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Decoupling of the N-soliton solution for a new discrete MKdV equation
Chaos,
Pergamon
.Solrton.t & Fracrals Vol. 4, No. 12, pp. 2237-2244, 1994 Copyright 0 1994 Elsetier Science Ltd Printed m Great Britain All raghts reserved 0960-0779/94$7.00 + .oO
0960-0779(94)E0130-H
Decoupling of the N-soliton Solution for a New Discrete MKdV Equation KAZUAKI NARITA BlOlO CI Heights,
1-31 Yamada-Nishi, (Received
10 March
Suita-Shi,
Osaka
565, Japan
1994)
Abstract-We prove that the N-soliton solution for a new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete MKdV equation, and approach single soliton solutions as t -+ km. We present concrete expressions of soliton components in the case of N = 2, and illustrate the results.
1. INTRODUCTION
Recently, we studied a series of highly nonlinear difference-differential equations [l-4]. We derived N-soliton solutions in [l-3], and a single soliton solution in [4]. These equations are akin to one another, and are presumed to appear in future contexts of particle physics. On the other hand, it has been shown by Yoneyama and us that N-soliton solutions for the well-known nonlinear difference-differential equations can be decoupled to linear superpositions of interacting single soliton solutions [5,6]. These differencedifferential equations include the Toda lattice [5], the Volterra equation [6] and the discrete KdV equation [6]. These results suggest a possibility of decoupling the alreadyknown N-soliton solution for the highly nonlinear new discrete MKdV equation [l]. In the following sections, we shall present a method of its decoupling and the resulting findings. 2. DECOUPLING OF N-SOLITON SOLUTION
In this section, we attempt to find a decoupling for the new discrete MKdV equation [l] tii = 4(2&l
- U,+J[(l
of the already
known
N-soliton
- U,_l - U,)“2(1 + U, + u,+#2
+ (1 + u;_, + u;)1’2(1 - Uj - ui+1)1’2]2. The decoupling
should
solution
satisfy the following
three
(1)
conditions.
(i) u, = &in) That is, the N-soliton solution should be expressed components. (ii) ui”’ should satisfy the following equation: a’“’ = 4@“‘,
- &\ )/[(l
- U,_l - UJ’2(1
(2) as a linear
of N
f U, + U,+1)1’2
+ (1 + CL-1 + U,)“2(1 - U, - U,+1)1’*]2. 2237
superposition
(3)
2238
K. NARITA
Hereafter we call (3) the interacting new discrete MKdV equation. (iii) u!“’ should approach a single soliton solution as t + &a. It is subsequently proven that a decoupling satisfying the conditions following equations:
(i-iii)
is given by the
(6) x, = k,i - 2sinh k;t exp @(m, n) = - sinh’ [(k,
+ tanh (k,/2).r,,
- k,)/2]/sinh2
[(k,
+ a,,,
(7)
+ k,)/2],
(8)
5 ;rL, and
~~$, imply the summation over all possible combination of p1 = 0, 1, p,v = 0,l under the condition ~~=,,nn = even integer and ~j!=+,~ = odd p2 = 0, and xi,“!, implies the summation over all possible pairs chosen from integer . re&d;ively, N elements. we first substitute (4) into (3). Then, as a consequence of the To prove (4)-(8), decomposition, we find seven bilinear equations as listed below: where
N
c
Drncosh$
- sinh$
=
(9)
0,
r,=...=ru=O
n=l
5
Drn
sinh
+(l;..f
+
ZZ
g;d
(10)
0,
T,=...=T,~=o
n=l
sinh’+U.L +g,*d
= 0,
(11)
=o
(12)
r,=...=r,&=c
D,(D, + 2 sinh Di>A’Sr
r,=...=r,=o
D,,,(Q + 2sinh DJ(L.J;- g,+d sinh +
sinh:
(
ND,,, cash %
NDTn cash:
*,=.. =r,,=o
= 0,
(13)
= 0,
- sinh:
- sinh?
T,=...=Ty-o
(14) _- 0,
(13
in which D,, DTn and D, are the usual Hirota’s bilinear operators [ 11. For N = 1,2 we can directly prove that (4) exactly satisfies (9)-(15). For general N, we can prove it by the use of mathematical induction, but we omit its details [l]. In the following, we shall see how the equations (9)-(15) arise. First, we consider the case of the interacting new discrete MKdV equation. We assume
u;“’ = Then,
Gj"'/fi;/.
,_= r,=,,
=
0.
(16)
from (4) we find F; = f;
+ gf.
Gj”’ = 2DT,,g;L.
(17) (18)
2239
Decoupling of the N-soliton solution
From (2) and (16), we find
+Etli
G;"'
1 k (ui + u~+~)=
rl=...=r,=O
n=l
(19)
For (19), we find from (17) that &&+I = (.LL+I + gigl+l)2 +
Uki+l
-
(20)
.h+lgi)*7
and also require that &$ G::‘, + F;,, 5 G!“’
= 2(iL+I +
‘,=...=“N=O
n=l
ll=l
giSi+l>(XSi+l
-
C21)
.f+lgi)lrl=...=rN=O~
Substituting (20), (21) into (19), we find l
*
Therefore. [(l
+
Ui_1
(4
+
&+I)
=
[(.h.t+l
+
gikTi+l>
*
(J;gi+l
(22)
~+lgi)12/F;~+llr,=...=r,=0.
-
we have +
Ui)l’*(l - Ui - U~+l)l’~+ (1 - Ui-1 - Ui)1’2(1+ U,+ Ui+1)1’2]2
= (4/fi-IFT~+1)[(A-I$
+
gi-lgi)(.fX+l
+
gigi+l>
-
CLlSi
-
f;fT,-l)Uki+l
-
f;+l~i)121~1=...=~N=O~
(23) = (4/Kl&+l)(.Llf;+1
+
(24)
gi-lgi+l)2/r,=...=rN=0.
From (16) and (24) we find (25) Also, from (16) we find 1.h.s. of (3) = Therefore,
(26)
we obtain the following two equations:
.Llh+l + d,‘$“’
;F,
dt
_
’
gi-lgi+llrl=,..=tN=O
G!“‘dF, ’
dt
=
(27)
FIIT,=.,,=T~=O~
= C+lGi-1 -
KIGi+lIrl=...=r,,,=o.
r,=...=rN=O
Equations (21), (27), (28) constitute decomposed hereafter to reduce to bilinear equations. First, substituting (17), (18) into (21), we find cash $(f;$
equations for (3), which will be shown
+ gi.gi) (29)
Using the formula cash
D1(D2a~b)~a2 = (cash D,a.a)(D2
cash D,a.b)
- (D, sinh D,a.a)(sinh
D,a*b),
(30)
2240
K. NARITA
we can transform cosh$(f.J
(29) into the following
form:
+ g;g,)
[
-
i,
’ D,,,sinh+(J.t,
+ s,g)](sinh~~.~~~l~,=,,.~~~~,,
= 0.
(31)
Equation (31) produces the first two bilinear equations (9), (10). The substitution of (17) into (27) leads to the third bilinear equation (11). The substitution of (17), (18) into (25) leads to the quartilinear equation (Dr + 2sinh Here,
D,,)(ff
+ gf).(D,,,~;.g,)l,,=...=~~=,,
(32)
= 0.
using the formula
sinh
D,a'.(D,a.b) = (cash D,a.a)(Dzsinh
D,a.b)- (D,sinh D,a.a)(cosh
D,a.h),(33)
and the equality
D = lim(l/b)sinh?jD,
(34)
&+I)
we can transform
(32) into the following
(ff - sf)(DT,?Di.L.gJ
- (DT,!D,(f;.A
form:
- g;g,))k
- 2(Drrfsinh This equation
can be deformed
(fy - sf)(DT,s(Df
+ 2sinh
+ (S/N)
further
DJJ+g,) sinh’+(s.i
+ 2(cosh D0.f;
D,(J.f;
- g;g,))(D,,,
- g;g,))(cosh
D,f;.g,)l,,=
ND,,, sinh +
-T,=o = 0.
(35)
into the form
- (D,,,(D!
+ 2 sinh D,)(~.J? - g;g,))f;g,
ND,,sinh 1
- g;g,)
cash $J.g,
I( -
sinh D,.A.g,)
cash +(j.i
sinh’ :)‘L.g,
- g;g,)
Ill =0.
(36)
T,= ,,=r,=l)
Necessary conditions for the satisfaction of (36) are four bilinear equations (12)-( 15). Thus we have succeeded in deriving seven bilinear equations. Since the solution of these we have completed the proof that our decoupling actually equations is given by (4)-(S). satisfies the conditions (i), (ii). Condition (iii) is directly proved to be satisfied in the case of a few-soliton solution, and this proof is easily extended to the case of the N-soliton solution.
3. DECOUPLING
OF 2-SOLITON
SOLUTION
In this section, we apply the result obtained in the previous section to the case of the 2-soliton solution. Putting N = 2 in (4)-(S) and rearranging the result, we find concrete expressions of two components of U, expresed in the following forms:
D = edA sinh’[(k,
‘A(‘I I = tanh(k,/2)cosh(k?i
- 2sinhk?.t)/D.
(37)
11I(‘I = tanh(k2/2)cosh(k,i
-.- 2sinhk,.r)/D.
(38)
+ k,)i/2
- (sinh k, + sinh k,)t] + eAcosh’[(k,
-
k,)i/2
- (sinhk,
- sinh k,)t].
(39)
Decoupling
of the N-soliton
solution
2241
(40)
in which we have assumed aI = 6, = A.
(41)
Profiles of ui’), ~1’) and ui are plotted in Figs 1, 2. Figure 1 illustrates a collision of two positive solitons in the case where kl is extremely large. Similarly to the previous two cases [2,3], it is observed that ui forms three peaks
(a)
(b)
(d)
Fig. 1. Collision
of two positive solitons; abscissa i; longitudinal axis (1) uj’), (2) uj*), (3) u;; k, = 12, k2 = 1; (a) t = -2 x lo-+‘, (b) I = -8 x 10-5, (c) t = -2 x lo+, (d) t = 0.
K. NARITA
2242
which are unobserved when k, and k2 have comparable temporarily in the interaction, small magnitudes. Figure 2 illustrates a collision of a positive soliton (wavenumber: k,) and a negative soliton (wavenumber: k2) when the conditions k,, / kZ( >> 1 and k, - lk21 << k,, 1k71 are satisfied. In this case, u:” mtttally forms a large positive peak and successively a positive 3-peak at the middle stage of the interaction. Similarly, ~1~) initially forms a large negative peak and successively a negative 3-peak at the middle stage of the interaction. As a result, U, initially subjects to a complex process, and forms a sharp positive peak at the different in magnitudes middle stage of the interaction. When k, and \kz\ are considerably or both small, this behaviour becomes unobserved.
(a)
- 3
(b) -3 3
--I
3
I 3
Fig. 2. (a)-(c)
Decoupling
of the N-soliton
2243
solution
Cd)
3
\
I
II
I I
‘I
\
\
\
II \
::
!
:’
3
I
I
I I
\ ’ \J
JQ, : I
3
JC'
I
‘_’
‘\ ‘._
I 3
I
1
\ \
Fig. 2. Collision (3) ui; k, =6,
I
of a positive soliton and a negative soliton; abscissa i; longitudinal axis (1) ui”, (2) u12’ k2 = -5.5; (a) t = -1.5 x 10e2, (b) t = -7 x 10M3, (c) I = -3 x 10-3, (d) t = -1.2 x lo-‘: (e) t = -4 X 10m4, (f) t = 0.
4.
CONCLUSION
We have proved that the N-soliton solution for the new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete
2244
K. NARITA
MKdV equation, and approach single soliton solution as t + +@J. We presented expressions of soliton components in the case of iV = 2, and illustrated the results. Acknowledgement--The the study.
author
wishes to thank
Mr. Yoshibumi
Narita
of Wakatake
School
concrete
for his continual
aid to
REFERENCES 1. K. Narita, New discrete modified KdV equation, Prog. Theor. Phys. 86, 817-824 (1991). equation, Chaos, Solirons & Fractals 3, 2. K. Narita, Soliton solution for a highly nonlinear difference-differential 279-283 (1993). difference-differential equation related to the Volterra equation, J. Math. Anal. 3. K. Narita, New nonlinear Appl. 186, 120-131 (1994). 4. K. Narita, Triple-humped soliton solution for a lattice equation related to the discrete KdV equation, J. Phys. A25, L1167-L1168 (1992). 5. T. Yoneyama, Interacting Toda equations, J. Phys. Sot. Japan 55, 753-761 (1986). 6. K Narita, Decoupling of N-soliton solutions for Volterra and Volterra-like circuit equations, Jupan. J. Appl. Phys. 27, 679-683 (1988).
Pergamon
.Solrton.t & Fracrals Vol. 4, No. 12, pp. 2237-2244, 1994 Copyright 0 1994 Elsetier Science Ltd Printed m Great Britain All raghts reserved 0960-0779/94$7.00 + .oO
0960-0779(94)E0130-H
Decoupling of the N-soliton Solution for a New Discrete MKdV Equation KAZUAKI NARITA BlOlO CI Heights,
1-31 Yamada-Nishi, (Received
10 March
Suita-Shi,
Osaka
565, Japan
1994)
Abstract-We prove that the N-soliton solution for a new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete MKdV equation, and approach single soliton solutions as t -+ km. We present concrete expressions of soliton components in the case of N = 2, and illustrate the results.
1. INTRODUCTION
Recently, we studied a series of highly nonlinear difference-differential equations [l-4]. We derived N-soliton solutions in [l-3], and a single soliton solution in [4]. These equations are akin to one another, and are presumed to appear in future contexts of particle physics. On the other hand, it has been shown by Yoneyama and us that N-soliton solutions for the well-known nonlinear difference-differential equations can be decoupled to linear superpositions of interacting single soliton solutions [5,6]. These differencedifferential equations include the Toda lattice [5], the Volterra equation [6] and the discrete KdV equation [6]. These results suggest a possibility of decoupling the alreadyknown N-soliton solution for the highly nonlinear new discrete MKdV equation [l]. In the following sections, we shall present a method of its decoupling and the resulting findings. 2. DECOUPLING OF N-SOLITON SOLUTION
In this section, we attempt to find a decoupling for the new discrete MKdV equation [l] tii = 4(2&l
- U,+J[(l
of the already
known
N-soliton
- U,_l - U,)“2(1 + U, + u,+#2
+ (1 + u;_, + u;)1’2(1 - Uj - ui+1)1’2]2. The decoupling
should
solution
satisfy the following
three
(1)
conditions.
(i) u, = &in) That is, the N-soliton solution should be expressed components. (ii) ui”’ should satisfy the following equation: a’“’ = 4@“‘,
- &\ )/[(l
- U,_l - UJ’2(1
(2) as a linear
of N
f U, + U,+1)1’2
+ (1 + CL-1 + U,)“2(1 - U, - U,+1)1’*]2. 2237
superposition
(3)
2238
K. NARITA
Hereafter we call (3) the interacting new discrete MKdV equation. (iii) u!“’ should approach a single soliton solution as t + &a. It is subsequently proven that a decoupling satisfying the conditions following equations:
(i-iii)
is given by the
(6) x, = k,i - 2sinh k;t exp @(m, n) = - sinh’ [(k,
+ tanh (k,/2).r,,
- k,)/2]/sinh2
[(k,
+ a,,,
(7)
+ k,)/2],
(8)
5 ;rL, and
~~$, imply the summation over all possible combination of p1 = 0, 1, p,v = 0,l under the condition ~~=,,nn = even integer and ~j!=+,~ = odd p2 = 0, and xi,“!, implies the summation over all possible pairs chosen from integer . re&d;ively, N elements. we first substitute (4) into (3). Then, as a consequence of the To prove (4)-(8), decomposition, we find seven bilinear equations as listed below: where
N
c
Drncosh$
- sinh$
=
(9)
0,
r,=...=ru=O
n=l
5
Drn
sinh
+(l;..f
+
ZZ
g;d
(10)
0,
T,=...=T,~=o
n=l
sinh’+U.L +g,*d
= 0,
(11)
=o
(12)
r,=...=r,&=c
D,(D, + 2 sinh Di>A’Sr
r,=...=r,=o
D,,,(Q + 2sinh DJ(L.J;- g,+d sinh +
sinh:
(
ND,,, cash %
NDTn cash:
*,=.. =r,,=o
= 0,
(13)
= 0,
- sinh:
- sinh?
T,=...=Ty-o
(14) _- 0,
(13
in which D,, DTn and D, are the usual Hirota’s bilinear operators [ 11. For N = 1,2 we can directly prove that (4) exactly satisfies (9)-(15). For general N, we can prove it by the use of mathematical induction, but we omit its details [l]. In the following, we shall see how the equations (9)-(15) arise. First, we consider the case of the interacting new discrete MKdV equation. We assume
u;“’ = Then,
Gj"'/fi;/.
,_= r,=,,
=
0.
(16)
from (4) we find F; = f;
+ gf.
Gj”’ = 2DT,,g;L.
(17) (18)
2239
Decoupling of the N-soliton solution
From (2) and (16), we find
+Etli
G;"'
1 k (ui + u~+~)=
rl=...=r,=O
n=l
(19)
For (19), we find from (17) that &&+I = (.LL+I + gigl+l)2 +
Uki+l
-
(20)
.h+lgi)*7
and also require that &$ G::‘, + F;,, 5 G!“’
= 2(iL+I +
‘,=...=“N=O
n=l
ll=l
giSi+l>(XSi+l
-
C21)
.f+lgi)lrl=...=rN=O~
Substituting (20), (21) into (19), we find l
*
Therefore. [(l
+
Ui_1
(4
+
&+I)
=
[(.h.t+l
+
gikTi+l>
*
(J;gi+l
(22)
~+lgi)12/F;~+llr,=...=r,=0.
-
we have +
Ui)l’*(l - Ui - U~+l)l’~+ (1 - Ui-1 - Ui)1’2(1+ U,+ Ui+1)1’2]2
= (4/fi-IFT~+1)[(A-I$
+
gi-lgi)(.fX+l
+
gigi+l>
-
CLlSi
-
f;fT,-l)Uki+l
-
f;+l~i)121~1=...=~N=O~
(23) = (4/Kl&+l)(.Llf;+1
+
(24)
gi-lgi+l)2/r,=...=rN=0.
From (16) and (24) we find (25) Also, from (16) we find 1.h.s. of (3) = Therefore,
(26)
we obtain the following two equations:
.Llh+l + d,‘$“’
;F,
dt
_
’
gi-lgi+llrl=,..=tN=O
G!“‘dF, ’
dt
=
(27)
FIIT,=.,,=T~=O~
= C+lGi-1 -
KIGi+lIrl=...=r,,,=o.
r,=...=rN=O
Equations (21), (27), (28) constitute decomposed hereafter to reduce to bilinear equations. First, substituting (17), (18) into (21), we find cash $(f;$
equations for (3), which will be shown
+ gi.gi) (29)
Using the formula cash
D1(D2a~b)~a2 = (cash D,a.a)(D2
cash D,a.b)
- (D, sinh D,a.a)(sinh
D,a*b),
(30)
2240
K. NARITA
we can transform cosh$(f.J
(29) into the following
form:
+ g;g,)
[
-
i,
’ D,,,sinh+(J.t,
+ s,g)](sinh~~.~~~l~,=,,.~~~~,,
= 0.
(31)
Equation (31) produces the first two bilinear equations (9), (10). The substitution of (17) into (27) leads to the third bilinear equation (11). The substitution of (17), (18) into (25) leads to the quartilinear equation (Dr + 2sinh Here,
D,,)(ff
+ gf).(D,,,~;.g,)l,,=...=~~=,,
(32)
= 0.
using the formula
sinh
D,a'.(D,a.b) = (cash D,a.a)(Dzsinh
D,a.b)- (D,sinh D,a.a)(cosh
D,a.h),(33)
and the equality
D = lim(l/b)sinh?jD,
(34)
&+I)
we can transform
(32) into the following
(ff - sf)(DT,?Di.L.gJ
- (DT,!D,(f;.A
form:
- g;g,))k
- 2(Drrfsinh This equation
can be deformed
(fy - sf)(DT,s(Df
+ 2sinh
+ (S/N)
further
DJJ+g,) sinh’+(s.i
+ 2(cosh D0.f;
D,(J.f;
- g;g,))(D,,,
- g;g,))(cosh
D,f;.g,)l,,=
ND,,, sinh +
-T,=o = 0.
(35)
into the form
- (D,,,(D!
+ 2 sinh D,)(~.J? - g;g,))f;g,
ND,,sinh 1
- g;g,)
cash $J.g,
I( -
sinh D,.A.g,)
cash +(j.i
sinh’ :)‘L.g,
- g;g,)
Ill =0.
(36)
T,= ,,=r,=l)
Necessary conditions for the satisfaction of (36) are four bilinear equations (12)-( 15). Thus we have succeeded in deriving seven bilinear equations. Since the solution of these we have completed the proof that our decoupling actually equations is given by (4)-(S). satisfies the conditions (i), (ii). Condition (iii) is directly proved to be satisfied in the case of a few-soliton solution, and this proof is easily extended to the case of the N-soliton solution.
3. DECOUPLING
OF 2-SOLITON
SOLUTION
In this section, we apply the result obtained in the previous section to the case of the 2-soliton solution. Putting N = 2 in (4)-(S) and rearranging the result, we find concrete expressions of two components of U, expresed in the following forms:
D = edA sinh’[(k,
‘A(‘I I = tanh(k,/2)cosh(k?i
- 2sinhk?.t)/D.
(37)
11I(‘I = tanh(k2/2)cosh(k,i
-.- 2sinhk,.r)/D.
(38)
+ k,)i/2
- (sinh k, + sinh k,)t] + eAcosh’[(k,
-
k,)i/2
- (sinhk,
- sinh k,)t].
(39)
Decoupling
of the N-soliton
solution
2241
(40)
in which we have assumed aI = 6, = A.
(41)
Profiles of ui’), ~1’) and ui are plotted in Figs 1, 2. Figure 1 illustrates a collision of two positive solitons in the case where kl is extremely large. Similarly to the previous two cases [2,3], it is observed that ui forms three peaks
(a)
(b)
(d)
Fig. 1. Collision
of two positive solitons; abscissa i; longitudinal axis (1) uj’), (2) uj*), (3) u;; k, = 12, k2 = 1; (a) t = -2 x lo-+‘, (b) I = -8 x 10-5, (c) t = -2 x lo+, (d) t = 0.
K. NARITA
2242
which are unobserved when k, and k2 have comparable temporarily in the interaction, small magnitudes. Figure 2 illustrates a collision of a positive soliton (wavenumber: k,) and a negative soliton (wavenumber: k2) when the conditions k,, / kZ( >> 1 and k, - lk21 << k,, 1k71 are satisfied. In this case, u:” mtttally forms a large positive peak and successively a positive 3-peak at the middle stage of the interaction. Similarly, ~1~) initially forms a large negative peak and successively a negative 3-peak at the middle stage of the interaction. As a result, U, initially subjects to a complex process, and forms a sharp positive peak at the different in magnitudes middle stage of the interaction. When k, and \kz\ are considerably or both small, this behaviour becomes unobserved.
(a)
- 3
(b) -3 3
--I
3
I 3
Fig. 2. (a)-(c)
Decoupling
of the N-soliton
2243
solution
Cd)
3
\
I
II
I I
‘I
\
\
\
II \
::
!
:’
3
I
I
I I
\ ’ \J
JQ, : I
3
JC'
I
‘_’
‘\ ‘._
I 3
I
1
\ \
Fig. 2. Collision (3) ui; k, =6,
I
of a positive soliton and a negative soliton; abscissa i; longitudinal axis (1) ui”, (2) u12’ k2 = -5.5; (a) t = -1.5 x 10e2, (b) t = -7 x 10M3, (c) I = -3 x 10-3, (d) t = -1.2 x lo-‘: (e) t = -4 X 10m4, (f) t = 0.
4.
CONCLUSION
We have proved that the N-soliton solution for the new discrete MKdV equation can be decoupled into N components. These components satisfy the interacting new discrete
2244
K. NARITA
MKdV equation, and approach single soliton solution as t + +@J. We presented expressions of soliton components in the case of iV = 2, and illustrated the results. Acknowledgement--The the study.
author
wishes to thank
Mr. Yoshibumi
Narita
of Wakatake
School
concrete
for his continual
aid to
REFERENCES 1. K. Narita, New discrete modified KdV equation, Prog. Theor. Phys. 86, 817-824 (1991). equation, Chaos, Solirons & Fractals 3, 2. K. Narita, Soliton solution for a highly nonlinear difference-differential 279-283 (1993). difference-differential equation related to the Volterra equation, J. Math. Anal. 3. K. Narita, New nonlinear Appl. 186, 120-131 (1994). 4. K. Narita, Triple-humped soliton solution for a lattice equation related to the discrete KdV equation, J. Phys. A25, L1167-L1168 (1992). 5. T. Yoneyama, Interacting Toda equations, J. Phys. Sot. Japan 55, 753-761 (1986). 6. K Narita, Decoupling of N-soliton solutions for Volterra and Volterra-like circuit equations, Jupan. J. Appl. Phys. 27, 679-683 (1988).