability detector for U-D equations basedon singularity analysis

What about Cellular Automata?

Classification of Painleve equations

Bilinearisation of Painleve equations