Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos,...
Transcript of Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos,...
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