J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro and M. Torii- Toda-type Cellular Automaton...

8/3/2019 J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro and M. Torii- Toda-type Cellular Automaton and its N-soliton S

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Toda-type Cellular Automaton and its N-soliton

Solution

J. Matsukidaira a, J. Satsuma b, D. Takahashi a, T. Tokihiro b

and M. Torii b

a Department of Applied Mathematics and Informatics, Ryukoku University, Seta,

Ohtsu 520-21, Japan

b Graduate School of Mathematical Sciences, University of Tokyo, Meguro-ku,

Tokyo 153, Japan

Abstract

In this letter, we show that the cellular automaton proposed by two of the authors(D.T and J.M) is obtained from the discrete Toda lattice equation through a speciallimiting procedure. Also by applying a similar kind of limiting procedure to the

N-soliton solution of the discrete Toda lattice equation, we obtain the N-solitonsolution for this cellular automaton.

Keywords: Soliton; Discrete; Cellular Automaton; Nonlinear; Toda Lattice

The phenomena we observe in nature have been described in many ways.Among several methods to analyze the behavior of nature, differential equa-tions have been traditionally the most powerful and often used. However manysystems in the fields of biology, statistical physics, etc., are difficult to describeusing differential equations. These systems are rather easy to deal with usingdiscrete methods, such as discrete equations, coupled map lattices and cellularautomata(CAs)[0].

Due to the enormous growth of computer power in recent years, we have beenable to analyze these discrete systems even though they have large degreesof freedom and strong nonlinearity. Among them, CAs are most suited forcomputer simulation because all of the variables are discrete, including fieldvariables, and round off error does not occur. Therefore CAs are extensivelystudied and various statistical results are obtained, though traditional meth-ods used in differential calculus could not have been applied due to the strongnonlinearity.

On the other hand, in the field of nonlinear physics, soliton theory has suc-ceeded as an analytical tool for nonlinear evolution equations for almost 30

Preprint submitted to Elsevier Science 24 October 1996


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years and has been applied to several fields: hydrodynamics, plasma physics,optical physics and so on. Moreover recent development of soliton theory tellsus it can be also applicable to discrete equations[0].

The notion of soliton for CAs was first introduced by Park et al[0]. After this

work, soliton-like structures have been found in several CAs and attemptsto apply soliton theory to CAs have been made by several groups [0,0,0,0].However direct relation of CA to soliton equations has not been clear.

Recently we proposed a general method to obtain CAs from discrete solitonequations through a limiting procedure[0]. By using this method, we clarifiedthe relation between the CA which was proposed by two of the authors (D.T.and J.S. )[0] and the Korteweg de-Vries equation. This is our answer to oneof the unsolved problems listed in the paper of Wolfram[0].

In this letter, we apply this method to the Toda lattice equation. We showthat the CA, which is proposed in the previous paper[0], is obtained from thediscrete Toda lattice equation through the limiting procedure. Also we obtainN-soliton solutions of this CA from those of the discrete Toda lattice equation.

The starting point is the discrete Toda lattice equation which was introducedby Hirota[0],

log(1 + Vt+1n ) 2 log(1 + Vtn) + log(1 + V

t1n )

= log(1 + 2Vtn+1

) 2 log(1 + 2Vtn

) + log(1 + 2Vtn

1

). (1)

By introducing Vtn = eUtn 1, we obtain

Ut+1n 2Utn + U

t1n

= log(1 + 2(eUtn+1 1)) 2 log(1 + 2(eU

tn 1)) + log(1 + 2(eU

tn1 1)).

(2)

One can easily obtain the continuous Toda lattice equation,

d2rndt2

= ern+1 2ern + ern1, (3)

from Eq.(2) by the relation Utn = rn(t) and taking 0.

The N-soliton solution of Eq.(2) is given by

Utn = 2n log f

tn, (4)

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with

ftn =

i=0,1

exp[Ni=1

ii +(N)ij denotes the summation over all possible pairs chosen from N elements.

Now we introduce a positive parameter defined by = eL2 where L is a

positive integer, and set Utn

= utn

/. Then noticing the fact

lim+0

log(1 + eX ) = max(0, X) =

X if X 0,

0 else,(11)

we obtain from Eq.(2) in the limit +0

ut+1n 2utn + u

t1n

= max(0, u

t

n+1 L) 2max(0, u

t

n L) + max(0, u

t

n1 L), (12)

where the equivalent equation was proposed in the previous paper[0]. Eq.(12)thus obtained is considered to be Toda-type CA, which shares the commonalgebraic properties with Toda lattice equation as shown below.

Let us look at whether the N-soliton solution can survive by this limitingprocedure. The one-soliton solution of Eq.(2) is expressed by

Utn = 2n log(1 + e

Pnt+0), (13)

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where

sinh(/2) = sinh(P/2), (14)

= 1 or 1, (15)

Here we set P = p/, = /, and 0 = 0/, then we obtain

utn = 2n log(1 + e

pnt+0

). (16)

Taking +0, Eqs.(16) and (14), respectively, become

utn = 2n max(0, pn t +

0), (17)

=

0 if n t0

p 1,

|p|(n + 1) t + 0 if t0

p 1 n t

0

p,

|p|(n 1) t + 0 if t0

p n t

0

p+ 1,

0 if n t0

p+ 1,

(18)

and

=

(p L) if p > L,

0 if L p L,

(p L) if p < L,

(19)

= (max(0, p L) max(0, p L)). (20)

This is the one-soliton solution of Eq.(12), which is identical to the one shownin the previous paper if we take p, 0 as integers and L = 1. It is easy to seethe speed and the maximum amplitude of soliton is expressed by /p and |p|from Eq.(18) respectively.

By setting Pi = pi/, i = i/,0i = 0i /, Aij = aij/ and noticing the fact

lim+0

log(Mi=1

eXi ) = max(X1, X2, . . . , XM1, XM), (21)

we also obtain the N-soliton solution in the limit +0

utn = 2n

tn, (22)

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with

tn = maxi=0,1

[Ni=1

ii +(N)i p2 L 1,

1 = p1 L, 2 = p2 L,

01 = 0, 02 = 0,

(27)

From Eq.(27) and Eq.(25), Eq.(26) is written by

tn = max(0, p1n (p1 L)t, p2n (p2 L)t,

(p1 + p2)n (p1 + p2 2L)t (2p2 L)). (28)

Fig.1 show the interaction of solitons where we take p1 = 3, p2 = 2 and L = 1.At the time t = and around the region 1 0, i.e. n

p1L

p1t, we have

2 L(p1 p2)

p1t , (29)

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and therefore the solution is written as

tn = max(0, 1). (30)

This expresses that one of the solitons exists around the region 1 0 att = . Similarly, at the time t = and around the region 1 0, we have

tn = max(0, 1, 2, 1 + 2 + a12),

= 2 + max(0, 1 + a12, 2, 1 2), (31)

max(0, 1 + a12, 2, 1 2), (32)

= max(0, 1 + a12), (33)

where denotes l.h.s and r.h.s give same solution, because the first term of

l.h.s 2 vanishes under the operation of difference operator 2

n. This expressesthat the soliton also exists around the region 1 0 at t = but its positionis shifted due to the term a12. Similarly around the region 2 0, the solutionis expressed by

tn =

max(0, 2 + a12) at t = ,

max(0, 2) at t = ,(34)

and this expresses the other soliton. The value of the phase shift of this solution

in the case ofL = 1 is given as follows. Eq.(33) can be written

tn = max(0, p1n (p1 1)t (2p2 1)), (35)

= max(0, p1(n (2p2 1)) (p1 1)(t (2p2 1))), (36)

and Eq.(34) at t = can be written

tn = max(0, p2n (p2 1)t (2p2 1)), (37)

= max(0, p2(n 1) (p2 1)(t + 1)). (38)

Therefore the solitons shifts its position

(2p2 1, 2p2 1) for the soliton near 1 0,

(1, 1) for the soliton near 2 0,(39)

in n-t plain.

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Similarly in the case 2-soliton solution with

L = 1, p1 > p2 L,

1 = p1 + 1, 2 = p2 + 1,

01 = 0, 02 = 0,

(40)

the phase shift is given by

(2p2 + 1, 2p2 1) for the soliton near 1 0,

(1, 1) for the soliton near 2 0,(41)

(See Fig.2 where we take p1 = 3, p2 = 2) and in the case with

L = 1, p1 L, p2 L,

1 = p1 1, 2 = p2 + 1,

01 = 0, 02 = 0,

(42)

the phase shift is given by

(1, 1) for the soliton near 1 0,

(1, 1) for the soliton near 2 0.

(43)

(See Fig.3 where we take p1 = 3, p2 = 2) These coincident with the resultof previous paper[0]. However in cases other than L = 1, similar estimationcannot always be applied because it can happens that the pattern of solitonchanges after the interaction. See Fig.4 where we take L = 3, p1 = 9, p2 =4, 1 = 6, 2 = 1. Although we showed only two soliton cases, we can also dealwith the interaction of N-soliton. For example, the 4-soliton solution shownby Fig.5 in the previous paper is obtained by setting,

p1 = 7, p2 = 3, p3 = 1, p4 = 2, L = 1,

1 = 6, 2 = 2, 3 = 0, 4 = 1,

01 = 10, 02 = 1,

03 = 0,

04 = 2.

(44)

It can also be shown that the phase shift of the N-soliton solution is given bya summation of that of 2-solitons, which each soliton have had through theinteraction with other solitons.

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Finally, it should be noted that tn satisfies

t+1n + t1n = max(2

tn,

tn+1 +

tn1 L), (45)

which is obtained from Eqs.(12) and (22), and this equation may be consideredan analogue of bilinear identity.

In this paper, we have derived a Toda-type CA from the discrete Toda latticeequation and given a formula for the N-soliton solution. This CA inherits theproperties of the Toda lattice equation including solitary waves and solitoninteractions. We are currently investigating physical properties, such as con-served quantities[0], and physical meaning, in terms of a dynamical system,and will report our results in forthcoming papers. Also, the algebraic structureof this class of CAs is to be studied in detail, and remains an open questionfor the future.

Acknowledgement

The authors would like to thank to Dr. James Keiser for his useful commentsand suggestions on this manuscript. And authors also would like to thank toMs. Meg Rowland for her editorial assistance. J. Matsukidaira acknowledgesthe financial support of Ryukoku University.

Appendix

In the cases i = 1 or j = 1, arguments of cosh in Eq.(10) do not go to 0if we take pi = pj. Therefore when taking +0, cosh terms dominate andaij tend to

aij = log

ij cosh((pi + i pj j)/2)

ij cosh((pi + i + pj + j)/2) , (A.1)

logexp((pi + i pj j)/2) + exp((pi i + pj + j)/2)

exp((pi + i + pj + j)/2) + exp((pi i pj j)/2), (A.2)

max

pi + i pj j2

,pi i + pj + j

2

max

pi + i + pj + j2

,pi i pj j

2

, (A.3)

=max

pi + i pj j2

max

pi + i + pj + j2

,pi i pj j

2

,

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pi i + pj + j2

max

pi + i + pj + j2

,pi i pj j

2

,

(A.4)

= max( max(pj + j, pi i), max(pi + i, pj j)), (A.5)

= max(min(pi + i, pj j), min(pi i, pj + j)). (A.6)

In the case of i = 1 and j = 1, we need careful estimation because thenumerator or the denominator of Eq.(10) goes to 0.

Let us take pi > L and pj > L as an example. From Eq.(8), we have

i2

= arcsinh

exp(L

2) sinh

pi2

. (A.7)

Considering the argument of arcsinh diverges and using the asymptotic ex-

pansion arcsinhx log2x +14x2 for x 0, we have

i2

log

2 exp(L

2)sinh

pi2

1

4 exp(L/) sinh2(pi/2). (A.8)

Therefore

pi + i pj j2

pi2

log(sinhpi2

) 1

4exp(L/) sinh2(pi/2)

pj2

+ log(sinh pj2

) + 14 exp(L/)sinh2(pj/2)

.

(A.9)

Here consider expanding each term in series of exponential function and es-timating an asymptotic behavior. First and second term of r.h.s. of Eq.(A.9)become

pi2

log(sinhpi2

) = logexp(pi/2) exp(pi/2)

2exp(pi/2),

= log1 exp(pi/)

2 log 2 + exp(

pi

) + exp(2pi

) + , (A.10)

and third term becomes

1

4exp(L/)sinh2(pi/2)

1

exp(L/)(exp(pi/2) exp(pi/2))2,

exp(pi L

). (A.11)

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Thus we obtain

pi + i pj j2

exp(pi

) + exp(2pi

) + exp(pi L

)

exp(pj ) exp(2

pj ) + exp(

pj

L

),

exp(pj L

) exp(

pi L

). (A.12)

Substituting this info the cosh of numerator and using Taylor expansion 1

cosh2 x = 2sinh2 x2

2x2

2for x 1,

aij log

2

pi + i pj j4

2 log

cosh

pi + i + pj + j2

1

,

log

exp(pj L

) exp(

pi L

)2

log

coshL

1

,

2max(pj + L, pi + L) L,

= 2min(pi, pj) + L. (A.13)

Similarly considering all other possible case, we obtain

aij = 2 min(|pi|, |pj|) + L. (A.14)

References

[1] See for example, Theory and Applications of Cellular Automata, ed. by S.Wolfram, (World Scientific, Singapore, 1986).

[2] R. Hirota and S. Tsujimoto, J. Phys. Soc. Jpn. 64 (1995) 3125.

[3] K. Park, K. Steiglitz, and W.P. Thurston, Physica D 19 (1986) 423.

[4] T.S. Papatheodorou, M.J. Ablowitz, and Y.G. Saridakis, Stud. in Appl. Math.79 (1988) 173.

[5] T.S. Papatheodorou and A.S. Fokas, Stud. in Appl. Math. 80 (1989) 165.

[6] M.J. Ablowitz, J.M. Keiser and L.A. Takhtajan, Phys. Rev. A 44 (1991) 6909.

[7] M. Bruschi, P.M. Santini and O. Ragnisco, Phys. Lett. A 169 (1992) 151.

[8] T.Tokihiro, D.Takahashi, J.Matsukidaira and J.Satsuma, Phys. Rev. Lett. 76(1996) 3247.

[9] D. Takahashi and J. Satsuma, J. Phys. Soc. Jpn. 59 (1990) 3514.

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[10] See Problem 9 in S. Wolfram, Physica Scripta T9 (1985) 170or S. Wolfram, Cellular Automata and Complexity (Addison Wesley) Page 457.

[11] D. Takahashi and J. Matsukidaira, Phys. Lett. A 209 (1995) 184.

[12] R. Hirota, J. Phys. Soc. Jpn. 43 (1977) 2074.

[13] M. Torii, D. Takahashi and J. Satsuma, Physica D 92 (1996) 209-220

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3 1 2 4 1 1

1 3 2

2 2 1 1

3 1 2

4 1 1

1 3 2

2 2 1 1

3 1 2

4 1 1

1 3 2

2 3 1

1 1 4

2 1 3

1 1 2 2

2 3 1

1 1 4

2 1 3

1 1 2 2

2 3 1

1 1 4

2 1 3

Fig. 1. A 2-soliton solution where p1 = 3, p2 = 2, 1 = 2, 2 = 1. expresses 0.

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1 1 2 1 2 1 2

1 1 3

2 2 1

1 1 1 2

2 3

1 1 2 1

2 1 2

1 1 3

2 2 1

1 2 2

3 1 1

2 1 2

1 2 1 1

3 2

2 1 1 1

1 2 2

3 1 1

2 1 2

1 2 1 1

3 2

2 1 1 1

Fig. 2. A 2-soliton solution where p1 = 3, p2 = 2, 1 = 2, 2 = 1.

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2 1 1 1

3 2

1 2 1 1

2 1 2

3 1 1

1 2 2

2 1 1 1

3 2

1 2 1 1

2 1 2

4 1

1 1 3

2 1 2

1 1 2 1 2 3

1 1 1 2

2 2 1

1 1 3

2 1 2

1 1 2 1

2 3 1 1 1 2

Fig. 3. A 2-soliton solution where p1 = 3, p2 = 2, 1 = 2, 2 = 1

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9 4

3 6 3 1

6 3 2 2

9 1 3

3 6 4

6 3 3 1

9 2 2

3 6 1 3

6 3 4

9 3 1

3 8 2

2 3 8

1 3 4 5

4 7 2 3 1 1 8

2 2 4 5

1 3 7 2

4 1 8

3 1 4 5

2 2 7 2

1 3 1 8 4 4 5

Fig. 4. A 2-soliton solution for the case L = 3

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J. Matsukidaira, J. Satsuma, D. Takahashi, T. Tokihiro and M. Torii- Toda-type Cellular Automaton...

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