singularity analysis for ultra- discrete equations and cellular … · ref:ramani, grammaticos,...
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![Page 1: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/1.jpg)
Department of Mathematics\
Singularity Analysis for Ultra-discrete Equations and
Cellular AutomataN. Joshi
S. L.
Department of MathematicsCollege of Charleston
![Page 2: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/2.jpg)
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
$
%
![Page 3: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/3.jpg)
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.
![Page 4: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/4.jpg)
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
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Singularity Confinement
!
xn
= 0
!
xn
= "Singularity if
Actually:
!
xn
= "1/k + #
!
xn+1xn"1 = k +
1
xn
![Page 6: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/6.jpg)
Singularity Confinement
!
xn+1xn"1 = k +
1
xn
![Page 7: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/7.jpg)
Singularity Confinement
!
xn+1xn"1 = #
n+1
xn
![Page 8: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/8.jpg)
Singularity Confinement
Condition
Painleve I equation
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Ultra-discrete limit: KdV
KdV
!
cij
= edij
/", # = e
$ /"
!
lim"#0+
" ln eA /" + eB /"( ) =max(A,B)
Ref: Tokihiro, Takahashi, Matsukidaira, Satsuma, PRL, 1996
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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
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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 /"
![Page 12: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/12.jpg)
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
![Page 13: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/13.jpg)
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)
![Page 14: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/14.jpg)
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
![Page 15: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/15.jpg)
Singularity Analysis: Ultra-D.
Singularity?
Mapping not differentiable at X=-K!
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ K,0)
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Singularity Analysis: Ultra-D.
Confinement?
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ K,0)
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Singularity Analysis: Ultra-D.
Deautonomization!
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ #n,0)
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Singularity Analysis: Ultra-D.
Confinement condition
Ultra-discrete PI equation and its asymmetric version
!
"n+5 #"n+3 #"n+2 + "
n= 0
!
"n
=# + $n + %(&1)n
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Singularity Analysis: Ultra-D.
Partial difference case
!
" =1 corresponds to KdV
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Singularity Analysis: Ultra-D.
Singularity Pattern
Confinement for σ=1
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Singularity Analysis: Ultra-D.
Complete classification of Painleve equations
1+1-D KdV and sine-Gordon
Cellular automata associated with KdV
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Bilinearisation P equations andsingularity analysis
Continuous case: Hietarinta and Kruskal, ‘92
Discrete case: Ramani, Grammaticos, and Satsuma, ‘95
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Bilinearisation of d-PI
Ref:Ramani, Grammaticos, Satsuma, JPA ’95
!
xn+1xn"1 =#$n +
1
xn
!
0,",",0}{
Introduce an entire function in such a way that the singularityPattern is reproduced
![Page 24: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/24.jpg)
Bilinearisation of d-PI
Ref:Ramani, Grammaticos, Satsuma, JPA ’95
!
xn+1xn"1 =#$n +
1
xn
!
0,",",0}{
Introduce an entire function in such a way that the singularityPattern is reproduced
!
" # function "n
It is entire but goes through zeroes
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Bilinearisation of d-PI
Ref:Ramani, Grammaticos, Satsuma, JPA ’95
!
xn+1xn"1 =#$n +
1
xn
!
0,",",0}{
!
xn
= "nD
![Page 26: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/26.jpg)
Bilinearisation of d-PI
Ref:Ramani, Grammaticos, Satsuma, JPA ’95
!
xn+1xn"1 =#$n +
1
xn
!
0,",",0}{
!
xn
="n
"n#1
D2
![Page 27: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/27.jpg)
Bilinearisation of d-PI
Ref:Ramani, Grammaticos, Satsuma, JPA ’95
!
xn+1xn"1 =#$n +
1
xn
!
0,",",0}{
!
xn
="n
"n#1" n#2
D3
![Page 28: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/28.jpg)
Bilinearisation of d-PI
Ref:Ramani, Grammaticos, Satsuma, JPA ’95
!
xn+1xn"1 =#$n +
1
xn
!
0,",",0}{
!
xn
="n"n#3
"n#1" n#2
![Page 29: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/29.jpg)
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
![Page 30: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/30.jpg)
Bilinearisation of U-PI
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ #n,0)
!
"n
=#n + $
![Page 31: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/31.jpg)
Bilinearisation of U-PI
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ #n,0)
!
"n
=#n + $
!
Xn
= "n( )
++ D
![Page 32: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/32.jpg)
Bilinearisation of U-PI
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ #n,0)
!
"n
=#n + $
!
Xn
= "n( )
+# "
n#1( )+
+ D2
![Page 33: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/33.jpg)
Bilinearisation of U-PI
!
Xn+1 + X
n+ X
n"1 =max(Xn
+ #n,0)
!
"n
=#n + $
!
Xn
= "n( )
+# "
n#1( )+# "
n#2( )+
+ D3
![Page 34: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/34.jpg)
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( )
+
![Page 35: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/35.jpg)
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 $n
+ "n( )
++ "
n#3( )+, "
n#1( )+
+ "n#2( )
+( )
![Page 36: Singularity Analysis for Ultra- discrete Equations and Cellular … · Ref:Ramani, Grammaticos, Satsuma, JPA ’95! x n+1 x n"1 =#$ n+ 1 x n! 0,",",0} {Introduce an entire function](https://reader034.vdocuments.us/reader034/viewer/2022050409/5f85c35a586cbf420071ccd5/html5/thumbnails/36.jpg)
Conclusions
Integrability detector for U-D equations basedon singularity analysis
What about Cellular Automata?
Classification of Painleve equations
Bilinearisation of Painleve equations