数理解析研究所講究録1020 巻 1997 年 143-159

143

Discretization of Coupled Modified

Ryogo HIROTA$\mathrm{K}\mathrm{d}\mathrm{V}$Equations$(\text{ノ^{}\frac{\backslash }{A}}, \hslash k^{\mathrm{A}_{\nu\triangleright}}|)$

Department of Information and Computer Science,

School of Science and Engineering,
Waseda University,.,::

3-4-1, Ohkubo, Shinjuku-ku, Tokyo 169

ABSTRACT:The coupled modified$\mathrm{K}\mathrm{d}\mathrm{V}$equation$\frac{\partial v:}{\partial t}+3[\sum_{1j,k=}^{N}c_{j,j}kvv_{k}]\frac{\partial v:}{\partial x}+\frac{\partial^{3}v:}{\partial x^{3}}=0$,$i=1,2,$$\cdots,$$N$,

is discretized in the formガ$v^{m+1} \dot{.},-nv^{m}.+\delta*|n[1+\sum C_{j,,,n}kv_{j}v_{k}j,k=1mnm]\Gamma_{n}^{m}[v^{m}.\cdot,-v^{m+}.-1]n+1*,n=10$$\Gamma_{n+1}^{m}=[1+\sum_{1\mathrm{j},k=}^{N}C_{\mathrm{j}},kv^{m}j,nvkm,n]\Gamma m/n[1+\sum_{j,k=1}^{N}c_{j},kv^{m+}\mathrm{j},nk,n1m+1v]$

where$\Gamma_{n}^{m}$,$i=1,2,$$\cdots,$

$N$

,

,

is an auxiliary variable. We integrate the difference equation numerically andcompare the results with exact solutions.

KEYWORDS: solitons,coupled system,exact solutions,difference scheme

144

1

Introduction-

We have proposed ) method of constructing nonlinear partial differential equations with$1$$\mathrm{a}$out destroying. The metho. uses the$\mathrm{d}$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{b}\cdot \mathbb{A}\mathrm{t}\mathrm{y}$ear formalism and follows 3 steps.

$\mathrm{b}4\dot{\mathrm{m}}$First, a given nonlinear partial differential equation is transformed into the bilinear formby the dependent variable transformation. Secondly the bilinear differential equation is

discretized. Thirdly the bilinear difference equation is transformed back into the nonlinear
difference equation by the associated dependent variable transformation.
In this paper we discuss how to discretize the bilinear differential equation taking the mod-ified$\mathrm{K}\mathrm{d}\mathrm{V}$equation as an example. Then we construct a system of coupled modified$\mathrm{K}\mathrm{d}\mathrm{V}$difference-difference equations.
The difference equation obtained is found to be of importance in integrating the coupledmodified2$\mathrm{K}\mathrm{d}\mathrm{V}$equations numerically.

$\mathrm{W}\Leftrightarrow-\mathrm{c}\mathrm{o}\mathrm{m}\mathrm{p}\mathrm{a}\mathrm{r}\overline{\mathrm{e}}$the results with exact solutions.

Discretization of the Modified$\mathrm{K}\mathrm{d}\mathrm{V}$Equation$*$

We have the...m$\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{i},\mathrm{e}\mathrm{d},\mathrm{K}\mathrm{d}\mathrm{V}$equation$\frac{\partial v}{\partial \mathrm{t}}+6v^{2_{\frac{\partial v}{\partial x}+\frac{\partial^{3}v}{\partial x^{3}}}}=0$,(1)which is transformed into the bilinear form 2)$(D_{\iota+}D_{x}3)g\cdot f=0$

,

(2)

145

$D_{x}^{2}f\cdot f=2g2$,(3)

through the dependent variable transformation $v=g/f$ . The bilinear differential equationhas$\mathrm{N}$-soliton solution. We write 2-soliton solution among them explicitly$f=1+a(1,1)e^{2\eta_{1}}+a(1,2)e^{\eta_{1}}+\eta_{2}+a(2,2)e2\eta_{2}+a(1,1,2,2)e2\eta_{1}+2\eta 2$

,

(4)

,: .–

$-$

$g=e^{\eta 1}+e^{\eta 2}+a(1,1,2)e^{2}\eta 1+\eta_{2}+a(1,2,2)e\eta 1+2\eta \mathrm{a}$

,

(5)

where
. $=p:(x-x_{0})-\omega t$ ,(6)$\eta_{*}$$\omega:=p_{:}^{3}$.

,$i=1,2$ ,$\cdot$$a(1,1)= \frac{1}{(2p_{1})^{2}}$,$a(1,2)= \frac{2}{(p_{1}+p2)^{2}}$$a(2,2)= \frac{1}{(2p_{2})^{2}}$

(7)
(8)

$p$

tively.

$x_{0}$$\cdot$$\tau_{\sim}$,$c(1,2)=[ \frac{(p_{1}-p_{2})}{(p_{1}+p_{2})}]^{2}$

where : and

.

,

(9)
(10)

,

(11)

$a(1,1,2)=a(1,1)*c(1,2)$ ,(12)$a(1,2,2)=a(2,2)*_{C}(1,2)$ ,(13)$a(1\cdot, 1,2,2)=a(1,1)*a(2,2)*c(1,2)2$ ,(14)

are parametes related to an amplitude and a position of a soliton respec-

146

A discrete analogue of the modified$\mathrm{K}\mathrm{d}\mathrm{V}$equation is obtained 3)(15)$\frac{\partial v_{n}}{\partial t}+(1+v^{2}n)(v_{n+}1^{-v_{n})=}-10,$

which is reduced to eq.(l) in the limit of small$\epsilon(x=n\epsilon)$.

Equation(15) is transformed into the bilinear form
$D_{t}g_{n}\cdot f_{n}+g_{n+1f1^{-}g_{n-1}f_{n+1}}n-=0$$f_{n+1}fn-1-f_{n}^{2}=g_{n}^{2}$

(16)

,

(17)

,

through the same dependent variable transformation $v_{n}=g_{n}/f_{n}$ . The bilinear differentialdifference equation has -soliton solution. We write 2-soliton solution among them explic$\mathrm{N}$itly$f=1+a(1,1)e^{2\eta_{1}}+a(1,2)e^{\eta 1}+\eta_{2}+a(2,2)e^{2\eta_{2}}+a(\mathrm{I}, 1,2,2)e^{2}\eta_{1}+2\eta 2$$g=e^{\eta_{1}}+e^{\eta 2}+a(1,1,2)^{2\eta\eta_{2}}e1++a(1,2,2)e\eta 1+2\eta 2$

,

,

(18)
(19)

where
$e^{\eta:}=P_{i}^{(0)}n-ne^{-vt}:$.

$\omega:=P_{1}$ $-\dot{1}/P:$

,

,

(20)

$i=1,2$ ,

(21)

,

(22)

$a(1,1)= \frac{P_{1}^{2}}{(P_{1}^{2}-1)^{2}}$$a(1,2)= \frac{P_{1}P_{2}}{(P_{1}P_{2}-1)2}$$a(2,2)= \frac{P_{2}^{2}}{(P_{2}^{2}-1)^{2}}$

,

(23)

,

$c(1,2)=[ \frac{(P_{1}-P_{2})}{(P_{1}*P_{2}-1)}]^{2}$

(24)

,

(25)

147

and $a(1,1,2),$ $a(1,2,2)$ and $a(1,1,2,2)$ have the same forms asand$n_{0}$$\mathrm{e}\mathrm{q}\mathrm{s}.(12)(13)$and (14).

$P_{i}$

are parametes related to an amplitude and a position of a soliton respectively.

We note that the bilinear equations (2) and (3) are invariant under the gauge transformation

where$\alpha$and$\beta$$farrow fe^{\alpha x+\beta t}$

,

(26)

$garrow ge^{\alpha x+\beta t}$

,

(27)

are constants. Also invariant are the$\mathrm{b}\dot{\mathrm{i}}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{a}\mathrm{r}$equations (16) and (17) under

the transformation$\mathrm{B}\mathrm{a}s$$f_{n}arrow f_{n}e^{\alpha n+\rho t}$

,

(28)

$g_{n}arrow g_{n}e^{\alpha n+\rho t}$

.

(29)

ed on these facts we postulate that the discrete-time bilinear equations are invariant

under the gauge transformationwhere$t=m\delta,\mathrm{m}$We ave by$\dot{\mathrm{h}}$$f_{n}^{m}arrow f_{n}^{m}e^{\alpha n}+^{\rho m}$

,

(30)

$g_{n}^{m}arrow g_{n}^{m}e\alpha n+\beta m$

,

(31)

being integers.

$\mathrm{d}\mathrm{e}\mathrm{f}\mathrm{i}\dot{\mathrm{n}}$ition$D_{t}g_{n} \cdot f_{n}=\frac{dg}{dt}f-g\frac{df}{dt}$

.

(32)

Replacing the differential operator by the forward difference operator$\frac{df}{dt}arrow\delta^{-1}[f^{m+1}n-f_{n}^{m}]$

,

(33)

148

$\frac{dg}{d\}arrow\delta^{-1}[g_{n}^{m+1}-g_{n}^{m}]$

(34)

,

we obtain a difference analogue of the bilinear operator$D_{t}g\cdot farrow\delta^{-1}[g_{n}^{m+1}fnm-gnmf^{m+}n1]$

(35)

.

Accordingly we have a discrete-time bilinear equation.

(36)$g_{n}^{m+1}f^{m}n-gnfmnm+1+\delta[gnm+1f_{n}^{m}-1^{-}gn-1f_{n+1}^{m}m]=0$

However this equation is not invariant under the gauge transformation (30) and (31). We
find that following two forms are gauge invariant$g_{n}^{m+1}f_{n}m-gnmf_{n}^{m}+1+\delta[g_{n}^{m_{\dagger 1}}f^{m+1}n-1-gn-1f^{m}m+1n+1]=0$

,

(37)

.

(38)

$g_{n}^{m+1}f_{n}m-g_{n}mf_{n}m+1\delta[+gn+1f^{m}m+1nn-1^{-}g^{m}-1f_{n+1}m+1]=0$

Tsujimoto 4) has pointed out that$\mathrm{e}\mathrm{q}.(37)$is transformed into$\mathrm{e}\mathrm{q}.(38)$by the coordinates

trasformation:$marrow m-n$ ,

(39)

.

(40)

$narrow n$

Hereafter we use$\mathrm{e}\mathrm{q}.(17)$$\backslash \mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}$$\mathrm{e}\mathrm{q}.(37)$

as a difference analogue of$\mathrm{e}\mathrm{q}.(16)$. As a difference analogue of

we use the bilinear eqaution (17) as it is because it does not concerned with the-development. Then we have the time-discretization of the bilinear forms (16) and (17)$g_{n}^{m\dagger 1}f^{m}n-gnmf_{n}m+1+\delta[g^{m}n+1fnm+-11-g_{n}^{m+1}-1f^{m}n+1]=0$$f_{n+1}^{m}f_{n-}^{m}1-(f^{m}n)^{2}=(g^{m}n)^{2}$

.

,

(41)
(42)

149

We have found that$\mathrm{e}\mathrm{q}\mathrm{s}.(41)$and (42) exhibit 2-soliton solution of the following$f=1+a(1,1)e^{2\eta_{1}}+a(1,2)..e^{\eta}.1..+\eta_{2}+$ $a(2,.2)e^{2\eta_{2}}...+a(1,1,2,2)e2.\eta_{1}.+2\eta 2$.

$g=e^{\eta_{1}}+e^{\eta_{2}}+a(1,1,2)e2\eta 1+\eta_{2}+a(1,2,2)e\eta 1+2\eta 2$

,

(43)
(44)

,

where
’

(45)

$.i=1,2$ ,(46)$e^{\eta:}=P^{(n_{0})}:/n-\Omega^{m}:$$\Omega:=\frac{1-\delta/P:}{1-\delta P:}$

and $a(1,1),$ $a(1,2),$ $a(2,2),$ $c(1,2),$ $a(1,1,2),$,$a(1,2,2)\overline{\mathrm{a}}\mathrm{n}\mathrm{d}a(1,1,2,2)$

have the same forms

as before.
Now we transform the bilinear$\mathrm{e}\mathrm{q}\mathrm{s}.(41)$and (42) back into$\mathrm{t}\mathrm{h}\mathrm{e}.\cdot \mathrm{n}\mathrm{o}\mathrm{n}\mathrm{l}- \mathrm{i}\mathrm{n}\mathrm{e},\mathrm{a}.\mathrm{r}\mathrm{d}\mathrm{i}\mathrm{f}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}$,nce-difference

equations. Let$g_{n}^{mm}=v_{n}f^{m}n$Substituting this into$\mathrm{e}\mathrm{q}\mathrm{s}.(41)$

.

(47)

and (42) we find$v_{n}^{m+1}-v^{m}n+ \delta\frac{f_{n+1}^{m}f_{n-1}^{m}+1}{f_{n}^{m+1}f_{n}^{m}}[v^{m}-n+1v^{m}n-+11]=0$

,

(48)

.

(49)

$\frac{f_{n+1}^{m}f_{n-1}^{m}}{(f_{n}^{m})^{2}}=1+(v_{n}^{m})^{2}$

The term
l.h.s of$f_{n+1}^{m}f_{n-1}^{m}-1/f_{n}^{m+1}f_{n}^{m}$$\mathrm{e}\mathrm{q}.(49)$in$\mathrm{e}\mathrm{q}.(48)$

is very$\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{l}\mathrm{i}\mathrm{l}\mathrm{a}\mathrm{r}$to the term

. Let us define the ratio of the two terms by$\Gamma_{n}^{m}$variable$\Gamma_{n}^{m}=\frac{f_{n+1}^{m}f_{n-1}^{m}-1}{f_{n}^{m+1}f_{n}^{m}}\div\frac{f_{n+1}^{m}f_{n-1}^{m}}{(f_{n}^{m})^{2}}$$f_{n+1}^{m}f_{n-1}^{m}/(f_{n}^{m})^{2}$

which iswhich becomes an auxiliary

150$= \frac{f_{n-1}^{m+1}f_{n}^{m}}{f_{n}^{m+1}f_{n-}m1}$

Substituting$\mathrm{e}\mathrm{q}.(49)$and$\Gamma_{n}^{m}$into$\mathrm{e}\mathrm{q}.(48)$

.

(50)

we obtain$v_{n}^{m+1}-v^{m}n+\delta[1+(v^{m})n2]\mathrm{r}_{n}^{m}[v_{n}v_{n}mm++1^{-}-11]=0$

On the other hand taking the ratio of$\Gamma_{n+1}^{m}$and$\Gamma_{n}^{m}$

.

(51)

we obtain$\frac{\Gamma_{n+1}^{m}}{\Gamma_{n}^{m}}=\frac{f_{n}^{m+1}f_{n+}^{m}1}{f_{n+1}^{m+}1f_{n}^{m}}\div\frac{fn-1f^{m}m+1n}{f_{n}^{m+1}f_{n-1}^{m}}$$= \frac{f_{n+1}^{m}f_{n-}m1}{(f_{n}^{m})^{2}}\div\frac{f_{n+1}^{m+1m+1}fn-1}{(f_{n}^{m+\mathrm{i}})^{2}}$$= \frac{1+(v_{n}^{m})^{2}}{1+(v_{n}^{m+1})2}$

.

(52)

Hence the following two equations$v_{n}^{m+1}-v_{n}^{m}+\delta[1+(v_{n}^{m})^{2}]\mathrm{r}nm[v_{n}^{m}-+1v^{m}-1]n=^{0}+1$$\frac{\Gamma_{n+1}^{m}}{\Gamma_{n}^{m}}=\frac{1+(v_{n}^{m})^{2}}{1+(v_{n}^{m+1})2}$

constitute of the modified$\mathrm{K}\mathrm{d}\mathrm{V}$

,

(53)

.

(54)

difference-difference equation.

Here we remark that the nonlinear Schr\"odinger equation(55)$i \frac{\partial}{\partial l}\psi+\frac{\partial^{2}}{\partial x^{2}}\psi_{+}2|\psi|2\psi=0$

has been discretized 5) by the same$\mathrm{f}\mathrm{a}_{1}\mathrm{s}$hion in the form

$i\delta^{-1}[\psi_{n}^{m}+1-\psi^{m}n]-[\psi_{n}^{m+1}+\psi_{n}m]+[1+|\psi n|m2]\Gamma_{n}m[\psi_{n}^{m}+1+\psi^{m}n-1]+1=0$$\Gamma_{n+1}^{m}=\frac{1+|\psi_{n}m|2}{1+|\psi_{n}^{m}+1|^{2}}\Gamma_{n}^{m}$

.

,

(56)
(57)

151

Multi-soliton solutions to the nonlinear Schr\"odinger difference-difference equation are obtained by Tsujimoto 6).

In the limit of samll$\delta \mathrm{e}\mathrm{q}.(56)$is reduced to a differential-difference equation obtained by

Ablowitz and Ladik 7).

3

System of Coupled Modified

Svinolupov 8) has shown that a system of coupled modified,$c_{\mathrm{j},k}$$\mathrm{K}\mathrm{d}\mathrm{V}$equations$i=1,2,$ $\cdots.,N$ ,$\frac{\partial v:}{\partial t}+3[\sum_{\mathrm{j},k=1}^{N}cj,kv_{\mathrm{j}}v_{k}]\frac{\partial v_{i}}{\partial x}+\frac{\partial^{3}v_{i}}{\partial x^{3}}=0$

where the coefficients

Equation$\mathrm{K}\mathrm{d}\mathrm{V}$(58)

are arbitrary constants, possesses an infinite series of general-ized symmetries and local conservation laws. Iwao and the author 9) have shown that it

p\‘Osseses multi-soliton solutions expressed by pfaffians under the conditions on the coefficients,namely$c_{\mathrm{j}\mathrm{j}}=0$

and $c_{jk}=c_{kj}$ .

Habibullin and Svinolupov 10) have found that the boundary conditions imposed at thepoint$\mathrm{x}=0$

are compatible with the integrability ofa system of the coupled modified
conditions$v_{n}^{(\cdot)}=0$$\mathrm{K}\mathrm{d}\mathrm{V}$at $n=0,$ $i=1,2,$$\mathrm{e}\mathrm{q}.(58)$. The author has shown 11) thatdifferential-difference equations under the boundary$\cdots,$

$N$

.

)

$\frac{\partial v_{n}^{(:)}}{\partial t}-[\sum_{kj,=1}^{N}cj,kvv_{n}^{k}]j[nv_{n}^{(}*\cdot$$+1-v_{n}(i)-1\mathrm{J}=0$,$i=.\dot{1},$$2,$$\cdot\cdot$$.,\cdot\dot{N}$(59)exhibits exact solutions expressed by pfaffians of the following form
$v_{n}^{(:)}=g^{()}n./*fn$’$i=1,2,$$\cdots,$

$N$

,

(60)

152

where
$f_{2n}=\mathrm{p}\mathrm{f}(0,1,2, \cdots, 2n-1)$$g_{2n}^{(:)}=\mathrm{p}\mathrm{f}(p:, 1,2, \cdots, 2n-1)$$f_{2n+1}=\mathrm{p}\mathrm{f}(1,2,3, \cdots, 2n)$

(61)

,

(62)

,

(63)

,

$g_{2n+}^{(:)}1=\mathrm{P}^{\mathrm{f}}(p:,..0,1, \cdots, 2n)$

(64)

,

$r$

where the elements of the pfaffians are defined as follows$\mathrm{p}\mathrm{f}(p:, m)=h(:)(t, m)$

(65)

,

$\frac{1}{2}[\mathrm{p}\mathrm{f}(l, m+1)-\mathrm{p}\mathrm{f}(l+1, m)]=\sum^{N}j,k=1Cj,kh(j)(\mathrm{t}, l)h^{()}k(t, m)$for $i=1,2,$$\cdots,$$N$$m)+\mathrm{p}\mathrm{f}(l, m+1)$

and for non-negative integers

(66)

(67)

$m+)$ ,$\frac{d}{dt}\mathrm{p}\mathrm{f}(p:, m)=_{\mathrm{P}^{\mathrm{f}}(1}p_{i},$$\frac{d}{dl}\mathrm{p}\mathrm{f}(l, m)=\mathrm{P}^{\mathrm{f}(l1}+,$

,

$l$

(68)

,

and

$m$

, and $h^{(:)}(t, m)$ are arbitrary

functions of and satisfy the relations$t$$\frac{d}{dt}h^{(:)}(t, m)=h^{(:)}(\mathrm{t}, m+1)$

.

(69)

Following the same procedure developed in the previous section we transform the systemof coupled modified$\mathrm{K}\mathrm{d}\mathrm{V}$equations$\frac{\partial v:}{\partial t}+3[\sum_{1j,k=}c\mathrm{j},kvjvk]N\frac{\partial v:}{\partial x}+\frac{\partial^{3}v_{i}}{\partial x^{3}}=0$,$i=1,2,$$\cdots,$

$N$

,

(70)

153

into the bilinear forms$(D_{t}-D^{3})xg:\cdot f=0,$,$D_{x}^{2}f \cdot f=\sum_{j,k=1}^{N}C_{j,g\mathrm{j}}kg_{k}$$\mathrm{t}\mathrm{h}\dot{\mathrm{r}}\mathrm{o}\mathrm{u}\mathrm{g}\mathrm{h}$$\cdot i=1,2,$ $\cdots,$ $N$

(71)

,

(72)

,

the dependent variable transformation$v:=g:/f$ ,We have a difference analogue of$D_{\iota}g_{n}^{(}:)$$\mathrm{e}\mathrm{q}\mathrm{s}.(71)$$i=1,2,$$\cdots$

(73)

, N.

and (72). $f_{n}-gn+1f_{n}(:)(*\cdot)-1+g_{n-1}fn+1=0$,$f_{n+1}f_{n-1^{-}}f_{n}2^{\cdot}.

\sum_{1}=.\cdot cj,kg_{n}g----j,k=N(j)(k)n$

(74)

.

(75)

Following the procedure described in the previous section we obtain a time-discretizationof the bilinear forms$g_{:}^{m+1},nf_{n}m-g_{:},nf^{m}m+1sn+[g:,mn+^{f^{m+}-}1n-11g^{m+}:,n-1fnm1+1]=^{0}$$f_{n+1}^{m}f^{m}n-1^{-(f_{n})^{2}}m= \mathrm{j},k=\sum_{1}^{N}c_{j,g,n}kjmg_{k}^{m},n$

,

(76)
(77)

’

which are transformed back into the coupled nonlinear difference-difference equations$v^{m+1}:,n-v^{m}:,n+ \delta[1+\mathrm{j},k\sum_{1=}\mathrm{c}_{\mathrm{j},,nk_{)}}kv_{j}v]mm\mathrm{r}_{n}m[nn1-v_{:,1}^{m+}-]v_{i,+}=0Nnn1$$\Gamma_{n+1}^{m}=[1+\sum_{j)k=1}^{N}C_{\mathrm{j},k}vj,n]mm\Gamma_{n}^{m}/v_{k,n}[1+\sum_{kj)=1}^{N}c_{j,knk^{+1}}v_{\mathrm{j}},v,]m+1mn$’,$i=1,2,$$\cdots,$

$N$

,

(78)
(79)

154

through the dependent variable transformation$g^{m}:,n=v^{m}i,nf.\cdot mn|$$i=1,2,$’$\Gamma_{n}^{m}=\frac{f_{n-1f^{m}}^{m+1}n}{f_{n}^{m+1}f_{n-1}^{m}}$We have found12) that multi-soliton$\cdots,$

$N$

,

(80)

.

(81)
. to the coupl.ed modified$\mathrm{s}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}}$$\mathrm{K}\mathrm{d}\mathrm{V}\vee$diffe-rence-difference equations are expressed by pfaffians,which have the same structure as those 9) of
the coupled modified$\mathrm{K}\mathrm{d}\mathrm{V}$differential equations.

Let $N=2$ and $c_{11}=c_{22}=0,$ $c_{12}=c_{21}=1/2$ . in

simplest form of the coupled modified$\mathrm{K}\mathrm{d}\mathrm{V}$$\mathrm{e}\mathrm{q}\mathrm{s}.(78)$and (79). Then we have thedifference-difference equations$v_{1,n}^{m+1}-v1,nm+\delta[1+v_{1n1}^{mm}v_{2,n}]\mathrm{r}_{n}m[v_{1},-mn+1v_{1}^{m+1},-1]n=0$

,

(82)

,

(83)

$v_{2,n}^{m+1}-v_{2}^{\dot{m}},\overline{n}+\delta[1+v_{1,n}^{\overline{m}m}v_{2,n}]’\Gamma_{n}m[v_{2,+}^{m}-n1v^{\overline{m}+1}-1]2,n=0$,$\Gamma_{n+1}^{m}=[1+v_{1_{1}n}^{mm}v2,n]\mathrm{r}_{n}^{m}/[1+v_{1,n}^{m+}v_{2,n}^{m+}11]$(84)

which exhibit the following 1+1 soliton solution.
$f=1+a_{12}\exp \mathfrak{k}\eta 1+\eta 2]$

(85)

(86)

$- g_{1}=\mathrm{e}\mathrm{x}.\mathrm{p}’[\eta 1]-,--$$g_{2}=\exp[\eta_{2}]$

,

,

(87)

where
$\exp[\eta:]=P^{n-}:n:/\Omega_{:}^{m}$

,

(88)

155

$\Omega:=\frac{1-\delta/P:}{1-\delta P_{1}}.$’$i=1,2$ ,

(89)

.

(90)

$a_{12}= \frac{P_{1}P_{2}}{(P_{1}P_{2}-1)^{2}}$This expression of the 1+1 soliton solution shows that a soliton $v_{1}=g_{1}/f$ is damping andanother soliton $v_{2}=g_{2}/f$ is growing as time $m$ increases.
The difference-difference equations (82),(83) and (84) are integrated numerically using theinitial conditions of the 1+1 soliton solution with the following parameters: $\delta=1/5$ ,$2$,$P_{2}=3$,$n_{1}=12$,$n_{2}=7$$P_{1}=$. The numerical results agree exactly with the theoreticalones except round-off errors (see Fig.1).

References
1) R. Hirota, S. Tsujimoto and T. Imai: “Difference Scheme of Soliton Equations”:
inFuture Directions of Nonlinear Dynamics in Physical and Biological Systems , ed. P.
L. Christiansen, J. C. Eilbeck and R. D. Parmentier (Plenum, New York, 1993) p.7.
2) Ryogo Hirota: J. Phys. Soc. Jpn. 33(1972) 1456.
3) Ryogo Hirota:J.Phys.Soc.Jpn. 35(1973)289.
4) Satoshi Tujimoto: private communication.

1565) Kazuo Horiuchi,Shin’ichi Oishi,Ryogo Hirota and Satoshi Tsujimoto: ”High Preci-sion Simulation of Soliton Problems” in Ultrafast and Ultra-Parallel Optoelectronics ,ed.T.Sueta and T.Okoshi (John Wiley&Sons, and Ohmsha,Tokyo, $1995$ )6) Satoshi Tsujimoto: ”Ph.
7)$\mathrm{M}.\mathrm{J}$.Ablowitz and$\mathrm{J}.\mathrm{F}$.

$\mathrm{D}$Thesis, Waseda University,1997”.

. Phys.16(1975)598.

$\mathrm{L}\mathrm{a}\mathrm{d}\mathrm{i}\mathrm{k}:\mathrm{J}.\mathrm{M}\mathrm{a}\mathrm{t}\mathrm{h}$8) S. I. Svinolupov: Commun. Math. Phys. 143 (1992)559.

9) Masataka Iwao and Ryogo Hirota: J.Phys.Soc.Jpn. 66(1997)577.

10) I. T. Habibullin and S. I. Svinolupov: Physica11) R. Hirota: submitted to J.Phys.Soc.Jpn.
12) R. Hirota: in preparation.

$\mathrm{D}87$(1995)134.

$\mathrm{p}.445$

.

157

Figure Captions

Fig.1 Numerical integration of$\mathrm{e}\mathrm{q}\mathrm{s}.(82),(83)$soliton solultion expressed by$\mathrm{e}\mathrm{q}\mathrm{s}.(85)-(90)$other soliton$g_{2}/f$and (84) with the initial condition of the 1+1. The soliton $v_{1}=g_{1}/f$ is damping and theis growing as time $m$ increases from 1 to 50. Numerical results agree

exactly with the theoretical ones except round-off errors.

158$\overline{\mathrm{A}}_{17}\backslash .1$

159$\overline{\}t/}?$.

$1\mathrm{c}_{\mathrm{M}}\mathfrak{U}^{\backslash },$$\eta_{\mathrm{K}e}$A-