Discrete and Ultradiscrete Mixed Soliton Solutions
aa r X i v : . [ n li n . S I] D ec Discrete and Ultradiscrete MixedSoliton Solutions
HIDETOMO NAGAI † Department of Mathematics, Tokai University, 4-1-1,Kitakaname, Hiratsuka, Kanagawa, 259-1292, Japan † Corresponding author. Email:[email protected] SHINZAWA
Department of Engineering, Nishinippon Institute of Technology,1-11, Aratsu, Kandamachi, Miyako-gun, Fukuoka, Japan
We propose a new type of soliton equation, which is obtained fromthe generalized discrete BKP equation. The obtained equation admitstwo types of soliton solutions. The signs of amplitude and velocity ofthe soliton solution are opposite to the other. We also propose the ultra-discrete analogues of them. The ultradiscrete equation also admits thesimilar properties. In particular it behaves the original Box-Ball systemin a special case.
Key words : Discrete BKP equation; Ultradiscrete soliton equation; Box-BallSystem.
Soliton equations are known as nonlinear equations which have exact solutions.Among these equations, the soliton equations are classified in some hierarchies.For example, the KP equation, BKP equation[1]. In the KP equation, the KdVequation or the Toda equation have been researched especially. This is becausethey have good properties and to easy to handle. Similarly, the Sawada-Koteraequation is also known as a good example in the BKP equation.Soliton equations can be divided into continuous, discrete and ultradiscreteequations by its discreteness of dependent and independent variables. In discretesoliton equations, there are also the discrete KP, BKP equations similar to thecontinuous equations. Recently, one of the authors proposed the generalizeddiscrete BKP equation and its soliton solution[2]. The equation is expressed by z τ ( p + 1 , q, r ) τ ( p, q + 1 , r + 1) + z τ ( p, q + 1 , r ) τ ( p + 1 , q, r + 1)+ z τ ( p, q, r + 1) τ ( p + 1 , q + 1 , r ) − z τ ( p, q, r ) τ ( p + 1 , q + 1 , r + 1) = 0 , (1)1here z , z , z , z are arbitrary parameters satisfying z + z + z − z = 0.We note (1) reduces to the discrete KP equation when z = 0. Moreover, whenwe set z = ( a + b )( a + c )( b − c ) , z = ( b + c )( b + a )( c − a ) ,z = ( c + a )( c + b )( a − b ) , z = − ( a − b )( b − c )( c − a ) , (2)then (1) reduces to the discrete BKP equation[3]. This fact means the gener-alized discrete BKP equation includes the discrete KP, BKP equations. Con-versely, as mentioned in [2], (2) have solutions for a , b , c when z z z z = 0. Inother words, the generalized discrete BKP equation (1) coincides with the dis-crete BKP equation for most values of z i . However, the advantage of adoptingthe expression of (1) than that of the discrete BKP equation is the coefficientscan be taken freely. This freedom gives us a simple expression of soliton so-lutions. Furthermore it enables us to ultradiscretize equations and solutionseasily.In this paper we present a new type of soliton equation from (1) through thereduction. The obtained equation possesses exact solution which contains twodifferent types of soliton solutions. The signs of amplitudes and the velocities areopposite each other. We also give its ultradiscrete analogues[5]. The obtainedultradiscrete solution also contains two different types of solutions. One behavesas the original Box-Ball system and the other does negative soliton solution.This paper composed below. In Section 2, we derive a certain discrete solitonequation from (1). Also we give the ultradiscrete equation. In Section 3, wegive the discrete soliton solutions and show its properties. In Section 4, weultradiscretize the discrete soliton solutions. In Section 5, we take the continuouslimit and show the relation between our equation and the KdV-Sawada-Koteraequation. We give concluding remarks in Section 6. In this Section, we derive new discrete and ultradiscrete equations from (1)by imposing some conditions. First we introduce new independent variabletransformations p = n − m − l, q = − l, r = m + l. (3)Then (1) is rewritten by z ˆ τ ( l, m + 1 , n )ˆ τ ( l + 1 , m − , n + 1) + z ˆ τ ( l, m, n − τ ( l + 1 , m, n + 2)+ z ˆ τ ( l, m, n )ˆ τ ( l + 1 , m, n + 1) − z ˆ τ ( l + 1 , m − , n )ˆ τ ( l, m + 1 , n + 1) = 0 , (4)where ˆ τ depends on l , m , n . In particular, we set z = d , z = d , z = 1 − d , z = 1 + d , (5)with 0 < d < d <
1. By imposing ˆ τ does not depend on l , that is, ˆ τ ( l +1 , m, n ) = ˆ τ ( l, m, n ) holds, then we finally obtain(1 + d ) f m − n f m +1 n +1 = d f m +1 n f m − n +1 + d f mn − f mn +2 + (1 − d ) f mn f mn +1 , (6)2here f mn denotes ˆ τ ( l, m, n ). We note (6) is reduced to the discrete KdV equa-tion in bilinear form when d = 0[4]. In addition we note (6) can be also obtainedfrom the discrete BKP equation by putting suitable parameters a, b, c ∈ C into(2), however the expressions of them are complicated to deal with.If we introduce dependent variable transformations u mn = f mn +1 f m +1 n f mn f m +1 n +1 , v mn = f m +1 n f m − n ( f mn ) , x mn = f mn +1 f mn − ( f mn ) , (7)for (6), we obtain u mn u m − n = v mn v mn +1 , (8a)(1 + d ) v mn +1 = (1 − d ) u m − n + d v mn ( u m − n ) + d x mn +1 x mn u m − n , (8b) x mn x m − n = u m − n − u m − n . (8c)We regard m and n as time and space variables. Then the time evolutions of u mn , v mn , x mn are described by (8a), (8b) and (8c) under the boundary conditionlim n →−∞ v mn = const. Next, we shall derive the ultradiscrete analogues[5]. By introducing trans-formations d i = e − δ i /ε , f mn = e F mn /ε , and taking the limit ε → +0 for (6), weobtain the ultradiscrete equation. F m +1 n +1 + F m − n = max( F mn +1 + F mn , F m +1 n + F m − n +1 − δ , F mn +2 + F mn − − δ ) , (9)where 0 < δ < δ . Here we use the key formula lim ε → +0 ε log( e A/ε + e B/ε ) =max(
A, B ). We note (9) reduces to the ultradiscrete KdV equation in bilinearform when δ is sufficiently large[6]. We can also obtain the ultradiscrete ana-logues of (8) with the transformations u mn = e U mn /ε , v mn = e V mn /ε , x mn = e X mn /ε by the similar procedure. U mn = U m − n + V mn − V mn +1 , (10a) V mn +1 = max(0 , V mn + U m − n − δ , X mn +1 + X mn − δ ) + U m − n , (10b) X mn = X m − n + U m − n − − U m − n . (10c)Notice that the equations (10a) and (10c) take the form of the conservationlaw. This procedure determines X mn from the values of X m − n and U m − n from(10c). Then, if we assume boundary condition lim n →−∞ V mn we can determinethe values of V mn +1 from the values of U m − n , V mn , X mn using (10b). Finally U mn is determined from the values of U m − n and V mn from (10a). Repeating thisprocedure to the new values U mn , X mn , we can obtain the time evolution.Eq.(10) can be considered as a kind of Box-Ball systems. Let us consider U mn and V mn as a number of balls in n th box at time m and a carrier of ballsrespectively. Also let us consider δ as the capacity of the boxes. Assume theboundary conditionslim n →−∞ U mn = lim n →−∞ V mn = lim n →−∞ X mn = 0 . m − m , the carrier moves from the −∞ site to the ∞ site. It passes each box fromthe left to the right. At n th box, the carrier loads the balls as many as possiblefrom the n th box and at the same time, unloads the balls from the carrier asmany as the vacant spaces of the n th box, which corresponds to δ − U m − n .Moreover, if the maximum value of RHS in (10b) is given by X mn +1 − X mn − δ ,then the carrier loads more balls even if the number of balls becomes negative.In this case we consider the box has ‘negative balls’ as a debt.As a special case, if the condition − δ + X mn + X mn +1 < V mn +1 = max(0 , V mn + U m − n − δ ) + U m − n . (11)Using the conservation law (10a) to the left hand side of this equation, we obtainthe following. U mn = min( V mn , δ − U m − n ) . (12)We can also rewrite the dependent variable V mn as V mn = n X k = −∞ ( V mk − V mk − ) = n X k = −∞ ( U m − k − − U mk − ) , (13)using the conservation law (10a) together with the boundary condition lim n →−∞ V mn =0. Substituting this representation to the equation (12), the equations (11) andthe conservation law (10a) turns out to be U mn = min n X k = −∞ ( U m − k − − U mk − ) , δ − U m − n ! . (14)This is nothing but the Box-Ball system with the box capacity δ .Exact solutions and examples for (10) are given in Section 4. The soliton solution of (1) is expressed by τ ( p, q, r ) = X ≤ k ≤ N X I k ⊂ [ N ] Y i ∈ I k c i ϕ ( t i ) ϕ ( s i ) Y i,j ∈ Iki Set t ′ i = − z z z z s i , s ′ i = − z z z z t i . Then ϕ ( t i ) ϕ ( s i ) = ϕ ( t ′ i ) ϕ ( s ′ i ) , b ij = b i ′ j = b ij ′ = b i ′ j ′ (18) hold for any i, j = 1 , , . . . , N ( i = j ) . Here b i ′ j denotes b i ′ j = cf ( t ′ i , t j ) cf ( s ′ i , s j ) cf ( s ′ i , t j ) cf ( t ′ i , s j ) . (19) Proof. The relations (18) can be obtained by confirming a k ( t ) a k ( s ) = a k ( t ′ ) a k ( s ′ ) , cf ( t i , t j ) = − z z z z cf ( t i , s ′ j ) = cf ( s ′ i , s ′ j ) (20)for k = 1 , , t i or s i is always positive forany i without loss of generality. Actually, the solution with t i , s i and theone with t ′ i , s ′ i correspond. Moreover, considering the term ( a ( t i ) /a ( s i )) r =( t i /s i ) r in ϕ ( t i ) /ϕ ( s i ), we have the condition both t i and s i should be positiveso that ϕ ( t i ) /ϕ ( s i ) > 0. Similarly, by considering the positivity of a ( t i ) /a ( s i ), a ( t i ) /a ( s i ), we obtain the following proposition. Proposition 3.2 If parameters t , s satisfy one of the conditions,1. < t, s < z ,2. z < t, s < z ,3. z < t, s ,where z and z denote min( z , z z z ) , max( z , z z z ) respectively, then ϕ ( t ) /ϕ ( s ) take positive values for any p, q, r ∈ Z . We also have the following. Proposition 3.3 Suppose t i , s i are positive for any i = 1 , , . . . , N . If param-eters t i , s i satisfy one of the conditions,1. Both t i and s i take values between t j and s j ,2. Both t j and s j take values between t i and s i ,3. max( t i , s i ) < min( t j , s j ) , t s s t t s t s t s FIG. 1. An example of parameters t i and s i . max( t j , s j ) < min( t i , s i ) ,for any combination i , j ( i < j ) , then b ij takes a positive value. Proof. Figure 1 shows an example of the parameters satisfying the conditions.The conditions in the proposition mean each line connected between t i and s i does not intersected with others. The conditions derive ( t i − t j )( s i − s j )( t i − s j )( s i − t j ) > i , j and it gives b ij > Theorem 3.1 Suppose c i > and parameters t i and s i satisfy the conditionsgiven in Proposition 3.2 and 3.3 for i = 1 , , . . . , N , then the solution τ ( p, q, r ) take positive values for any p, q, r ∈ Z . Now let us derive the soliton solution of (6) under the assumptions given inTheorem 3.1. Hereafter we set z k as (5). By applying the transformations (3),the solution (15) is transformed toˆ τ ( l, m, n ) = X ≤ k ≤ N X I k ⊂ [ N ] Y i ∈ I k c i φ ( t i ) φ ( s i ) Y i,j ∈ Iki 3) such that ddt ψ ( α i ) = 0 on 0 < t .Each α i satisfies α ∈ ( z z z , ∞ ), α = z , α ∈ (0 , z ) respectively. • ψ ( t ) is monotone increasing on ( z z /z , α ), ( z , z z /z ) and (0 , α ).6 ψ ( t ) - z z α t ψ ( t ) FIG. 2. Plot of ψ ( t ). The right figure shows near the origin. • ψ ( t ) is monotone decreasing on ( α , ∞ ) and ( α , z ). • ψ ( t ) ≥ , z z /z ). ψ ( t ) < z z /z , ∞ ).Therefore if t belongs to either J = (cid:16) z z z , ∞ (cid:17) or J = (0 , z ), except for α , α , then there exists a unique value s in the same interval such that s = t andsatisfying (23). In particular, these parameters t , t , . . . , t N , s , s , . . . s N holdthe conditions in Theorem 3.1. These results are summerized as the followingtheorem. Theorem 3.2 The solution of (6) is expressed by f mn = X ≤ k ≤ N X I k ⊂ [ N ] Y i ∈ I k c i φ ( t i ) φ ( s i ) Y i,j ∈ Iki 3) and s so that satisfying (23). The solitary wave moves tothe right side. In Fig.4, we set t = 3 / 10, ( d , d ) = (1 / , / 3) and s satisfies(23). The solitary wave moves to the left side slowly. In each case, the velocity ofsolitary wave is given by 1 − log( a ( t ) /a ( s )) / log( a ( t ) /a ( s )). Figure 5 showsthe interaction of two different soliton solutions. We set ( t , t ) = (3 , / d , d ) = (1 / , / s and s satisfy (23) respectively.It is noted distinct parameters t , s in J do not exist when d = 0 since ψ ( t )is monotone decreasing on J . It means the discrete KdV equation does notadmit the solution with t , s in J . 7 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 m = − m = 0 m = 10 FIG. 3. 1-soliton solution u mn with t, s ∈ J . ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 m = − m = 0 m = 10 FIG. 4. 1-soliton solution u mn with t, s ∈ J . ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● - - 20 20 400.91.01.11.21.31.4 m = − m = 0 m = 10 FIG. 5. Mixed (1 + 1)-soliton solution u mn with t , s ∈ J and t , s ∈ J .8 Ultradiscrete Soliton Solutions In order to derive ultradiscrete solutions for (9), we introduce new parametersfor each J , J : t = t (1) + z z z for t ∈ J ,t = t (2) z t (2) for t ∈ J . (26)These replacements enable us to take t ( i ) for any positive values. Then for t ∈ J , by using z + z + z − z = 0, we have a ( t ) = z z + z tz − t = − z ( z ( z + z ) + z t (1) ) z ( z + z ) + z t (1) . (27)We may neglect the signature since it can be cancelled with a ( s ). Thus, from(5) and d = e − δ /ε , d = e − δ /ε , t ( i ) = e − T ( i ) /ε , we obtain the ultradiscreteanalogue of | a ( t ) | as | a ( t ) | → max( − δ , T (1) ) − max( − δ , T (1) )=: A ( T (1) ) . (28)Similarly, we also obtain the ultradiscrete analogues of | a ( t ) | , | a ( t ) | as follows. | a ( t ) | → A ( T (1) ) = T (1) − max( T (1) , − δ ) , | a ( t ) | → A ( T (1) ) = max( T (1) , − δ ) . (29)Hence we obtain the ultradiscrete analogue of the one soliton solution with t, s ∈ J . F mn = max(0 ,m ( A ( T (1) ) − A ( S (1) ) − A ( T (1) ) + A ( S (1) ))+ n ( A ( T (1) ) − A ( S (1) )) + C ) , (30)where T (1) , S (1) satisfy the dispersion relation, A ( T (1) ) − A ( T (1) ) − A ( T (1) ) = A ( S (1) ) − A ( S (1) ) − A ( S (1) ) , (31)which is obtained by ultradiscretizing (23). The above is simplified as | T (1) + 2 δ | = | S (1) + 2 δ | . (32)This leads S (1) = − T (1) − δ due to T (1) = S (1) . If we denote Ω, K as thecoefficients of m , n :Ω = A ( T (1) ) − A ( S (1) ) − A ( T (1) ) + A ( S (1) ) ,K = A ( T (1) ) − A ( S (1) ) , (33)9hen Ω can take any value since Ω = T (1) + 2 δ and we find K = ( | Ω − δ | −| Ω + δ | ) holds. Therefore, (30) is rewritten by F mn = max(0 , Ω m + Kn + C ) . (34)This is the one soliton solution for (9). It is noted this solution corresponds tothe one of the ultradiscrete KdV equation[6].For t ∈ J , we obtain | a ( t ) | → A ( T (2) ) = max( T (2) − δ , − δ ) + δ , | a ( t ) | → A ( T (2) ) = max( T (2) − δ , − δ ) − max( T (2) − δ , − δ ) , | a ( t ) | → A ( T (2) ) = T (2) − δ − max(0 , T (2) ) . (35)The dispersion relation A ( T (2) ) − A ( T (2) ) − A ( T (2) ) = A ( S (2) ) − A ( S (2) ) − A ( S (2) ) (36)holds by setting S (2) = − T (2) + δ − δ − max( T (2) − δ , − 12 max( − T (2) − δ , . (37)If we denote Q , P as the coefficients of m , n , Q = A ( T (2) ) − A ( S (2) ) − A ( T (2) ) + A ( S (2) ) ,P = A ( T (2) ) − A ( S (2) ) , (38)then P can take any value and Q = max(0 , P − δ ) + min(0 , P + δ ) holds.Therefore, we obtain another one soliton solution for (9). F mn = max(0 , Qm + P n + C ) . (39)It is noted this solution corresponds to the one of the ultradiscrete Toda equation[8]. We consider the ultradiscrete analogue of b ij . We define B ( T ( α ) i , T ( β ) j )= Cf ( T ( α ) i , T ( β ) j ) + Cf ( S ( α ) i , S ( β ) j ) − Cf ( T ( α ) i , S ( β ) j ) − Cf ( S ( α ) i , T ( β ) j ) (40)for α, β = 1 , 2, where Cf ( T ( α ) i , T ( β ) j ) is the ultradiscrete analogue of | cf ( t i , t j ) | ,in which t i and t j are replaced as t ( α ) i , t ( β ) j . For example, α = 1, β = 2, we have cf ( t (1) , t (2) ) = 1 z z t (2) t (1) + z t (1) + z t (2) ( z + z ) + z z ( z t (1) + z z ) z t (2) + (1 + t (2) ) z z z → max( T (1) , T (1) + T (2) , T (2) − δ , − δ ) − max( T (1) + T (2) , T (2) − δ , − δ ) + δ =: Cf ( T (1) , T (2) ) . (41)10otice Cf ( S (1) , S (2) ) can be expressed by T (1) and T (2) through the dispersionrelations. We can calculate B ( T (1) i , T (1) j ) =2 max(min(Ω i , − Ω j ) , min( − Ω i , Ω j )) ,B ( T (2) i , T (2) j ) = max(min( − P i − Q i , P j + Q j ) , min(2 P i + Q i , − P j − Q j )) , (42)and B ( T (1) i , T (2) j ) = − max(min(Ω i + 2 K i , Q j ) , min(Ω i + 2 K i − P j + δ , i > , P j > i + 2 K i , − Q j ) , min(Ω i + 2 K i + P j + δ , i > , P j < − Ω i − K i , Q j ) , min( − Ω i − K i − P j + δ , i < , P j > − max(min( − Ω i − K i , − Q j ) , min( − Ω i − K i + P j + δ , i < , P j < i , K i , P i , Q i are defined by (33) and (38). Due to | b ij | = | b ji | , B ( T (2) i , T (1) j ) can be also obtained. Thus the following holds. Theorem 4.1 The solution of (9) is expressed by F mn = max ≤ k ≤ N + M max I k ⊂ [ N + M ] X i ∈ I k Φ i ( m, n ) + X i,j ∈ Iki 4) and ( δ , δ ) = (1 , δ . The middle figure in Fig.6 shows (0 + 3) soliton solution with( P , P , P ) = (3 , , − 1. We can observe U mn takenegative values and move to the left side as discrete solutions. The right figurein Fig.6 shows (2 + 1)-mixed soliton solution with (Ω , Ω , P ) = (2 , , In this Section, we investigate the continuous limit of the equation (6). Let usstart with rewriting the equation (6) using the Hirota’s D-derivative. (cid:26) − (1 + d ) cosh (cid:18) − D m − D n (cid:19) + d cosh (cid:18) D m − D n (cid:19) + d cosh (cid:18) D n (cid:19) + (1 − d ) cosh (cid:18) D n (cid:19)(cid:27) f · f = 0 (50)Here, D m , D n are the Hirota’s D-derivative corresponding to the variables m , n respectively. 12o take the continuous limit we introduce the new variables x , x and relatethem with the original variables as D m = 2 ǫD + 23 a ǫ D + 25 a ǫ D ,D n = 2 kǫD + 23 b ǫ D + 25 b ǫ D . Here, D , D are the Hirota’s D-derivative of the variables x , x , and k, a , a , b , b are arbitrary complex parameters. ǫ is the real parameter which represents thelattice spacing of the discrete variables. Furthermore, we replace the parameters d , d as d = − 12 + − k + b ǫ k , d = a ǫ + 14 k . Then equation (50) becomes as follows (cid:26) ( k (2 k − 1) + b ǫ ) cosh (cid:18) − D m − D n (cid:19) + (cid:0) k (2 k + 1) − b ǫ (cid:1) cosh (cid:18) D m − D n (cid:19) +( − − a ǫ ) cosh (cid:18) D n (cid:19) + (1 − k + a ǫ ) cosh (cid:18) D n (cid:19)(cid:27) f · f = 0Taking the small ǫ limit we obtain the following equation as the coefficient ofthe sixth order term of the parameter ǫ . (cid:8) k ( a k − b ) D D − k ( k − k − D − k ( k − a k + 2 b ) D + 40( a k − b ) D (cid:9) f · f = 0 (51)This equation contains the terms of D D f · f, D f · f, D f · f and D f · f .As a special case, let us assume the following form of the parameters a i , b i . a = c − p ( k − k − − k + 13 ! ,b = ck p ( k − k − − k + 13 ! ,a = − 23 + 103 k − k ,b = 0Substituting these expressions into the equations (51), we obtain the followingequation. − k ( k − k − D D + D − cD − c D ) f · f = 0This is nothing but the bilinear form of the KdV-Sawada-Kotera equation ap-pearing in the paper [10]. 13 Concluding Remarks We have derived an soliton equation and its solution from the generalized dis-crete BKP equation. The equation admits two types of solitary wave solutions.One is the solution which moves to the positive direction and has the positiveamplitude while the other does the negative direction and amplitude. The typeof the solution depends on whether the parameter t belongs to J or J . 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