156
The N ‐soliton formulas for a multi‐component

modified nonlinear Schrödinger system with nonzero boundary conditions
Yoshimasa Matsuno

Division of Applied Mathematical Science
Graduate School of Sciences and Technology for Innovation
Yamaguchi University
E‐mail address: matsuno@yamaguchi‐u.ac.jp
Abstract

The present paper provides the dark N ‐soliton solution for a multi‐component modified nonlinear

Schrödinger (NLS) system with plane‐wave boundary conditions, as well as the bright‐dark N‐
soliton solution with mixed zero and plane‐wave boundary conditions. The N ‐soliton formulas
obtained here include as special cases the existing soliton solutions of the NLS and derivative NLS
equations and their integrable multi‐component analogs. The new features of soliton solutions
are discussed. In particular, it is shown that the N constraints must be imposed on the amplitude
parameters of solitons in constructing the dark N ‐soliton solution with plane‐wave boundary
conditions. For mixed type boundary conditions, the structure of soliton solutions is found to be
more explicit than that of dark soliton solutions since no constraints are imposed on the soliton
parameters. Last, we comment on an integrable multi‐component system associated with thefirst negative flow of the multi‐component derivative NLS hierarchy.
1. Introduction

The nonlinear Schrödinger (NLS) equation is a milestone in the theory of nonlinear waves.
One remarkable feature is that it is a completely integrable equation, allowing us to solve it

by means of several exact methods of solution such as the inverse scattering transform (IST)[1‐3], Bäcklund‐Darboux (BD) transformation [4‐6] and Hirota’s direct method [7‐9]. In recent
years, much attention has been paid to the multi‐component generalizations of the NLS equation
because of their wide applicability in many physical contexts such as nonlinear optics, nonlinear

water waves and plasma physics and so on [10‐12].
In this paper, we consider the following multi‐component system of nonlinear partial dif‐ferential equations (PDEs) which is a hybrid of the multi‐component NLS system and multi‐
component derivative NLS system

iq_{j,t}+q_{j,x }+\mu(\sum_{s=1}^{n}\sigma_{s}|q_{s}|^{2})q_{j}+i\gamma[(\sum_ {s=1}^{n}a_{s}|q_{s}|^{2})q_{j}]_{x}=0,

j=1,2, . ., , (1.1)
nwhere q_{j}=q_{j}(x, t)(j=1,2, \ldots, n) are complex‐valued functions of x and t, \mu and \gamma are realconstants, n is an arbitrary positive integer, and subscripts x and t appended to q_{j} denote partial

differentiations. The coefficients \sigma_{s}=\pm 1(s=1,2, \ldots, n) specify the sign of the nonlinearity.
For the multi‐component NLS system for which \gamma=0 , we can deal with the three types of cubic

157
nonlinearities, i.e., focusing (\sigma_{s}=1, s=1,2, \ldots, n) , defocusing (\sigma_{s}=-1, s=1,2, \ldots, n) andmixed focusing‐defocusing (\sigma_{s}=1, s=1,2, \ldots, m;\sigma_{s}=-1, s=m+1, m+2, \ldots, n) nonlinearities
with \mu>0 , where m is an arbitrary positive integer such that 1\leq m<n . The special

systems reduced from (1.1) have been summarized in a previous paper [13]. The system (1.1)
has been shown to be completely integrable [14] and hence the exact methods mentioned above
can be applied to it to obtain various types of soliton solutions. The properties of solutions,however, depend essentially on the boundary conditions. We emphasize that most works havebeen concerned with the analysis of the system under vanishing boundary conditions at spatial

infinity [13].
The main purpose of the present paper is to construct N‐soliton solutions (N : arbitrary
positive integer) of the system (1.1) with the following two types of boundary conditions:
1) Plane‐wave boundary conditions.

q_{j}\sim\rho_{j}\exp i(k_{j}x-\omega_{j}t+\phi_{j}^{(\pm)}) , xarrow\pm\infty, j=1,2, n, (1.2a)\omega_{j}=k_{j}^{2}-\mu\sum_{s=1}^{n}\sigma_{s}\rho_{s}^{2}+\gamma(\sum_{s=1} ^{n}\sigma_{s}\rho_{s}^{2})k_{j} , j=1,2, n. (1.2b)2) Mixed type boundary conditions.

q_{j}\sim 0, xarrow\pm\infty, j=1,2, m, (1.3a)q_{m+j}\sim\rho_{j}\exp i(k_{j}x-\omega_{j}t+\phi_{j}^{(\pm)}) , xarrow\pm\infty, j=1,2, n-m, (1.3b)\omega_{j}=k_{\dot{j} ^{2}-\mu\sum_{s=1}^{n-m}\sigma_{s}\rho_{s}^{2}+ \gamma(\sum_{s=1}^{n-m}\sigma_{s}\rho_{s}^{2})k_{j} , j=1,2, n-m. (1.3c)\phi_{j}^{(\pm)}Here, \rho_{\dot{j} , k_{j} andare arbitrary real constants and \omega_{j} from (1.2b) and (1.3c) represent thelinear dispersion relations for components with the plane‐wave boundary conditions. For case1), all the components approach the specified plane waves at infinity whereas for case 2), thefirst m components vanish at infinity and the remaining n-m components have the specified
plane waves. It is important that for the nonzero background fields like (1.2a) and (1.3b) , eachcomponent has different wavenumber and frequency. This causes difficulties in constructing

soliton solutions. In particular, in the case of 1) above, one must impose certain constraints
between the real and imaginary parts of the complex parameters characterizing the amplitude(or the velocity) of solitons. This peculiar feature has never been encountered in the case of zeroboundary conditions [13].
The approach which will be employed in our analysis is the direct method [7‐9]. While boththe IST and BD transformation are based on the Lax representations of completely integrable
PDEs, the direct method does not need the knowledge of the IST, and it can also be applicable
to nonintegrable equations. It provides a powerful tool to obtain special solutions such as
soliton and periodic solutions irrespective of the boundary conditions. The N ‐soliton formulas
presented in this paper include as special cases the existing N‐soliton solutions of the NLS andderivative NLS equations as well as the Manakov system with nonzero boundary conditions.
The compact expressions of the soliton solutions are particularly useful for investigating their

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structures, asymptotic behaviors and dynamics. On the other hand, the IST is technicallyinvolved for nonzero boundary conditions, and hence it has been applied mainly to the multi‐

component system such as (1.1) under the restricted class of boundary conditions in which all thecomponents have the same plane‐wave boundary condition at infinity, for instance [15, 16]. The
development of the IST for the multi‐component systems with the general nonzero boundary
conditions is still in progress. In particular, the derivation of the N ‐soliton solutions by meansof the IST is an important issue left to the future. In the following, we describe only the main
results, and the details will be reported in a separate paper.
2. Notation and basic formulas for determinants

First of all, we introduce for convenience the notation and some basic formulas for determinantsassociated with the N ‐soliton solutions.
2.1. Notation

Let D=(d_{jk})_{1\leq j,k\leq N} be an N\cross N matrix and a=(a_{j})_{1\leq j\leq N}, b=(b_{j})_{1\leq j\leq N}, c=(c_{\dot{j}})_{1\leq j\leq N}, d=(d_{j})_{1\leq j\leq N} be the N‐component row vectors, where all the entries in the matrices and vectors
are complex‐valued functions of x and t . The bordered marices associated with the matrix D
are defined byD(a;b)=(\begin{ar ay}{l} D b^{T} a 0 \end{ar ay}),D(a,b;c,d)=(\begin{ar ay}{l } D c^{T} d^{T} a 0 0 b 0 0 \end{ar ay})

,

(2.1)

where the symbol T denotes transpose. The first cofactor of d_{ij} is defined by D_{\dot{i}j}=\partial|D|/\partial d_{ij},(|D|=\det D) and the second cofactor by D_{ij,pq}=\partial^{2}|D|/\partial d_{ip}\partial d_{jq}(i<j, p<q) .
2.2. Basic formulas for determinants

The following formulas are well‐known in the theory of determinants and will be used frequently

in our analysis [17]:\frac{\partial|D}{\partialx}=\sum_{i,j=1}^{N}\frac{\partiald_{ij}{\partial x}D_{\dot{\iota}j|\begin{ar ay}{l } D a^{T} b z \end{ar ay}|=D|z-\sum_{i,j=1}^{N}D_{ij}a_{i}b_{j}
,

(2.2)

,

(2.3)

|D(a, b;c, d)||D|=|D(a;c)||D(b;d)|-|D(a;d)||D(b;c)| ,(2.4)\delta_{ij}|D|=\sum_{k=1}^{N}d_{ik}D_{jk}=\sum_{k=1}^{N}d_{ki}D_{kj}(2.5)

D_{ij}= \sum_{p=1}^{N}d_{pq}D_{ip,jq} (j\neq q)= \sum_{q=1}^{N}d_{pq}D_{ip,jq} (i\neq p),(2.6a),(2.6b)

159|D( a, b;c, d)|=\sum_{i,j p,q=1}^{N}a_{p}b_{q}c_{i}d_{j}D_{ij,pq}

.

(2.7)

(i\neq j,p\neq q)

3. The modified NLS system with plane‐wave boundary conditions

3.1. Gauge transformation

We consider the system of PDEs (1.1) subjected to the plane‐wave boundary conditions (1.2).
As in the case of zero boundary conditions considered in [13], we first apply the following gaugetransformationq_{j}=u_{j} \exp[-\frac{\dot{ \imath} \gamma}{2}\int_{-\infty}^{x}\sum_{s=1} ^{n}\sigma_{s}(|u_{s}|^{2}-\rho_{s}^{2})dx] , j=1,2, n

,

(3.1)

to the system, where u_{j}=u_{j}(x, t)(j=1,2, \ldots, n) are complex‐valued functions of x and t . The

system (1.1) is then transformed to the system of nonlinear PDEs for u_{j} :iu_{\dot{j} ,t+u_{\dot{j},x }+i\lambda\gam a u_{\dot{J}^{x} ,+i\gam a(\sum_{s= 1}^{n}\sigma_{s}u_{s}^{*}u_{s,x}-i\sum_{s=1}^{n}\sigma_{s}k_{s}\rho_{s}^{2}) u_{j}+[( \mu-\frac{\lambda\gam a^{2} {2})\sum_{s=1}^{n}\sigma_{s}|u_{s}|^{2}+\frac{( \lambda\gam a)^{2} {2}]u_{j}=0,

j=1,2 , ..., n .

(3.2)

Here, we have put \lambda=\sum_{s=1}^{n}\sigma_{s}\rho_{s}^{2} for simplicity and the asterisk appended to u_{s} denotes complex
conjugate.
3.2. Bilinearization

The following proposition is the starting point in our analysis.
Proposition 3.1. By means of the dependent variable transformationsu_{j}= \rho_{j}e^{i(k_{j}x-\omega_{j}t)}\frac{h_{j} {f},j=1,2 , ..., n ,(3.3)

the system of nonlinear PDEs (3.2) can be decoupled into the following system of bilinear equa‐tions for f and h_{j}iD_{t}h_{j}\cdot f+i(2k_{j}+\lambda\gamma)D_{x}h_{j}\cdot f+D_{x}^{2}h_{j}\cdot f=0, j=1,2, n ,D_{x}f \cdot f^{*}-\frac{\dot{ \imath} \gamma}{2}\sum_{s=1}^{n}\sigma_{s} \rho_{s}^{2}(h_{s}h_{s}^{*}-f ^{*})=0
D_{x}^{2}f \cdot f^{*}-\frac{i\gam a}{2}\sum_{s=1}^{n}\sigma_{s}\rho_{s}^{2} D_{x}h_{s}\cdot h_{s}^{*}+\gam a\sum_{s=1}^{n}\sigma_{s}k_{s}\rho_{s}^{2}(h_{s} h_{s}^{*}-f ^{*})
,

(3.4)
(3.5)

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+( \frac{\lambda\gamma^{2} {4}-\mu)\sum_{s=1}^{n}\sigma_{s}\rho_{s}^{2}(h_{s}h_ {s}-f ^{*})=0

.

(3.6)

Here, f=f(x, t) and h_{j}=h_{j}(x, t)(j=1,2, \ldots, n) are complex‐valued functions of x and t , andthe bilinear operators D_{x} and D_{t} are defined by

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} ,(3.7)

where m and n are nonnegative integers.

The fundamental quantities f and h_{j} characterize completely solutions. They are sometimes
called the tau functions. While the expressions (3.3) for u_{j} give the solutions of the system of
PDEs (3.2) in terms of the tau functions f and h_{j}(j=1,2, \ldots, n) , the original variables q_{j} areexpressible by them as well. To show this, we use (3.3) and (3.5) to obtain the relation\frac{\partial}{\partialx}\ln\frac{f} ^{*}=-\frac{\dot{\imath}\gam a}{2} \sum_{s=1}^{n}\sigma_{s}(|u_{s}|^{2}-\rho_{s}^{2})

.

(3.8)

If we introduce (3.8) into (3.1), integrate with respect to x and note (3.3), we find the desired
expressions

q_{j}= \rho_{\dot{j} e^{i(k_{j}x-\omega_{j}t)}\frac{h_{j}f^{*} {f^{2} ,j=1,2 , ..., n .

(3.9)

Remark 3.1. If \gamma=0 , then the bilinear equation (3.5) reduces to D_{x}f\cdot f^{*}=0 , implyingthat f^{*}=c(t)f . An arbitrary function c(t) can be set to 1 by imposing a boundary condition,
f=1, xarrow-\infty , for instance. Thus, q_{j} from (3.9) simplify toq_{j}= \rho_{j}e^{i(k_{j}x-\omega_{j}t)}\frac{h_{j} {f},j=1,2 , ..., n ,(3.10)and they satisfy the n ‐component NLS system with the plane‐wave boundary conditions (1.2)

iq_{j,t}+q_{j,x }+\mu(\sum_{s=1}^{n}\sigma_{s}|q_{s}|^{2})q_{j}=0,

j=1,2 , ..., n .

(3.11)

3.3. The dark N ‐soliton solution

Here, we establish the following theorem.

Theorem 3.1. The N‐soliton solution of the system of bilinear equations (3.4)‐(3.6) is given interms of the following determinants

f=|D|, h_{s}=|H_{s}|, s=1,2, n, (3.12a)

D=(d_{jk})_{1\leq j,k\leq N}, d_{\dot{j} k= \delta_{jk}-\frac{ip_{j}+\frac{\mu} {\gamma} {p_{\dot{j} +p_{k} *z_{\dot{j} z_{k}^{*}, (3.12b)

161
161

H_{s}=(h_{jk}^{(s)} _{1\leq j,k\leq N}, h_{jk}^{(s)}= \delta_{jk}+\frac{(\dot{ \imath} p_{\dot{j} +\frac{\mu}{\gamma})(p_{j}-ik_{s}) {(p_{j}+p_{k}^{*})(p_{k}^{ *}+ik_{s}) z_{j}z_{k}^{*}, (3.12c)z_{j}= \exp[p_{j}x+(ip_{\dot{j} ^{2}-\lambda\gamma p_{j})t+\zeta_{j0}] , j=1,2, N, \lambda=\sum_{s=1}^{n}\sigma_{s}\rho_{s}^{2}. (3.12d)
Here, p_{j} and \zeta_{\dot{j}}0(j=1,2, \ldots, N) are arbitrary complex parameters characterizing the amplitude
and phase of the solitons, respectively, and the N constraints are imposed on the parameters p_{j}\frac{\gam a}{2\sum_{s=1}^{n}\sigma_{s}p_{s}^{2}\frac{\dot{\imath}(p_{j}-p_ {\dot{j}^{*})+k_{s}+\frac{\mu}{\gam a}{(p_{\dot{j}-\dot{\imath}k_{s})(p_{j} +\dot{\imath}k_{s})=-1,

j=1,2 , ..., N .

(3.13)

Remark 3.2. By means of the transformation of the variables, f=\tilde{f}, h_{s}=\tilde{h}_{s}(s=1,2, \ldots, n), x=\tilde{x}+\frac{2\mu}{\gamma}\tilde{t} combined with the transformation of the parameters, p_{j}= \tilde{p}_{j}+i\frac{\mu}{\gamma}(j=1,2, \ldots, N), k_{s}=\tilde{k}_{s}+\frac{\mu}{\gamma}(s=1,2, \ldots, n) , the form of the bilinear equations (3.4) and (3.5) is unchanged whereas
the bilinear equation (3.6) reduces to a simplified form with \mu=0 . The N ‐soliton solution(3.12) and the constraints (3.13) remain the same form with \mu=0 . Thus, the proof of the
N ‐soliton solution may be performed under the setting\mu=0 without loss of generality.

Remark 3.3. Unlike the bright N ‐soliton solution with zero boundary conditions [13], thereal part of p_{j} is related to its imaginary part by (3.13). In the general n ‐component system,
one needs to solve the algebraic equation of order n forsolutions are not available for n\geq 5 . If the wavenumbers({\rm Re} p_{j})^{2} . As well‐known, analyticalk_{j}(j=1,2, \ldots, n) take the same value

which implies that all the components of the system have the same asymptotic form except theamplitudes \rho_{j} and the phase constants \phi_{\dot{j} ^{(\pm)} (see (1.2)), then the constraints (3.13) reduce simply

to a single quadratic equation for {\rm Re} p_{j} . This special case has been dealt with by means of the

IST for the multi‐component NLS system [15, 16].

Remark 3.4. The multi‐component NLS system (i.e., \gamma=0 in (1.1)) is invariant under thetransformation t=-\tilde{t}, x=i\tilde{x}, \mu=-\tilde{\mu} . Suppose that\sigma_{s}=1(s=1,2, \ldots, n) and \tilde{\mu}>0 . Wechange the parameters in (3.12) according to the rule k_{s}=-i\tilde{k}_{s}, \omega_{s}=-\tilde{\omega}_{s}(s=1,2, \ldots, n) ,p_{j}=-i\tilde{p}_{j}, \zeta_{j0}=\tilde{\zeta}_{j0}+\ln\sqrt{\gamma}(j=1,2, \ldots, N) and then take the limit \gammaarrow 0 . The resulting
expression gives rise to the N ‐soliton solution of the focusing NLS system with plane‐wave
boundary conditions. The solutions thus constructed exhibit a rich mathematical structure.
Specifically, a reduction procedure applied to the soliton solutions would produce the breather
and rogue wave solutions, as already demonstrated for the soliton solutions of the focusing NLS

equation [18].
The proof of theorem 3.1 will be performed by using a sequence of lemmas, which we shallsummarize. In accordance with remark 3.2, we put \mu=0 in formulas that follow.
Lemma 3.1. The expression of h_{s} from (3.12a) is rewritten in the form

h_{s}=|D|-|D(z_{s}^{*};z_{x})|\overline{k_{s}}\dot{{\imath}}, s=1,2, n, (3.14a)

162
where

z=(z_{j})_{1\leq j\leq N},
are N ‐component row vectors.

z_{s}=(\frac{k_{s} {p_{j}-\dot{ \imath} k_{s} z_{j})_{1\leq j\leq N}

s=1,2

,

...,

(3.14b)n,

The following lemma provides the differentiation rules of f and h_{s} with respect to t and x.
Lemma 3.2.

f_{t}=i|D(z^{*};z_{t})|+|D(z_{x}^{*};z_{x})| ,(3.15)

f_{x}=i|D(z^{*};z_{x})| ,(3.16)

f_{xx}=i|D(z_{x}^{*};z_{x})|+i|D(z^{*};z_{xx})| ,(3.17)

h_{s,t}= i|D(z^{*};z_{t})|+|D(z_{x}^{*};z_{x})|-\overline{k_{s} \dot{ \imath} |D(z_{s,t}^{*};z_{x})|-\overline{k_{s} \dot{ \imath} |D(z_{s}^{*};z_{xt})|+\frac {1}{k_{s} |D(z_{s}^{*}, z^{*};z_{x}, z_{t})| , (3.18)(3.19)
h_{s,x}=-|D( z_{\mathcal{S} ^{*};z_{x})|-\frac{i}{k_{\mathcal{S} }|D(z_{s}^{*}; z_{x })| ,h_{s,xx}= i|D(z_{x}^{*};z_{x})|+i|D(z^{*};z_{xx})|-\overline{k_{s} \dot{ \imath} |D(z_{s,xx}^{*};z)|-\frac{2\dot{ \imath} }{k_{s} |D(z_{s,x}^{*};z_ {xx})|‐(3.20)\overline{k_{S} \dot{ \imath} |D(z_{s}^{*};z_{x x})|+\frac{1}{k_{s} |D(z_{s}^{ *}, z^{*};z_{x }, z_{x})| ,Lemma 3.3. The complex conjugate expressions of f, f_{x} and h_{s} are given as follows.

f^{*}=|D|-i|D(z^{*};z)| ,(3.21)

f_{x}^{*}=-i|D(z_{x}^{*};z)| ,(3.22)

h_{s}^{*}=|D|- i|D(z^{*};z)|+\overline{k_{s} \dot{ \imath} |D(z_{x}^{*};z_{s})| +\frac{1}{k_{s} |D(z_{x}^{*}, z^{*};z_{s}, z)| .

(3.23)

When one tries to show that the N ‐soliton solution (3.12) solves the bilinear equations (3.5) and(3.6), the following lemma plays the central role together with Jacobi’s identity (2.4).
Lemma 3.4.

\sum_{s=1}^{n}\frac{\sigma_{s}\rho_{s}^{2} {k_{s}^{2} |D(z_{s}^{*};z_{s})|= |D(z^{*}B;z)|\sum_{s=1}^{n}\frac{\sigma_{s}\rho_{s}^{2} {k_{s}^{2} |D(z_{s}^{*}, a^{*};b, z_{s})|=|D(z^{*}B, a^{*};b, z)|-i(|D(a^{*};bB)|- D(a^{*}B;b)|)
,

Here,

B

(3.24).

(3.25)

is a diagonal matríx given by
B= diag(\beta_{1}, \beta_{2}, \ldots, \beta_{N}) \beta_{j}=\sum_{s=1}^{n}\frac{\sigma_{s}\rho_{s}^{2} {(p_{j}-\dot{ \imath} k_{s})(p_{j}^{*}+ik_{s}) ,,j=1,2 , ..., N ,(3.26)

163z^{*}B=(\beta_{\dot{j}}z_{j}^{*})_{1\leq j\leq N} is the N ‐component row vector, and a=(a_{j})_{1\leq j\leq N} and b=(b_{j})_{1\leq j\leq N} arearbitrary N ‐component row vectors.
3.4. Proof of theorem 3.1

We can show that the tau functions (3.12) associated with the dark N‐soliton solution solve thebilinear equations (3.4)-(3.6) . The proof will be performed by employing lemmas 3.1‐3.4 andsome basic formulas for determinants. In particular, Jacobi’s identity (2.4) plays the central
role, as in the case of the proof of the bright

N ‐soliton solution of the modified NLS

system(1.1) [13].
Remark 3.5. We remark that the constraints (3.13) has not been used for the proof of (3.4). On
the other hand, in establishing (3.5) and (3.6), one has relied on lemma (3.4) which is essentiallybased on the constraints.

3.5. One‐soliton solutions

The soliton solutions are characterized completely by the tau functions f and h_{j} given by (3.12).
Here, we describe the feature of one‐soliton solutions. The general N ‐soliton solutions will beconsidered elsewhere. To simplify the notation, we first put p_{1}=a+ib, \zeta_{10}=a(x_{0}+iy_{0})(a, b, x_{0}, y_{0}\in \mathbb{R}) , and\frac{b-\frac{\mu}{\gamma}-\dot{ \imath} a}{2a}=\beta e^{2i\phi}(\beta>0) , \frac{a+\dot{ \imath} (b-k_{\dot{j} )}{a-i(b-k_{\dot{j} )}=e^{2i\theta_{j} ,j=1,2 , ..., n .

(3.27)

Then, the tau functions for the one‐soliton solutions are written compactly in the form

f=1+e^{2a(\xi+\xi_{0})+2i\phi}, h_{j}=1-e^{2a(\xi+\xi_{0})+2i(\theta_{j}+\phi)} , j=1,2, n ,(3.28)

with \xi=x-(2b+\lambda\gamma)t+x_{0} , and \xi_{0}=(1/2a)\ln\beta . Formula (3.12) with (3.28) gives one‐soliton
solutions. The square modulas of the complex variable q_{j} is computed as\rho_{j}^{2}(2b-\mu/\gamma-k_{j})|q_{j}|^{2}= \rho_{j}^{2}-\frac{2a^{2}sgna}{\sqrt{a^{2}+(b-\mu/\gamma)^{2} a^{2}+(b-k_{j})^{2} \frac{1}{\cosh 2a(\xi+\xi_{0})+\frac{(b-\mu/\gam a)sgna}{\sqrt{a^{2}+(b- \mu/\gam a)^{2}.

(3.29)In the simplest one‐soliton case, the constraints (3.13) reduce simply to a single relation
which connectsawith b\frac{\gam a}{2}\sum_{s=1}^{n}\sigma_{s}\rho_{s}^{2}\frac{-2b+k_{s}+\frac{\mu} {\gam a}{a^{2}+(b-k_{s})^{2}=-1.

(3.30)For the n ‐component system, one must solve the algebraic equation of order n for a^{2} to express
a in terms of b .
Note, however, that analytical solutions are obtainable up to n=4 . This is
the main difficulty in constructing soliton solutions. Although one can deal with the degeneratecase for which all the wave numbers k_{s} have the same (possibly zero) value, the resulting one
soliton solutions coincide essentially with those of the one‐component system.

164
Last, we consider the soliton solutions of the 1‐component system. Using (3.30) with n=1,
the expression (3.29) simplifies to|q_{1}|^{2}=\rho_{1}^{2}-\frac{4a^{2}sgna}{\sigma_{1}\gam a\sqrt{a^{2}+(b-\mu/ \gam a)^{2} \frac{1}{\cosh2a(\xi+\xi_{0})+\frac{(b-\mu/\gam a)sgna} {\sqrt{a^{2}+(b-\mu/\gam a)^{2}.

(3.31)

When the condition sgn(\sigma_{1}\gamma a)>0 is satisfied, then this represents the dark soliton solution witha constant background. Taking the limit \gammaarrow 0 under the conditions sgn(\sigma_{1}\gamma a)>0, sgn(\gamma\mu a)<
0 , the expression (3.31) reduces to the dark soliton solution of the NLS equation|q_{1}|^{2}= \rho_{1}^{2}-\frac{2a^{2} {|\mu|} sech2 a(\xi+\xi_{0}) ,(3.32)

with a constraint a^{2}+(b-k_{1})^{2}=-(\mu/2)\sigma_{1}\rho_{1}^{2} imposed on the parameters a and b.

4. The modified NLS system with mixed type boundary conditions

In this section, we present the N‐soliton solution of the system of equations (1.1) with themixed zero and plane‐wave boundary conditions (1.3). We will see that solutions take the formof bright‐dark type solitons.
4.1. Gauge transformation
We first apply the gauge transformationq_{j}=u_{\dot{j} \exp[-\frac{\dot{ \imath} \gamma}{2}\int_{-\infty}^{x}\{\sum_ {s=1}^{n}\sigma_{s}|u_{s}|^{2}-\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}\}dx] , j =1,2, n

,

(4.1)

to the system (1.1) and obtain the system of nonlinear PDEs for u_{j} :iu_{j,t}+u_{j,x }+i\hat{\lambda}\gamma u_{j,x}+i\gamma(\sum_{s=1}^{n} \sigma_{s}u^{*}u_{s,x}-i\sum_{s=1}^{n-m}\sigma_{m+s}k_{s}\rho_{s}^{2})u_{j}+[(\mu-\frac{\hat{\lambda}\gam a^{2} {2})\sum_{s=1}^{n}\sigma_{s}|u_{s}|^{2}+ \frac{(\hat{\lambda}\gam a)^{2} {2}]u_{j}=0,
j=1,2 , ..., n ,

where

(4.2)

\hat{\lambda}=\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}.

4.2. Bilinearization

The bilinearization of the system of nonlinear PDEs (4.2) is accomplished by the following
proposition.

Proposition 4.1. By means of the dependent variable transformationsu_{j}= e^{i\hat{\lambda}\mu t}\frac{g_{j}}{f} , j=1,2, m, (4.3a)165

u_{m+j}= \rho_{j}e^{i(k_{j}x-\omega_{j}t)}\frac{h_{j} {f},

j=1,2

,

...,

(4.3b)n-m,

the system of nonlinear PDEs (4\cdot 2) can be decoupled into the following system of bilinear equa‐tions for f, g_{j} and h_{j}iD_{t}g_{j}\cdot f+i\hat{\lambda}\gamma D_{x}g_{j}\cdot f+D_{x}^{2}g_{j}\cdot f =0, j=1,2, m ,(4.4)

iD_{t}h_{j}\cdot f+i(2k_{j}+\hat{\lambda}\gamma)D_{x}h_{j}\cdot f+D_{x}^{2} h_{j}\cdot f=0, j=1,2, n-m ,(4.5)

D_{x}f \cdot f^{*}-\frac{\dot{ \imath} \gam a}{2}\{\sum_{s=1}^{m}\sigma_{s} g_{s}g_{s}^{*}+\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}(h_{s}h_{s}^{*}-f ^{*})\} =0
D_{x}^{2}f \cdot f^{*}-\frac{\dot{ \imath} \gam a}{2}(\sum_{s=1}^{m}a_{s}D_{x} g_{s}\cdot g^{*}+\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}D_{x}h_{s}\cdot h_{s}^{ *})+\gam a\sum_{s=1}^{n-m}\sigma_{m+s}k_{s}\rho_{s}^{2}(h_{s}h_{s}^{*}-f ^{*})+( \frac{\hat{\lambda}\gamma^{2} {4}-\mu)\{\sum_{s=1}^{m}\sigma_{s}g_{s}g_{s}^{ *}+\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}(h_{s}h_{s}^{*}-f ^{*})\}=0
,

(4.6)

.

(4.7)

Here, f=f(x, t), g_{\dot{j}}(x, t) and h_{j}=h_{j}(x, t) are complex‐valued functions of x and t.

Using (4.1) and (4.6), we can express q_{j} in terms of the tau functions f, g_{j} and h_{j} as

q_{\dot{j}}= e^{i\hat{\lambda}\mu t}\frac{g_{j}f^{*}}{f^{2}}, j=1,2, m, (4.8a)q_{m+j}= \rho_{j}e^{i(k_{j}x-\omega_{j}t)}\frac{h_{\dot{j}}f^{*}}{f^{2}}, j=1 , 2, n-m. (4.8b)
Remark 4.1. In the case of \gamma=0 , the expressions (4.8) becomeq_{j}= e^{i\hat{\lambda}\mu t}\frac{g_{j} {f},q_{m+j}= \rho_{j}e^{i(k_{j}x-\omega_{j}t)}\frac{h_{j} {f},

j=1,2

,

j=1,2

(4.9a)

m,

...,

,

...,

(4.9b)n-m,

and they satisfy the n ‐component NLS system (3.11).
4.3. The bright‐dark N ‐soliton solution

The main result in this section is provided by the following theorem.

Theorem 4.1. The N ‐soliton solution of the system of bilinear equations (4.4)‐(4. 7) is given
in terms of the determinants

f=|D|

,

g_{s}=-|D(a_{s}^{*};z)|,s=1,2,m,h_{s}=|D|+ \frac{1}{k_{s} |D(z_{s}^{*};z)|,s=1,2,

n-m,(4.10a)166

D=(d_{\dot{j} k)_{1\leq j,k\leq N}, d_{jk}= \frac{z_{j}z_{k}^{*}+\frac{1}{2} (\mu-i\gamma p_{k}^{*})c_{jk} {p_{j}+p_{k}^{*} , (4.10b)z_{j}=\exp[p_{j}x+(ip_{j}^{2}-\hat{\lambda}\gamma p_{j})t], j=1,2, N, (4.10c)wherec_{jk}=\frac{\sum_{s=1}^{m\sigma_{s}\alph_{sj}\alph_{sk}^{* 1+ \frac{\gam a}{2\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^2\frac{\dot{\imath} (p_{j}-p_{k}^*)+k_{s}+\frac{\mu}{\gam a}{(p_j}-\dot{\imath}k_{s})(p_{k} ^{*}+\dot{\imath}k_{s}) ,zj, k=1,2

,

...,

N,

(4.10d)and z_{s} are N ‐component row vectors defined by (3.14b) with z_{j} given by (4\cdot 1\theta c) anda_{s}=(\alpha_{s1}, \alpha_{s2}, \ldots, \alpha_{sN}) , s=1,2, m, (4.10e)
are row vectors with elements\alpha_{sj}\in \mathbb{C}(s=1,2, .., m;j=1,2, .., N) .

The N ‐soliton solution is characterized by the N complex parameters p_{j}(j=1,2, \ldots, N) and\alpha_{sj}(s=1,2, \ldots, m;j=1,2, \ldots, N) . The former parameters determine
the amplitude of solitons and the latter ones determine the polarization and the envelope phasesof solitons. Note that we have used the same symbol as that appears in theorem 3.1 for the taufunctions f and h_{s} since no confusion would be likely to arise from this convention.
First, we summarize the differentiation rules corresponding to those given by lemma 3.2 andlemma 3.3. Remark 3.2 is applied to the current problem, so that we can put \mu=0 without
loss of generality.
mN complex parameters

Lemma 4.1.

f_{t}=-i|D(z^{*};z_{x})|+i|D(z_{x}^{*};z)|+\hat{\lambda}\gamma|D(z^{*};z)| ,(4.11)

f_{x}=-|D(z^{*};z)| ,(4.12)

f_{xx}=-|D(z_{x}^{*};z)|-|D(z^{*};z_{x})| ,(4.13)

g_{s,t}=-|D(a_{s}^{*};z_{t})|+i|D(a_{s}^{*}, z^{*};z, z_{x})| ,(4.14)

g_{s,x}=-|D(a_{s}^{*};z_{x})| ,(4.15)

g_{s,xx}=-|D(a_{s}^{*};z_{xx})|+|D(a_{s}^{*}, z^{*};z_{x}, z)| ,(4.16)

h_{s,t}=- i|D(z^{*};z_{x})|+i|D(z_{x}^{*};z)|+\hat{\lambda}\gamma|D(z^{*};z)|+ \frac{1}{k_{s} |D(z_{s,t}^{*};z)|+\frac{1}{k_{s} |D(z_{s}^{*};z_{t})|(4.17)-|D(z_{s}^{*}, z^{*};z, z_{x})|\overline{k_{s} \dot{ \imath} ,(4.18)
h_{s,x}=- i|D(z_{s}^{*};z)|+\frac{1}{k_{s} |D(z_{s}^{*};z_{x})| ,h_{s,xx}=- i|D(z_{s,x}^{*};z)|-i|D(z_{s}^{*};z_{x})|+\frac{1}{k_{s} |D(z_{s,x}^ {*};z_{x})|+\frac{1}{k_{s} |D(z_{s}^{*};z_{xx})|-\frac{1}{k_{s} |D(z_{s}^{*}, z^ {*};z_{x}, z)| . (4.19)
The above formulas can be derived by using (2.2) and some basic properties of determinants.
The lemma 4.2 below provides the complex conjugate expressions of f, f_{x}, g_{s} and h_{s}.

167Lemma 4.2.

f^{*}=| \overline{D}|=|(\overline{d}_{jk})_{1\leq j,k\leq N}|, \overline{d} _{jk}=d_{jk}+\frac{i\gamma}{2}c_{jk} ,(4.20)

f_{x}^{*}=-|\overline{D}(z^{*};z)| ,(4.21)

g_{s}^{*}=-|\overline{D}(z^{*};a_{s})| ,(4.22)

h_{s}^{*}=| \overline{D}|+\frac{1}{k_{s} |\overline{D}(z^{*};z_{s})| .

(4.23)

The following two lemmas will be used effectively in the proof of (4.6) and (4.7).
Lemma 4.3.

\sum_{s=1}^{n-m}\frac{\sigma_{m+s}\rho_{s}^{2} {k_{s}^{2} |D(z_{s}^{*};z) |\overline{D}(z^{*};z_{s})|=-|D(\tilde{z}^{*};z)|\overline{D}|-\overline{D} (z^{*};\tilde{z})|D|+ \frac{2\dot{ \imath} {\gamma}(|D(z^{*};z)|\overline{D}|-\overline{D}(z^{*} ;z)|D|)-\sum_{s=1}^{m}\sigma_{s}|D(a_{s}^{*};z)|\overline{D}(z^{*};a_{s})|\sum_{s=1}^{n-m}\frac{\sigma_{m+s}\rho_{s}^{2} {k_{s}^{2} |D(z_{s}^{*};z_{x}) |\overline{D}(z^{*};z_{s})|=-|D(\tilde{z}^{*};z_{x})|\overline{D}|- |\overline{D}(z^{*};\tilde{z}_{x})|D|+ \frac{2i}{\gamma}(|D(z^{*};z_{x})|\overline{D}|-\overline{D}(z^{*};z_{x}) |D|)-\sum_{s=1}^{m}\sigma_{s}|D(a_{s}^{*};z_{x})|\overline{D}(z^{*};a_{s})|,.

Here, \tilde{z}=(\tilde{z}_{j})_{1\leq j\leq N} is an N ‐component row vector with elements(4.24)(4.25)\tilde{z}_{j}=\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}\frac{z_{j} {p_{j}-\dot{ {\imath} k_{s} .

Lemma 4.4.

|D(z_{x}^{*};z)||\overline{D}|+|D(z^{*};z)||\overline{D}(z^{*};z)|+ |\overline{D}(z^{*};z_{x})||D|=0 .

(4.26)

4.4. Proof of theorem 4.1

The proof of theorem 4.1 can be performed by using the differentiation rules for the tau functions
f, g_{s} and h_{s} given by lemmas 4.1 and 4.2 as well as lemmas 4.3 and 4.4.
4.5. One‐soliton solutions

If we putp_{1}=a+ib, z=e^{a\xi+i\{bx+(a^{2}-b^{2}-\hat{\lambda}\gamma b)t\}}, \xi=x-(2b+ \hat{\lambda}\gamma)t, (4.27a)the tau functions (4.10) for the one‐soliton solutions are written in the form

f= \frac{1}{2a}\{zz^{*}+\frac{1}{2}(\mu-\gamma b-i\gamma a)c_{11}\} , g_{j}= \alpha_{j1}^{*}z, j=1,2, m,

168h_{j}=f- \frac{z ^{*} {a-\dot{ \imath} (b-k_{j})},

j=1,2

,

...,

(4.27b)n-m.

The parameter c_{11} from (4.10d) is given byc_{1 }= \frac{\sum_{s=1}^{m}\sigma_{s}\alpha_{s1}\alpha_{s1}^{*} {1+ \frac{\gam a}{2}\sum_{s=1}^{n-m}\sigma_{m+s}\rho_{s}^{2}\frac{-2b+k_{s}+ \mu/\gam a}{a^{2}+(b-k_{s})^{2} . (4.27c)
Note that c_{11} is real. This also follows directly from the Hermitian nature of c_{jk} . If one introduces

the real quantities \beta, \phi and \theta_{j} by (3.27) and puts (a\beta\gamma)^{2}=e^{-4a\xi_{0}} , then one can express theone‐soliton solutions compactly in terms of these parameters. After some manipulations, one
finds that|q_{j}|^{2}= \frac{2a^{2}\alpha_{j1}1\alpha_{j}^{*}e^{2a\xi_{0} {\cosh 2a(\xi+ \xi_{0})-sgn(a\gamma c_{1 })\cos 2\phi},

j=1,2 , ..., m ,(4.28)|q_{j+m}|^{2}= \rho_{j}^{2}[1+\frac{sgn(a\gamma c_{11})\{\cos 2(\theta_{j}+ \phi)\cos 2\phi\} {sh2a(\xi+\xi_{0})-s,n(a\gamma c_{11})co,2\phi}], j=1,2, n-m .

(4.29)

The components q_{j} from (4.28) take the form of bright‐solitons with zero background whereas
those of (4.29) represent the dark‐ or bright‐solitons with nonzero background. A striking featureof the soliton solutions is that the parameters a and b can be chosen independently unlike thesoliton solutions with the pure plane‐wave boundary conditions discussed in section 3. It turns
out that in the case of the mixed type boundary conditions, the explicit form of the N ‐soliton
solution is available without solving algebraic equations.
If one takes the limit \gammaarrow 0 under the condition \mu c_{11}>0 , then the expressions (4.28) and(4.29) reduce to the one‐soliton solutions of the n‐component NLS system|q_{j}|^{2}=a^{2}\alpha_{j1}\alpha_{j1}^{*}e^{2a\xi_{0}}sech^{2}a(\xi+\xi_{0}) , j=1,2, m ,|q_{j+m}|^{2}= \rho_{\dot{j} ^{2}[1-\frac{a^{2} {a^{2}+(b-k_{j})^{2} sech^{2}a( \xi+\xi_{0})] , j=1,2, n-m(4.30).

(4.31)

Note that the condition \mu c_{11}>0 assures the regularity of the solutions. Actually, if \mu c_{11}<0,
then the solutions exhibit a singularity at \xi=-\xi_{0}.
Remark 4.2. The N‐soliton formula presented in theorem 4.1 have an alternative expressions.

Indeed, in accordance with the procedure developed in [13], we can show that the tau functions
given below satisfy the system of bilinear equations (4.4)-(4.7)

f=|\begin{ar ay}{l} \hat{A} I -I \hat{B} \end{ar ay}|, g_{s}=|\begin{ar y}{l \hat{A} I z^{T} -I \hat{B} 0^{T} 0 -a_{s}^* 0 \end{ar y}|,Âh_{s}=\^{A}= (âjk)l \leq j,k\leq N,‐I

I\hat{B}0^{T}z_{s}^{*}/k_{s}

0

1

s=1,2

,

...,

m,

z^{T}
s=1,2,n-m,\^{a} jk=\frac{z_{j}z_{k}^{*} {p_{j}+p_{k} *, \hat{B}=(\hat{b}_{jk})_{1\leq j,k\leq N}, \hat{b}_{jk}=\frac{\frac{1}{2}(\mu+\dot{ \imath} \gam ap_{k})c_{jk} {p_{j}+p_ {k} *,(4.32a)(4.32b)

169
where the N ‐component row vectors z, z_{s} and a_{s} are defined in (4.10) and I is an N\cross N unit
matrix. It is noteworthy that the tau functions f and g_{s} have the same forms as those of thebright N ‐soliton solution presented in [13]. Indeed, if one puts \rho_{s}=0(s=1,2, \ldots, n-m) , orequivalently

q_{m+s}=0(s=1,2, \ldots, n-m) , then q_{s}=g_{s}f^{*}/f^{2}(s=1,2, \ldots, m) solve the system

of PDEs (1.1) with the boundary conditions q_{s}arrow 0, |x|arrow\infty . Since m is an arbitrary positive
integer, this gives another proof of the bright N ‐soliton solution.
5. Concluding remark
In conclusion, it will be worthwhile to comment on an integrable system associated with thesystem (1.1). The multi‐component Fokas‐Lenells (FL) system

u_{j,xt}=u_{j}- i\{(\sum_{s=1}^{n}a_{s}u_{s,x}u_{s}^{*})u_{j}+(\sum_{s=1}^{n} \sigma_{s}u_{s}u_{s}^{*})u_{j,x}\},u_{j}=u_{j}(x, t)\in \mathbb{C},

j=1,2

,

...,

n,

(5.1)

is an integrable multi‐component generalization of the FL equation which describes the nonlinear

propagation of short pulses in a monomode fiber [19, 20]. It belongs to the first negative flowof the multi‐component derivative NLS hierarchy [21, 22]. The FL equation is a special caseof the system (5.1) with n=1 . Its N ‐soliton solutions have been obtained by employing
the direct method for both zero and plane‐wave boundary conditions [23, 24]. In view of theabove observation on an integrable hierarchy, the structure of the N‐soliton solution of themulti‐component FL system is closely related to that of the multi‐component derivative NLS

system. This statement has been confirmed for the one‐component system [24]. Thus, thebilinearization and the construction of the N ‐soliton solution of the system (5.1) with zero andnonzero boundary conditions will be performed in accordance with the procedure developed in
the present paper. This interesting issue is currently under study.
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