数理解析研究所講究録
第 1800 巻 2012 年 130-144130

多成分変形非線形シュレーディンガー方程式の多重ソリトン解Multisoliton solution of a multi-component modified
nonlinear Schr\"odinger equation
松野 好雅 (Yoshimasa Matsuno)
山口大学大学院理工学研究科
Division of Applied Mathematical Science
Graduate School of Science and Engineering

Yamaguchi University

Abstract
We develop a systematic method for constructing the bright N-soliton solutionof a multi-component modified nonlinear Schr\"odinger equation. We present thetwo different expressions of the solution both of which are expressed as a ratio of
determinants. We find a simple relation between them by employing the propertiesof the Cauchy matrix. Last, we propose a (2 1)-dimensional nonlocal modified
nonlinear Schr\"odinger equation arising from the multi-component system as thenumber of dependent variable tends to infinity and then obtain its bright N-soliton
solution. In this paper, we describe only the main results. The detail has been
published in Matsuno (2011) [1].
$+$1. Introduction

We consider the following multi-component system of nonlinear PDEs which is ahybrid of the coupled nonlinear Schr\"odinger (NLS) equation and coupled derivative
NLS equation$iq_{j,t}+q_{j,xx}+\mu(\sum_{k=1}^{n}|q_{k}|^{2})q_{j}+i\gamma[(\sum_{k=1}^{n}|q_{k}|^{2})q_{j}]_{x}=0$,$(j=1,2, \ldots, n),$where $q_{j}=q_{j}(x, t)(j=1,2, \ldots, n)$ are complex-valued functions of andand are real constants, is an arbitrary positive integer and subscripts
appended to denote partial differentiations.
$x$

.

$n$

$\gamma$$t,$$(1.1)$$\mu(\geq 0)$
$x$

and

$q_{j}$

Integrability of Eq. (1.1): Hisakado and Wadati (1995) [2]eSpecial cases:$\gamma=0$ : NLS equation, Zakharov and Shabat (1972) [3]1 $n=1,$: Derivative NLS equation, Kaup and Newell (1978) [4]2 $n=1,$ $\mu=0,$$\gamma=0$ : Manakov system, Manakov (1974) [5]3 $n=2,$: A model equation describing the propagation of short
4 $n=2,$
pulses in birefringent optical fiber, Hisakado, Iizuka and Wadati (1994) [6]5 Bright N-soliton solution of Eq. (1.1) with $n=2$ : Matsuno (2011) [7]$)$$\mu\neq 0,$$\gamma\neq 0$

$)$

$)$

$\mu\neq 0,$$)$$\mu\neq 0,$ $\gamma\neq 0$

$)$

$t$

131
2. Bilinearization and bright N-soliton solution

2. 1. Bilinearization
We first apply the gauge transformations$q_{j}=u_{j} \exp[-\frac{i\gamma}{2}\int_{-\infty}^{x}\sum_{k=1}^{n}|u_{k}|^{2}dx]$

,

(2.1)

$(j=1,2, \ldots, n)$ ,to the system (1.1) subjected to the the boundary conditions $q_{j}arrow 0,$ $u_{j}arrow 0$$(j=1,2, \ldots, n)$ as $|x|arrow\infty$ , where $u_{j}=u_{j}(x, t)(j=1,2, \ldots, n)$ are complexvalued functions of and . Then, we obtain the system of nonlinear PDEs for
$x$

$t$

$u_{j}$$iu_{j,t}+u_{j,xx}+\mu(\sum_{k=1}^{n}|u_{k}|^{2})u_{j}+i\gamma(\sum_{k=1}^{n}u_{k}^{*}u_{k,x})u_{j}=0$

where the asterisk appended to,$(j=1,2, \ldots, n),$$(2.2)$

denotes complex conjugate.

$u_{k}$

Proposition 2.1. By means of the dependent variable tmnsformations$u_{j}= \frac{g_{j}}{f}$,$(j=1,2, \ldots, n)$ ,(2.3)

the system of nonlinear PDEs (2.2) can be decoupled into the following system ofbilinear equations for $f$ and$g_{j}$$(iD_{t}+D_{x}^{2})g_{j}\cdot f=0$,$(j=1,2, \ldots, n)$$D_{x}f \cdot f^{*}=\frac{i\gamma}{2}\sum_{k=1}^{n}|g_{k}|^{2}$

,

(2.4)

,

$D_{x}^{2}f \cdot f^{*}=\mu\sum_{k=1}^{n}|g_{k}|^{2}+\frac{i\gamma}{2}\sum_{k=1}^{n}D_{x}g_{k}\cdot g_{k}^{*}$

(2.5)

.

(2.6)

Here, $f=f(x, t)$ and $g_{j}=g_{j}(x, t)(j=1,2, \ldots, n)$ are complex-valued functions ofand and the bilinear opemtors
andare defined by

$x$

$t$

$D_{x}$$D_{t}$$D_{x}^{m}D_{t}^{n}f \cdot g=(\frac{\partial}{\partial x}-\frac{\partial}{\partial x’})^{m}(\frac{\partial}{\partial t}-\frac{\partial}{\partial t’})^{n}f(x, t)g(x’, t’)|_{x’=x,t’=t}$where $m$ and

$n$

,

(2.7)

are nonnegative integers.

Proof. Substituting (2.3) into (2.2) and rewriting the resultant equation in termsof the. bilinear operators, equations (2.2) can be rewritten as$\frac{1}{f^{2}}(iD_{t}g_{j}\cdot f+D_{x}^{2}g_{j}\cdot f)+\frac{g_{j}}{f^{3}f^{*}}(-f^{*}D_{x}^{2}f\cdot f+\mu f\sum_{k=1}^{n}|g_{k}|^{2}+i\gamma\sum_{k=1}^{n}g_{k}^{*}D_{x}g_{k}\cdot f)=0$,

132$(j=1,2, \ldots, n)$

.

(2.8)

Insert the identity$f^{*}D_{x}^{2}f\cdot f=fD_{x}^{2}f\cdot f^{*}-2f_{x}D_{x}f\cdot f^{*}+f(D_{x}f\cdot f^{*})_{x}$

(2.9)

,

into the second term on the left-hand side of (2.8). Then, equations (2.8) become$\frac{1}{f^{2}}(iD_{t}g_{j}\cdot f+D_{x}^{2}g_{j}\cdot f)+\frac{g_{j}}{f^{3}f^{*}}[f\{-D_{x}^{2}f\cdot f^{*}+\mu\sum_{k=1}^{n}|g_{k}|^{2}-(D_{x}f\cdot f^{*})_{x}+i\gamma\sum_{k=1}^{n}g_{k}^{*}g_{k,x}\}$$+f_{x} \{2D_{x}f\cdot f^{*}-i\gamma\sum_{k=1}^{n}|g_{k}|^{2}\}]=0$,$(j=1,2, \ldots, n)$.

(2.10)

As easily confirmed by a direct calculation, the left-hand side of (2.10) becomeszero by virtue of equations $(2.4)-(2.6)$ .

$\square$

It now follows from (2.3) and (2.5) that$- \frac{i\gamma}{2}\sum_{k=1}^{n}|u_{k}|^{2}=\frac{\partial}{\partial x}\ln\frac{f^{*}}{f}$

(211)

,

which, substituted into (2.1), yields the solution of the system (1.1) in the form
$q_{j}= \frac{g_{j}f^{*}}{f^{2}}$,$(j=1,2, \ldots, n)$

.

(212)

Note that for the n-component NLS equation (the system (1.1) with $\gamma=0$ ), thesolution (2.12) simplifies to $q_{j}=g_{j}/f$ . Indeed, if $\gamma=0$ , then the bilinear equation(2.5) reduces to $D_{x}f\cdot f^{*}=0$ . Thus, the ratio $f^{*}/f$ turns out to be an arbitrary
function of which can be set to 1 under appropriate boundary condition.
$t$2.2. Bright N-soliton solution
We now state the main result:
Theorem 2.1. The bright N-soliton solution of the system of bilinear equations(2.4)-(2.6) is given by the determinants and $g_{j}(j=1,2, \ldots, n)$ where$f$$f=|\begin{array}{ll}A I-I B\end{array}|$,$g_{j}=|\begin{array}{lll}A I z^{T}-I B 0^{T}0 -a_{j}^{*} 0\end{array}|$,$(j=1,2, \ldots, n)$.

(2.13)

and are N-component row vectors
Here, $A,$ $B$ and I are $N\cross N$ matrtces anddefined below and the symbol denotes the tmnspose:$z,$ $a_{j}$

$0$

$T$

$A=(a_{jk})_{1\leq j,k\leq N}$,$a_{jk}= \frac{1}{2}\frac{z_{j^{Z_{k}^{*}}}}{p_{j}+p_{k}^{*}}$,$z_{j}=\exp(p_{j^{X}}+ip_{j}^{2}t)$,$(2.14a)$133,$B=(b_{jk})_{1\leq j,k\leq N}$$b_{jk}= \frac{(\mu+i\gamma p_{k})c_{jk}}{p_{j}+p_{k}}*$$I=(\delta_{jk})_{1\leq j,k\leq N}$$z=(z_{1}, z_{2}, \ldots, z_{N})$,,$c_{jk}= \sum_{s=1}^{n}\alpha_{sj}\alpha_{sk}^{*}$,$(2.14b)$, : $N\cross N$ unit matrix,$(2.14c)$,$(2.14d)$$a_{j}=(\alpha_{j1)}\alpha_{j2}, \ldots, \alpha_{jN})$$0=(0,0, \ldots, 0)$ .

The above bright N-soliton solution is characterized by $(n+1)N$ complex parameters $p_{j}(j=1,2, \ldots, n)$ and $\alpha_{sj}(s=1,2, .., n;j=1,2, \ldots, N)$ . The former
parameters determine the amplitude and velocity of the solitons whereas the latter ones determine the polarizations and the envelope phases of the solitons.
To simplify the proof of theorem 2.1, the following observation is useful:
Proposition 2.2. If we introduce the gauge tmnsformations$f=f$ ,

$g_{j}= \exp[i\{\frac{\mu}{\gamma}\tilde{x}+(\frac{\mu}{\gamma})^{2}\tilde{t}\}]\tilde{g}_{j}$$x= \tilde{x}+\frac{2\mu}{\gamma}\tilde{t}$,$(j=1,2, \ldots, n)$ ,,$t=\tilde{t}$,$(2.15a)$$(2.15b)$then the bilinear equations (2.4)-(2.6) recast to,$( iD_{\overline{t}}+D\frac{2}{x})\tilde{g}_{j}\cdot f=0$

$D_{\overline{x}}f$.

$(j=1,2, \ldots, n)$ ,

(216)

,

(2.17)$\tilde{f}^{*}=\frac{i\gamma}{2}\sum_{k=1}^{n}|\tilde{g}_{k}|^{2}$$D \frac{2}{x}f\cdot\tilde{f}^{*}=\frac{i\gamma}{2}\sum_{k=1}^{n}D_{\overline{x}}\tilde{g}_{k}\cdot\tilde{g};$,(2.18)

respectively.

Thus, the form of equations (2.4) and (2.5) is unchanged whereas equation (2.6)
becomes a simplified equation with $\mu=0$ . Consequently, the proof of the Nsoliton solution may be performed for the corresponding solution with $\mu=0$ .
Hence, in the analysis developed in the following sections, we put $\mu=0$ without
loss of generality.

3. Notation and some basic formulas for determinants
In this section, we first introduce the notation for matrices and then provide somebasic formulas for determinants.

134
3.1. Notation
We define the following matrices associated with the N-soliton solution (2.13) with
(2.14):$D=(\begin{array}{ll}A I-I B\end{array})$,$D(a^{*};b)=(\begin{array}{lll}A I 0^{T}-I B b^{T}0 a^{*} 0\end{array})$

(3.1)

,

$D(a^{*};z)=(\begin{array}{lll}A I z^{T}-I B 0^{T}0 a^{*} 0\end{array})$$D(z^{*};z)=(\begin{array}{lll}A I z^{T}-I B 0^{T}z^{*} 0 0\end{array})$

,

.

(3.2)

in the above expressions. The
Note the position of the vectors , b, z andmatrices which include more than two vectors will be introduced as well. For$z^{*}$$a^{*}$

example,$D(a^{*}, z^{*};b, z)=(\begin{array}{llll}A I 0 z^{T}-I B b^{T} 0^{T}0 a^{*} 0 0z^{*} 0 0 0\end{array})$,$D(a^{*}, z^{*};z, z’)=(\begin{array}{llll}A I z^{T} z^{\prime T}-I B 0^{T} 0^{T}0 a^{*} 0 0z^{*} 0 0 0\end{array})$

.

(3.3)

3.2. Formulas for determinants
Let $A=(a_{jk})_{1\leq j,k\leq M}$ be an $M\cross M$ matrix with $M$ being an arbitrary positive
andand $b_{j}(j=1,2, \ldots, n)$ be
integer,be the cofactor of the element
M-component row vectors. The following well-known formulas are used frequently
in our analysis:$A_{jk}$$a_{jk}$$a,$ $b,$ $a_{j}$

(3.4)

,

$\frac{\partial}{\partial x}|A|=\sum_{j,k=1}^{M}\frac{\partial a_{jk}}{\partial x}A_{jk}$$|\begin{array}{ll}A a^{T}b z\end{array}|=|A|z-\sum_{j,k=1}^{M}A_{jk}a_{j}b_{k}$

(3.5)

,

$|A(a_{1}, a_{2};b_{1}, b_{2})||A|=|A(a_{1};b_{1})||A(a_{2};b_{2})|-|A(a_{1};b2)||A(a_{2};b_{1})|$

.

(3.6)

The formula (3.4) is the differentiation rule of the determinant and (3.5) is the
expansion formula for a bordered determinant with respect to the last row andlast column. The formula (3.6) is Jacobi’s identity.
The following two formulas may not be popular but are very important in our
analysis. In particular, formula (3.7) gives rise to the expansion formulas for the135bordered determinant (see (3.9) and (3.10) below):$|A(a_{1};b_{1})|$$|A(a_{1};b_{n})|$:$|A(a_{1}, \ldots, a_{n};b_{1}, \ldots, b_{n})||A|^{n-1}=$

$..$

:.

.

,

$(3.7)$$(n\geq 2),$$|A(a_{n};b_{n})|$$|A(a_{\eta};b_{1})|$$|A+ \epsilon\sum_{s=1}^{n}b_{s}^{T}a_{s}|=|A|+\sum_{m=1}^{n’}(-\epsilon)^{m}\sum_{1\leq s_{1}<\ldots<s_{m}\leq n}|A(a_{s_{1}}, \ldots, a_{s_{m}};b_{s_{1}}, \ldots, b_{s_{m}})|$

.

(3.8)

$=|A|+ \sum_{m=1}^{n’}\frac{(-\epsilon)^{m}}{m!}\sum_{s_{1},\ldots,s_{m}=1}^{n}|A(a_{s_{1}}, \ldots, a_{s_{m}};b_{s_{1}}, \ldots, b_{s_{m}})|$

is an arbitrary parameter, the notationon the left-hand side of(3.8) represents an $M\cross M$ matrix whose $(j, k)$ element is given by
and$n’= \min(n, M)$ . The formula (3.7) is a variant of the Sylvester theorem in thetheory of determinants.
Suppose that $|A|\neq 0$ . Expanding the determinant on the right-hand side of(3.7) with respect to the first column and using (3.7) with replaced by $n-1$ , weHere,$b_{s}^{T}a_{s}$$\epsilon$$\beta_{sj}\alpha_{sk}$$n$then obtain an expansion formula$|A(a_{1}, \ldots, a_{n};b_{1}, \ldots, b_{n})|$$= \frac{1}{|A|}\sum_{j=1}^{n}(-1)^{j-1}|A(a_{j}, b_{1})||A(a_{1}, \ldots, a_{j-1}, a_{j+1}, \ldots, a_{n};b_{2}, \ldots, b_{n})|$

.

(3.9)

Similarly, the expansion with respect to the first row gives$|A(a_{1}, \ldots,$下$;b_{1}, \ldots, b_{n})|$$= \frac{1}{|A|}\sum_{j=1}^{n}(-1)^{j-1}|A(a_{1}, b_{j})||A(a_{2}, \ldots,$

砺 $;b_{1}, \ldots, b_{j-1}, b_{j+1}, \ldots, b_{n})|$.

(3.10)

4. Proof of the bright N-soliton solution

4.1. Formulas
In terms of the notation introduced in section 3.1 (see (3.1) and (3.2)),
are written in the form
$f=|D|$ ,The differentiation rules offollowing formulas:$g_{j}=-|D(a_{j}^{*};z)|$

$f$

and

$g_{j}$,$(j=1,2, \ldots, n)$with respect to

$t$

and

$x$

.

$f$

and

$g_{j}$(4.1)

are given by the

136Lemma 4.1.
$f_{t}=- \frac{i}{2}\{|D(z^{*};z_{x})|-|D(z_{x}^{*};z)|\}$$f_{x}=- \frac{1}{2}|D(z^{*};z)|$

(4.2)

,

(4.3)

,

$f_{xx}=- \frac{1}{2}\{|D(z^{*};z_{x})|+|D(z_{x}^{*};z)|\}$

(4.4)

,

$g_{j,t}=-|D( a_{j}^{*};z_{t})|+\frac{i}{2}|D(a_{j}^{*}, z^{*};z, z_{x})|$$g_{j,x}=-|D(a_{j}^{*};z_{x})|$

(4.5)

,

(4.6)

,

$g_{j,xx}=-|D( a_{j}^{*};z_{xx})|+\frac{1}{2}|D(a_{j}^{*}, z^{*};z_{x}, z)|$Here,$z_{t},$$ip_{N}^{2}z_{N}),$$z_{x}$

and$z_{xx}$are N-component row vectors given by$z_{x}=(p_{1}z_{1},p_{2}z_{2}, \ldots, p_{N}z_{N})$

and

.

(4.7)

$z_{t}=(ip_{1}^{2}z_{1},$ $ip_{2}^{2}z_{2},$$z_{xx}=(p_{1}^{2}z_{1},p_{2}^{2}z_{2}, \ldots,p_{N}^{2}z_{N})$$\ldots$,, respectively.

Lemma 4.2.
$f^{*}=|\overline{D}|$,$\overline{D}\equiv(\begin{array}{lll}A I-I B -i\gamma C\end{array})$$f_{x}^{*}=- \frac{1}{2}|D(z^{*};z)|-$$g_{j}^{*}=|\overline{D}(z^{*};a_{j})|$

(4.8)

,

(4.9)

,

.

(410)

Lemma 4.3.

$| \overline{D}|=|D|+\frac{1}{2}|D(z^{*};\tilde{z})|$$|D(b_{k}^{*};\tilde{z})|=|\overline{D}(a_{k}^{*};z)|$(4.11)

,

(412)

,

$| \overline{D}(a_{k}^{*};b_{k})=-|D(b_{k}^{*};a_{k})|-\frac{1}{2}|D(b_{k}^{*}, z^{*};a_{k},\tilde{z})|$$| \overline{D}(a_{k}^{*};z_{x})|=|D(b_{k}^{*};z)+\frac{1}{2}|D(b_{k}^{*}, z^{*};z,\tilde{z})|$$|D( z^{*};z)|=2i\gamma\sum_{k=1}^{n}|D(b_{k}^{*};a_{k})|$

,

.

,

are N-component row vectors given respectively byand.
..., $z_{N}/p_{N})$ andwhere$\tilde{z}$$b_{k}$$b_{k}=(\alpha_{k1}p_{1}^{*}, \alpha_{k2}p_{2}^{*}, \ldots, \alpha_{kN}p_{N}^{*})$(414)(4.15),$| \overline{D}(z^{*};z)|=-2i\gamma\sum_{k=1}^{n}|D^{-}(a_{k}^{*};b_{k})|$(413)(4.16)$\tilde{z}=(z_{1}/p_{1},$ $z_{2}/p_{2}$,137
4.2. Pmof of (2.4)

Let$P_{1}$

be
.

(417)

$P_{1}=(iD_{t}+D_{x}^{2})g_{j}\cdot f$

Substituting $(4.1)-(4.7)$ into (4.17),$P_{1}$

becomes$P_{1}=-|D(a_{j}^{*}, z^{*};z, z_{x})||D|+|D(a_{j}^{*};z)||D(z^{*};z_{x})|-|D(a_{j}^{*};z_{x})||D(z^{*};z)|$$-\{i|D(a_{j}^{*}; z_{t})|+|D(a_{j}^{*}; z_{xx})|\}$

Referring to Jacobi.

(4.18)

identity (3.6) and the fundamental formula $\alpha|D(a;b_{1})|+$$\beta|D(a;b_{2})|=|D(a;\alpha b_{1}+\beta b_{2})|$
simplifies to $P_{1}=-|D(a_{j}^{*}$ ; iz $t+$
. Since $iz_{t}+z_{xx}=0$ by $(2.14a)$ , the last column of the determinant consists
only of zero elements, implying that $P_{1}=0$ .
口$s$$(\alpha, \beta\in \mathbb{C}),$$P_{1}$$z_{xx})|$4.3. Proof of (2.5)

The equation to be proved is $P_{2}=0$ , where$P_{2}=D_{x}f$.

$f^{*}- \frac{i\gamma}{2}\sum_{k=1}^{n}|g_{k}|^{2}$

Substituting (4.1), (4.3) and $(4.8)-(4.10)$ into (4.19),.

$P_{2}$(419)

becomes.

$P_{2}=- \frac{1}{2}|D^{-}||D(z^{*};z)|+\frac{1}{2}|D||\overline{D}(z^{*};z)|+\frac{i\gamma}{2}\sum_{k=1}^{n}|D(a_{k}^{*};z)||\overline{D}(z^{*};a_{k})|$(4.20)

Further simplication is possible with use of (4.11), (4.15) and (4.16) with (4.13),
giving rise to$P_{2}= \frac{i\gamma}{2}\sum_{k=1}^{n}(-|D(b_{k}^{*};a_{k})||D(z^{*};\tilde{z})|+|D(b_{k}^{*}, z^{*};a_{k},\tilde{z})||D|+|D(a_{k}^{*};z)||\overline{D}(z^{*};a_{k})|)$.

(4.21)

Applying Jacobi’s identity (3.6) to the middle term and replacing
the right-hand side of (4.12) in the resultant expression,
reduces toby$|D(b_{k}^{*};\tilde{z})|$$P_{2}$$P_{2}= \frac{i\gamma}{2}\sum_{k=1}^{n}(-|\overline{D}(a_{k}^{*};z)||D(z^{*};a_{k})|+|D(a_{k}^{*};z)||\overline{D}(z^{*};a_{k})|)$.

(4.22)

It now follows from (3.8) that$| \overline{D}(a_{k}^{*};z)|=|D(a_{k}^{*};z)|+\sum_{m=1}^{n’’}\frac{(i\gamma)^{m}}{m!}\sum_{k_{1},\ldots,k_{m}=1}^{n}|D(a_{k}^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};z, a_{k_{1}}, \ldots, a_{k_{m}})|$,$(4.23a)$138$| \overline{D}(z^{*};a_{k})|=|D(z^{*};a_{k})|+\sum_{m=1}^{n’’}\frac{(i\gamma)^{m}}{m!}\sum_{k_{1},\ldots,k_{m}=1}^{n}|D(z^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k}, a_{k_{1}}, \ldots, a_{k_{m}})|$,$(4.23b)$where $n”= \min(n-1, N-1)$ . Referring to the expansion formulas (3.9) and(3.10), one has$|D(a_{k}^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};z, a_{k_{1}}, \ldots, a_{k_{m}})|=|D|^{-1}|D(a_{k}^{*};z)||D(a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k_{1}}, \ldots, a_{k_{m}})|$$+|D|^{-1} \sum_{l=1}^{m}(-1)^{l}|D(a_{k_{l}}^{*};z)||D(a_{k}^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{l-1}}^{*}, a_{k_{l+1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k_{1}}, \ldots, a_{k_{m}})|,$$(4.24a)$$|D(z^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k}, a_{k_{1}}, \ldots, a_{k_{m}})|=|D|^{-1}|D(z^{*};a_{k})||D(a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k_{1}}, \ldots, a_{k_{m}})|$$+|D|^{-1} \sum_{l=1}^{m}(-1)^{l}|D(z^{*};a_{k_{l}})||D(a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k}, a_{k_{1}}, \ldots, a_{k_{l-1}}, a_{k_{l+1}}, \ldots, a_{k_{m}})|$.

$(4.24b)$

By introducing (4.23) into (4.22) and then using (4.24),$P_{2}$

takes the form

$P_{2}= \frac{i\gamma}{2|D|}\sum_{m=1}^{n’’}\frac{(i\gamma)^{m}}{m!}\sum_{l=1}^{m}(-1)^{l}\cross$$\cross\sum_{k,k_{1},\ldots,k_{m}=1}^{n}[-|D(a_{k_{l}}^{*};z)||D(z^{*};a_{k})||D(a_{k}^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{l-1}}^{*}, a_{k_{l+1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k_{1}}, \ldots, a_{k_{m}})|$$+|D(a_{k}^{*};z)||D(z^{*};a_{k_{l}})||D(a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k}, a_{k_{1}}, \ldots, a_{k_{l-1}}, a_{k_{l+1}}, \ldots, a_{k_{m}})|]$.

(4.25)

Interchange the indices and in the first term and then shift the row vector, respectively. This leads
in front ofand the column vectorin front ofto the following relation$k$$a_{k_{l+1}}$$k_{l}$$a_{k_{l}}^{*}$$a_{k}$$a_{k_{1}}$$|D(a_{k}^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{l-1}}^{*}, a_{k_{l+1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k_{1}}, \ldots, a_{k_{m}})|$$arrow|D(a_{k_{l}}^{*}, a_{k_{1}}^{*}, \ldots, a_{k_{l-1}}^{*}, a_{k_{l+1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k_{1}}, \ldots, a_{k_{l-1}}, a_{k}, a_{k_{l+1}}, \ldots, a_{k_{m}})|$$=|D(a_{k_{1}}^{*}, \ldots, a_{k_{m}}^{*};a_{k}, a_{k_{1}}, \ldots, a_{k_{l-1}}, a_{k_{l+1}}, \ldots, a_{k_{m}})|$.

Note that the value of the determinant is not altered since the total signaturecaused by the above manipulation is $(-1)^{2(l-1)}=1$ . Thus, the first term on theright-hand side of (4.25) coincides with the second term and cosequently, $P_{2}=0$ .
口139
4.3. Proof of (2.6)
Instead of proving (2.6) directly, we differentiate (2.5) by and add the resultant
expression to (2.6) and then prove the equation $P_{3}=0$ , where$x$$P_{3}=f_{xx}f^{*}-f_{x}f_{x}^{*}- \frac{i\gamma}{2}\sum_{k=1}^{n}g_{k,x}g_{k}^{*}$(4.26).

This reduces the total amount of calculations considerably and the proof becomestransparent. It now follows from (4.1), (4.3), (4.4), (4.6) and $(4.8)-(4.10)$ that
$P_{3}=- \frac{1}{2}\{|D(z^{*};z_{x})|+|D(z_{x}^{*};z)|\}|\overline{D}|-\frac{1}{4}|D(z^{*};z)||\overline{D}(z^{*};z)|$$+ \frac{i\gamma}{2}\sum_{k=1}^{n}|D(a_{k}^{*};z_{x})||\overline{D}(z^{*};a_{k})|$Differention of (4.15) with respect to.

(4.27)gives$x$$|D( z^{*};z_{x})|+|D(z_{x}^{*};z)|=-i\gamma\sum_{k=1}^{n}|D(b_{k}^{*}, z^{*};a_{k}, z)|$Inserting (4.15) and (4.28) into (4.27),$P_{3}$(4.28).

can be put into the form

$P_{3}= \frac{i\gamma}{2}\sum_{k=1}^{n}\{|D^{-}||D(b_{k}^{*}, z^{*};a_{k}, z)|+|D(z^{*};z)||\overline{D}(a_{k}^{*};b_{k})|+|D(a_{k}^{*};z_{x})||\overline{D}(z^{*};a_{k})|\}$.

(4.29)

Note from (4.11), (4.13), (4.14) and Jacobi identity (3.6) that$s$$|\overline{D}||D(b_{k}^{*}, z^{*};a_{k}, z)|+|D(z^{*};z)||\overline{D}(a_{k}^{*};b_{k})|$$=-|D( z^{*};a_{k})|\{|D(b_{k}^{*};z)+\frac{1}{2}|D(b_{k}^{*}, z^{*};z,\tilde{z})|\}$$=-|D(z^{*};a_{k})||\overline{D}(a_{k}^{*};z_{x})|$After substituting (4.30) into (4.29),$P_{3}$.

(4.30)

becomes$P_{3}= \frac{i\gamma}{2}\sum_{k=1}^{n}\{-|\overline{D}(a_{k}^{*};z_{x})||D(z^{*};a_{k})|+|D(a_{k}^{*};z_{x})||\overline{D}(z^{*};a_{k})|\}$.

(4.31)

This expression reduces to (4.22) if one replacesby . Hence, the proof of therelation $P_{3}=0$ completely parallels that of $P_{2}=0$ withfrom (4.22).
$z_{x}$$z$$P_{2}$or140
5. Alternative expression of the bright N-soliton solution
Theorem 5.1. The determinants $f’$ and $g_{j}’(j=1,2, \ldots, n)$ given below satisfy thesystem of bilinear equations (2.4)-(2.6):$f’=|A’+B’|$ ,$g_{j}’=|\begin{array}{ll}A’+B’ y^{T}-a_{j}’* 0\end{array}|$

where $A’$ and $B’$ are $N\cross N$ matrices anddefined below:,$A’=(a_{jk}’)_{1\leq j,k\leq N}$$B’=(b_{jk}’)_{1\leq j,k\leq N}$

,

and

$y$

$a_{jk}’= \frac{1}{2}\frac{y_{j}y_{k}^{*}}{q_{j}+q_{k}^{*}}$,,

are N-component row vectors$a_{j}’$

へ

,

(5.1)

$(j=1,2, \ldots, n)$ ,$y_{j}=\exp(q$ $x+iq_{j}^{2}t)$$b_{jk}’= \frac{(\mu-i\gamma q_{k}^{*})d_{jk}}{q_{j}+q_{k}}*$$y=(y_{1}, y_{2}, \ldots, y_{N})$,,$c_{jk}’= \sum_{s=1}^{n}\alpha_{sj}’\alpha_{sk^{*}}’$$a_{j}’=(\alpha_{j1}’, \alpha_{j2}’, \ldots, \alpha_{jN}’)$$(5.2a)$,$(5.2b)$.

$(5.2c)$Here, $q_{j}(j=1,2, \ldots, N)$ and $\alpha_{sj}’(s=1,2, \ldots, n;j=1,2, \ldots, N)$ are complex pammeters chamcterizing the solution.

from (2.13) are closely related to.
Let us show that the determinants andthe determinants $f’$ and given by (5.1). The following lemma is useful for thispurpose:$f$$g_{j}$$g_{j}’$

Lemma 5.1. The determinants

$f$

and

$g_{j}$

given by (2.13) can be rewritten in theform$f=|I+AB|$ ,$(j=1,2, \ldots, n)$$g_{j}=|\begin{array}{ll}I+AB z^{T}-a_{j}^{*} 0\end{array}|$

.

(5.3)

We now establish the following theorem:
Theorem 5.2. Under the parameterization $q_{j}=-p_{j}^{*}(j=1,2, \ldots, N)$ and$-\alpha_{sj}/(2c_{j}^{*})(s=1,2, \ldots, n;j=1,2, \ldots, N)$ , the determinants $f,$ $f’,$and satisfy
the relations(5.4)$f=c|A|f’$ ,$\alpha_{sj}’=$$g_{j}’$$g_{j}$$g_{j}=c|A|g_{j}’$

,

(5.5)

$(j=1,2, \ldots, n)$ ,

where$c=(-1)^{N} \prod_{l=1}^{N}(4c_{l}^{*}c_{l})$,$c_{l}= \frac{\prod_{m--1}^{N}(p_{l}+p_{m}^{*})}{\prod_{(m\overline{\neq}l)}^{N}m-1(p\iota-p_{m})}$,$(l=1,2, \ldots, N)$

.

(5.6)

141
The pammeters $p_{j}(j=1,2, \ldots, N)$ are assumed to satisfy the conditions
for all and $m$ andfor $l\neq m$ .
$l$$p_{l}+p_{m}^{*}\neq 0$$p_{l}\neq p_{m}$Thus, we have obtained the two different expressions for the bright N-soliton
solution of the system of nonlinear PDEs (2.2). Explicitly, they read $u_{j}=g_{j}/f=$.
The following proposition provides an alternative proof of theorem 5.1:

$g_{j}’/f’(j=1,2, \ldots, n)$

Proposition 5.1. If andgiven respectively by (5.4) and (5.5) satisfy thesystem of bilinear equations (2.4)-(2.6), then $f’$ andsatisfy the same system of
equations, and vice versa.
$f$$g_{j}$$g_{j}’$

6. A continuum model
The n-component system (1.1) yields a continuum model when one takes a limit$narrow\infty$ . It represents a (21)-dimensional nonlocal modified NLS equation of the
form
$+$

$iq_{t}+q_{xx}+\mu(\int_{-\infty}^{\infty}|q|^{2}dy)q+i\gamma(\int_{-\infty}^{\infty}|q|^{2}dyq)_{x}=0$,$q=q(x, y, t)$ .

(6.1)

Recall that when $\gamma=0$ , this equation reduces to a (2 1)-dimensional nonlocal
NLS equation proposed by Zakharov [S]. The exact method of solution for equation(6.1) can be developed following the same procedure as that for the system ofnonlinear PDEs (1.1). Hence, we summarize the main results.
First, application of the gauge transformation$+$$q=u \exp[-\frac{i\gamma}{2}\int_{-\infty}^{x}\int_{-\infty}^{\infty}|u(x, y, t)|^{2}dxdy]$$u=u(x, y, t)$ ,

,

(6.2)

to the system (6.1) subjected to the the boundary conditions $qarrow 0,$ $uarrow 0|x|arrow$transforms (6.1) to a nonlocal nonlinear PDE for$\infty$$u$$iu_{t}+u_{xx}+\mu(\int_{-\infty}^{\infty}|u|^{2}dy)u+i\gamma(\int_{-\infty}^{\infty}u^{*}u_{x}dy)u=0$

.

(6.3)

The proposition below is an analog of proposition 2.1:
Proposition 6.1 By means of the dependent variable tmnsformation$u= \frac{g}{f}$

(6.4)

,

equation (6.3) can be decoupled into the following system of bilinear equations for$f=f(x, t)$ and $g=g(x, y, t)$$(iD_{t}+D_{x}^{2})g\cdot f=0$

,

(6.5)

142

$D_{x}f \cdot f^{*}=\frac{i\gamma}{2}\int_{-\infty}^{\infty}|g|^{2}dy$

(6.6)

,

$D_{x}^{2}f \cdot f^{*}=\mu\int_{-\infty}^{\infty}|g|^{2}dy+\frac{i\gamma}{2}\int_{-\infty}^{\infty}D_{x}g\cdot g^{*}dy$

.

(6.7)

Proof. The proof proceeds exactly as that of proposition 2.1. Formally, one may.
by the integral
simply replace the sum$\square$$\int_{-\infty}^{\infty}dy$$\sum_{k=1}^{n}$

It follows from (6.2), (6.4) and (6.6) that$q= \frac{gf^{*}}{f^{2}}$

(6.8)

,

which is just a continuum limit of (2.12).
The following theorem can be derived from a continuum limit of the brightN-soliton solution given by theorem 2.1 and theorem 5.1:

Theorem 6.1. The system of bilinear equations $(6.5)-(6.7)$ admits the following
and $f’,$ for the bright N-soliton solution:
two different expressions$g’$$f,$ $g$$f=|\begin{array}{ll}A I-I B\end{array}|$,$f’=|A’+B’|$ ,$g=|\begin{array}{lll}A I z^{T}-I B 0^{T}0 -a^{*} 0\end{array}|$

,

(6.9)

$g’=|\begin{array}{ll}A’+B’ y^{T}-a^{*} 0\end{array}|$.

(6.10)

Here,andare $N\cross N$ matntces given respectively by $(2.14a)$ and $(2.14b)$$A’$
and $B’$ are $N\cross N$ matrices given
being replaced bywithandbeing replaced byrespectively by $(5.2a)$ and $(5.2b)$ withare N-component row
andand $\alpha_{j}’(j=1,2, \ldots, N)$ are continuous functions of .
vectors where
$A$

$B$

$\int_{-\infty}^{\infty}\alpha_{j}(y)\alpha_{k}^{*}(y)dy,$$c_{jk}$$\int_{-\infty}^{\infty}\alpha_{j}’(y)\alpha_{k^{*}}’(y)dy$$c_{jk}’$$a=a(y)=(\alpha_{1}, \alpha_{2}, \ldots, \alpha_{N})$$a’=a’(y)=(\alpha_{1}’, \alpha_{2}’, \ldots, \alpha_{N}’)$$y$$\alpha_{j}$Proof. The proof can be done in the same way as that of theorem 2.1 and theorem
口
51.
and
Theorem 6.2. Under the pammeterization$N)$ , the determinants $f,$ $f’,$$1,2,$and satisfy the relations$q_{j}=-p_{j}^{*}$$\ldots,$$g$$\alpha_{j}’=-\alpha_{j}/(2c_{j}^{*})$

$(j=$

$g’$

$f=c|A|f’$ ,(611)$g=c|A|g’$ ,

(612)

143
$v]herec$ isthatdefined by (5.6) and the pammeters $p_{j}(j=1,2, \ldots, N)$ are specified suchfor all and andfor $l\neq m$ .

$p_{l}+p_{m}^{*}\neq 0$

$l$

$m$

$p_{l}\neq p_{m}$Proof. The proof parallels theorem 5.2.

$\square$Proposition 6.2. If and given by (6.9) satisfy the system of bilinear equations$(6.5)-(6.7)$ , then $f’$ andgiven by (6.11) and (6.12) satisfy the same system of
equations, and vice versa.
$f$

$g$

$g’$

Proof. The proof is completely parallel to that of proposition 5.1.

$\square$

7. Conclusion
1. We have obtained two different expressions of the bright N-soliton solution ofa multi-component modified NLS equation in terms of determinants.
2. We have proposed a continuum model arising from the multi-component system
as the number of dependent variables tends to infinity and presented its bright Nsoliton solution.
3. Our solutions include existing bright N-soliton solutions of the multi-component
NLS equation and its continuum model.
Acknowledgement
This work was partially supported by the Grant-in-Aid for Scientific Research(C) No. 22540228 from Japan Society for the Promotion of Science.

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