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General N-soliton solution to a vector nonlinear Schrödinger equation

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2014 J. Phys. A: Math. Theor. 47 355203

(http://iopscience.iop.org/1751-8121/47/35/355203)

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General N-soliton solution to a vectornonlinear Schrödinger equation

Bao-Feng Feng

Department of Mathematics, The University of Texas-Pan American, Edinburg, TX78541, USA

Received 21 March 2014, revised 12 July 2014Accepted for publication 18 July 2014Published 18 August 2014

AbstractWe consider a general N-soliton solution to a vector nonlinear Schrödinger(NLS) equation of all possible combinations of nonlinearities including all-focusing, all-defocusing and mixed types. Based on the KP hierarchy reduc-tion method, we firstly construct general two-bright-one-dark and one-bright-two-dark soliton solutions in a three-coupled NLS equation, then we extendour analysis to a vector NLS equation to obtain a general N-soliton solution inGram determinant form. This formula unifies the bright, dark and bright-darksoliton solutions, which have been widely studied in the literature. The con-ditions for the existence of all types of soliton solutions with all possiblecombinations of nonlinearities are elucidated.

Keywords: bright-dark soliton, vector nonlinear Schrödinger equation, KP-hierarchy reduction method, tau-functionPACS numbers: 02.30.Ik, 05.45.Yv

(Some figures may appear in colour only in the online journal)

1. Introduction

The study of the nonlinear Schrödinger (NLS) equation and coupled nonlinear Schrödinger(CNLS) equation is of great interest since they have been recognized as generic models fordescribing the evolution of slowly varying wave packets in general nonlinear wave systems[1–5]. They arise in a variety of physical contexts such as nonlinear optics [6, 7], Bose-Einstein condensates [8], water waves [9] and plasma physics [10]. The single NLS equation

σ+ + =q q q qi 2 0 (1)t xx2

was found to be integrable by Zakharov and Shabat [11, 12], and admits a bright solitonsolution in the focusing case (σ = 1) and a dark soliton solution in the defocusing case(σ = −1). Manakov [13] first recognized an integrable case (Manakov system) for the CNLS

Journal of Physics A: Mathematical and Theoretical

J. Phys. A: Math. Theor 47 (2014) 355203 (22pp) doi:10.1088/1751-8113/47/35/355203

1751-8113/14/355203+22$33.00 © 2014 IOP Publishing Ltd Printed in the UK 1

equation

σ σ

σ σ

+ + + =

+ + + =

⎧⎨⎪⎩⎪

( )( )

q q q q q

q q q q q

i 2 0 ,

i 2 0 ,(2)

t xx

t xx

1, 1, 1 12

2 22

1

2, 2, 1 12

2 22

2

which can be classified into three types depending on the signs of the nonlinear terms:focusing-focusing (σ σ= = 11 2 ), defocusing-defocusing (σ σ= = − 11 2 ) and focusing-defocusing (σ = 11 , σ = − 12 ). For the focusing-focusing case, the Manakov system admitsa bright-bright soliton solution [13, 14]; for the defocusing-defocusing case, the Manakovsystem admits a bright-dark and dark-dark soliton solution [14–16]. However, the focusing-defocusing Manakov system admits all types of soliton solutions; i.e., bright-bright solitons,dark-dark solitons and bright-dark solitons [17–19]. The Manakov system can be easilyextended to a multi-component case, the so-called vector NLS equation

∑σ+ + = = ⋯=

⎛⎝⎜⎜

⎞⎠⎟⎟q q q q j Mi 2 0 , 1, 2, , . (3)j t j xx

k

M

k k j, ,1

2

The vector NLS equation (3) has been widely studied [5, 16, 17, 20]. Particularly, the bright-dark soliton solution has been investigated in [19, 21]. It was also found that two or morebright soliton collisions usually exhibit inelastic (shape changing) features [22–24]. Theseinteresting collision features have opened a way for the implementation of all optical logic ina way that does not require fabrication of individual gates [25, 26].

The KP hierarchy reduction method was firstly developed by the Kyoto school [27], andwas later used to obtain soliton solutions in the NLS equation, the modified KdV equation,the Davey-Stewartson equation and coupled higher-order NLS equations [28, 29]. Recently,this method has been applied to derive dark-dark soliton solutions in a two-coupled NLSequation of the mixed type [18]. Compared to the inverse scattering transform method [16]and Hirotaʼs bilinear method [30], the KP hierarchy reduction method starts with the generalKP hierarchy, including the two-dimensional Toda hierarchy [31], and derives the generalsoliton solution in either determinant or Pfaffian form reduced directly from the tau functionsof the KP hierarchy.

In the present paper, we consider general N-soliton solutions to the vector NLSequation (3) with all possible combinations of nonlinearity (σ = ±1k for = ⋯k M1, ). It isknown that the bright soliton solution for the NLS and CNLS equations is derived from thereduction of the multi-component KP hierarchy, while the dark soliton is obtained from thesingle-component KP hierarchy but with shifted singular points [18]. However, the bright-dark soliton solution has not been obtained by this reduction method so far. It is the aim of thepresent paper to derive general N-soliton solutions including bright, dark and mixed types tothe above M-coupled NLS equation by the KP hierarchy reduction method.

The rest of the paper is organized as follows. In section 2, we derive general two-bright-one-dark and one-bright-two-dark soliton solutions in a three-coupled NLS equation. Then,we extend our analysis to the vector NLS equation (3) to find its general N-soliton solutionscontaining bright, dark and bright-dark solutions in section 3. We summarize the paper insection 4, and present one- and two-soliton solutions in the Appendix.

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

2

2. General bright-dark soliton solution to the three-coupled NLS equation

We consider a general bright-dark soliton solution to the three-coupled NLS equation

σ σ σ

σ σ σ

σ σ σ

+ + + + =

+ + + + =

+ + + + =

⎧

⎨⎪⎪

⎩⎪⎪

( )( )( )

q q q q q q

q q q q q q

q q q q q q

i 2 0 ,

i 2 0 ,

i 2 0 .

(4)

t xx

t xx

t xx

1, 1, 1 12

2 22

3 32

1

2, 2, 1 12

2 22

3 32

2

3, 3, 1 12

2 22

3 32

3

Since the mixed-type vector solitons for equation (4) consist of two types (two-bright-one-dark and one-bright-two-dark), we will construct these two types of soliton solutions,respectively, in the subsequent two subsections.

2.1. Two-bright-one-dark soliton solution

Assuming q1 and q2 are of bright type and q3 is of dark type, we introduce the dependentvariable transformations

ρ= = =σ ρ σ ρ β σ ρ β+ −( ( )qg

fe q

g

fe q

h

fe, , , (5)t t x t

11 2i

22 2i

3 11 i 2 )3 1

23 1

21 3 1

212

which transform the coupled NLS equation (4) into the following bilinear equations

∑

β

σ ρ σ σ ρ

+ = =

+ + =

+ = +=

⎜ ⎟

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

⎛⎝

⎞⎠

( )( )

D D g f j

D D D h f

D f f g h

i · 0 , 1, 2

i 2i · 0 ,

1

2· ,

(6)

t x j

t x x

xj

j j

2

21 1

23 1

2

1

22

3 12

12

where Hirotaʼs bilinear operator is defined as

= ∂∂

− ∂∂ ′

∂∂

− ∂∂ ′

′ ′′ ′= =

⎜ ⎟ ⎜ ⎟⎛⎝

⎞⎠

⎛⎝

⎞⎠ ( )D D f x t g x t

x x t tf x t g x t( , ) · ( , ) ( , ) , .x

ntl

n l

x x t t,

Firstly, we present the Gram determinant solutions satisfying the bilinear equations (6),thus providing a two-bright-one-dark soliton solution to the coupled NLS equation (4):

Φ=

−=

−= −f A I

I Bh A I

I Bg

A II B

C0

0, ,

0, (7)j

T

T

j

1(1)

where I is an N × N identity matrix; 0 is an N-component zero-row vector; A, A(1) and B areN × N matrices; and Φ and Cj are N-component row vectors whose elements are definedrespectively by

ββ

=+

=+

−−+

ξ ξ ξ ξ+ +⎛⎝⎜⎜

⎞⎠⎟⎟a

p pe a

p p

p

pe

1

¯,

1

¯

i

¯ i, (8)ij

i jij

i j

i

j

¯ (1) 1

1

¯i j i j

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

3

α σ α=

∑

+ +σ ρ

β β

=

+ −

⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( )( )

b

p p

¯

¯ 1

, (9)ijk i

kk j

k

i jp p

12 ( ) ( )

¯ i ii j

3 12

1 1

Φ α α α= ⋯ = − ⋯ξ ξ ξ( ) ( )e e e C, , , , , , , , (10)jj j

Nj

1( )

2( ) ( )N1 2

where ξ ξ= + +p x p tii i i i2

0, pi, αi and ξi0 ( = ⋯i N1, 2, , ) are complex constants.Now we proceed to show how the above soliton solution is constructed based on the KP

hierarchy reduction method. Let us start with a concrete form of the Gram determinantexpression of the tau function for a three-component KP hierarchy,

τ =−A II B

, (11)k0, 0

1

τΦ

Ψτ Ψ

Φ= −

−= −

−−

A II B

A I

I B00

0

0¯ 0

,¯ 0

, (12)k

T

T k

T

T1, 0 1, 01 1

τΦ

Υτ Υ

Φ= −

−= −

−−

A II B

A I

I B00

0

0¯ 0

,¯ 0

, (13)k

T

T k

T

T0, 1 0, 11 1

where A and B are N × N matrices, and Φ, Ψ , Υ , Φ, Ψ and Υ are N-component row vectorswhose elements are defined respectively as

=+

−−+

=+

++

ξ ξ η η χ χ+ + +⎛⎝⎜⎜

⎞⎠⎟⎟a

p p

p c

p ce b

q qe

r re

1

¯ ¯,

1

¯

1

¯,ij

k

i j

i

j

k

iji j i j

¯ ¯ ¯i j i j i j1

1

Φ Ψ Υ= ⋯ = ⋯ = ⋯ξ ξ ξ η η η χ χ χ( ) ( ) ( )e e e e e e e e e, , , , , , , , , , , ,N N N1 2 1 2 1 2

Φ Ψ Υ= ⋯ = ⋯ = ⋯ξ ξ ξ η η η χ χ χ( ) ( ) ( )e e e e e e e e e¯ , , , , ¯ , , , , ¯ , , , ,¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯ ¯N N N1 2 1 2 1 2

with

ξ ξ ξ ξ=−

+ + + =+

+ − +− −p c

x p x p xp c

x p x p x1

, ¯ 1

¯¯ ¯ ¯ ,i

ii i i j

ij j j1

(1)1

22 0 1

(1)1

22 0

η η η η χ χ χ χ= + = + = + = +q y q y r y r y, ¯ ¯ ¯ , , ¯ ¯ ¯ .i i i j j j i i j j j j1(1)

0 1(1)

0 1(2)

0 1(2)

0

Here pi, pj, qi, qj, ξr r, ¯ ,i j i0, ξ j0, ηi0, η j0, χ χ c, ¯ andj j0 0 are complex constants. Based on Satotheory for the KP hierarchy [27], the above tau functions satisfy the following bilinearequations

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

4

τ τ

τ τ

τ τ

τ τ τ τ

τ τ τ τ

τ τ τ τ

− =

− =

− − =

= −

= −

− = −

+

−

−

+ −−

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

( )( )( )

( )

D D

D D

D D cD

D D

D D

D D

· 0 ,

· 0 ,

2 · 0 ,

· 2 ,

· 2 ,

2 · 2 .

(14)

x xk k

x xk k

x x xk k

x yk k k k

x yk k k k

x xk k k k

21, 0 0, 0

20, 1 0, 0

20, 0

10, 0

0, 0 0, 0 1, 0 1, 0

0, 0 0, 0 0, 1 0, 1

0, 0 0, 0 0, 01

0, 01

2 11 1

2 11 1

2 1 11 1

1 1(1) 1 1 1 1

1 1(2) 1 1 1 1

1 1(1) 1 1 1 1

The proof of the above equations can be shown by using the Grammian technique [30, 31],which is omitted here. In what follows, we will perform reductions to the above bilinearequations. First, we apply the complex conjugate reduction by assuming −x x y y, , ,1 1

(1)1(1)

1(2) are

real, x2, c are pure imaginary and by letting pj, qj be the complex conjugates of pj and qj,respectively. Furthermore, we define

τ τ τ τ= = = =f g g h, , , .0, 00

1 1, 00

2 0, 10

1 0, 01

Under these assumptions, it is easy to check that

= =a a b b, ,ijk

jik

ij ji1 1

thus, f is real and

τ τ τ= − = − =− −−g g h¯ , ¯ , ¯ ,1 1, 0

02 0, 1

01 0, 0

1

where ¯ denotes the complex conjugate of the corresponding tau functions. In this way, thebilinear equations (14) read

− = =

− − =

= =

− = −−

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

( )( )

( )

D D g f j

D D cD h f

D D f f g g j

D D f f h h

· 0 , 1, 2

2 · 0 ,

· 2 ¯ , 1, 2

2 · 2 ¯ .

(15)

x x j

x x x

x y j j

x x

2

21

1 1

j

2 1

2 1 1

1 1( )

1 1(1)

Next, we show how the bilinear equations (15) are reduced to (6) by a dimension reductionrequiring

σ σ σ ρ= + −−

f f f f . (16)x y y x1 2 3 12

1 1(1)

1(2)

1(1)

Note that, by row operations, f can be rewritten as

= ′− ′

f A II B

,

where the entries of matrices ′A and ′B are

′ =+

′ =+

++

η η ξ ξ χ χ ξ ξ+ + + + + +ap p

bq q

er r

e1

¯,

1

¯

1

¯,ij

i jij

i j i j

¯ ¯ ¯ ¯i j i j i j i j

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

5

and where each exponent in ′bij can be divided into two parts

η ξ+ = ++

+ + ⋯−q yp c

x p x¯ 1

¯¯ ,i i i

ii1

(1)1

(1)1

η ξ+ = +−

+ + ⋯−q yp c

x p x¯ ¯1

,j j jj

j1(1)

1(1)

1

χ ξ+ = ++

+ + ⋯−r yp c

x p x¯ 1

¯¯ ,i i i

ii1

(2)1

(1)1

χ ξ+ = +−

+ + ⋯−r yp c

x p x¯ ¯1

.j j jj

j1(2)

1(1)

1

Therefore, under the reduction conditions

σσ ρ

σσ ρ

= ++

= +−

q pp c

q pp c

¯¯

, ¯ , (17)i ii

i ii

13 1

2

13 1

2

σσ ρ

σσ ρ

= ++

= +−

r pp c

r pp c

¯¯

, ¯ , (18)i ii

i ii

23 1

2

23 1

2

i.e.,

σ

σ ρ

σ

σ ρ

+=

+ ++ −

+=

+ ++ −

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

( ) ( )

( ) ( )

( )

( )

q q

p pp c p c

r r

p pp c p c

1

¯¯ 1

¯

,

1

¯¯ 1

¯

, (19)

i j

i j

i j

i j

i j

i j

1

3 12

2

3 12

the following relation holds

σ σ σ ρ∂ ′ = ∂ + ∂ − ∂ ′−( )b b ,x ij y y x ij1 2 3 1

21 1

(1)1(2)

1(1)

then (16) follows immediately.Differentiating (16) with respect to x1, we can also obtain

σ σ σ ρ= + −−

f f f f . (20)x x x y x y x x1 2 3 12

1 1 1 1(1)

1 1(2)

1 1(1)

On the other hand, the last two bilinear equations in (15) can be expanded as

− = − =f f f f g f f f f g, (21)x y x y x y x y12

22

1 1(1)

1 1(1)

1 1(2)

1 1(2)

and

− − = −− −

f f f f f h , (22)x x x x2

12

1 1(1)

1 1(1)

respectively. Multiplying (22) by σ ρ− | |3 12 and adding to the two equations in (21), we finally

arrive at

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

6

σ ρ σ σ σ ρ− + = + +f f f g g h , (23)x x x2

3 12 2

1 12

2 22

3 12

12

1 1 1

by using relations (16) and (20).Applying variable transformations

= =x x x t, i , (24)1 2

i.e.,

∂ = ∂ ∂ = − ∂, i , (25)x x x t1 2

the first two equations in (15) become the first two bilinear equation in (6) by assumingβ=c i 1. Equation (23) is nothing but the last bilinear equation in (6).In the last, we explain how the tau functions recover the bright-dark soliton solution of

the three-coupled NLS equation. Under the above variable transformations, the variables y1(1),

y1(2), −x 1

(1) become dummy variables, which can basically be treated as constants. Conse-quently, we could let α=ηe ¯i

(1)i , α=ηe i¯ (1)i , α=χe ¯i

(2)i , α= = ⋯χe i N( 1, 2, ,i¯ (2)i ); fur-

thermore, we let

Ψ Υ− = − =C C¯ , ¯ .1 2

Obviously (11)–(13) recover the two-bright-one-dark soliton solution (7).In what follows, we will illustrate one- and two-soliton solutions and make some

comments. By taking N = 1 in (7), or from (A.1)–(A.3) with m = 2, M = 3, we get the taufunctions for the one-soliton solution,

α= + = = +ξ ξ ξ ξ ξ+ +f c e g e h d e1 , , 1 , (26)jj

11¯

1( )

1 11(1) ¯

1 1 1 1 1

where

α σ α ββ

=∑

+ += −

−+σ ρ

β

=

−

⎛⎝⎜

⎞⎠⎟( )

c

p p

dp

pc

¯

¯ 1

,i

¯ i. (27)k

kk

k

p

111

21( )

1( )

1 12

i

11(1) 1 1

1 111

3 12

1 12

The above tau functions lead to the one-soliton solution as follows

αξ θ= + =ξ ( )q c e j

2sech , 1, 2 , (28)j

j

R1( )

11i

1 0I1

ρ ξ θ= + + − +ζ ϕ ϕ( )( ) ( )q e e e1

21 1 tanh , (29)R3 1

i 2i 2i1 01 1 1

where ξ ξ ξ= + iR I1 1 1 , ζ β σ ρ β= + −x t(2 | | )1 1 3 12

12 , =θe c2

110 , β β= − +ϕe p p(i ) (i ¯ )2i

1 1 1 11 .

Obviously, the amplitude of the bright soliton for qj is α a| |j1

2 1( )

11 (j = 1, 2). The darksoliton q3 approaches ρ| |1 as → ±∞x . In addition, the intensity of the dark solitonis ρ ϕ| | cos1 1.

By taking N = 2 in (7), or from (A.4)–(A.6) by having m = 2, M = 3, the tau functions forthe two-soliton solution are of the following form

= + + + + +ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ+ + + + + + +f c e c e c e c e c e1 , (30)11¯

21¯

12¯

22¯

1212¯ ¯

1 1 2 1 1 2 2 2 1 2 1 2

α α= + + +ξ ξ ξ ξ ξ ξ ξ ξ+ + + +g e e c e c e , (31)jj j j j

1( )

2( )

121( ) ¯

122( ) ¯

1 2 1 2 1 1 2 2

= + + + + +ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ+ + + + + + +h d e d e d e d e d e1 , (32)1 11(1) ¯

21(1) ¯

12(1) ¯

22(1) ¯

1212(1) ¯ ¯

1 1 2 1 1 2 2 2 1 2 1 2

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

7

where

α σ α ββ

=∑

+ += −

−+σ ρ

β β

=

− +

⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( )( )

c

p p

dp

pc

¯

¯ 1

,i

¯ i, (33)ij

k ik

k jk

i jp p

iji

jij¯

12 ( ) ( )

2

i ¯ i

¯(1) 1

1

¯

i j

3 12

1 1

= −+ +

−+ +

⎛⎝⎜⎜

⎞⎠⎟⎟( )( ) ( )( )

c p pc c

p p p p

c c

p p p p¯ ¯ ¯ ¯, (34)1212 2 1

2 11 22

1 2 2 1

12 21

1 1 2 2

α α β β

β β= −

+−

+=

− −

+ +

⎛⎝⎜⎜

⎞⎠⎟⎟

( )( )( )( )

c p pc

p p

c

p pd

p p

p pc( )

¯ ¯,

i i

¯ i ¯ i. (35)j

j

j

j

j12¯ 2 1

2(1)

1¯

2

1(1)

2¯

11212

1 1 2 1

1 1 2 1

1212

Several comments regarding the two-bright-one-dark soliton solution are in sequel.

Remark 2.1. In the three-coupled NLS equation (4), general two-bright-one-dark solitonsexist for any combinations of mixed type as long as the following condition

∑α σ α β σ ρ− + >=

⎛⎝⎜⎜

⎞⎠⎟⎟( )p¯ i 0 (36)

ki

kk i

ki

1

2( ) ( )

12

3 12

is satisfied. Specifically, the two-bright-one-dark soliton solution exists in both focusing-focusing-focusing (σ = 1j , =j 1, 2, 3) and defocusing-defocusing-defocusing cases(σ = − 1j , =j 1, 2, 3). For the former case there is no constraint since the condition (36)is automatically satisfied, whereas for the latter case a simpler condition

ρ β> −p i (37)i1 1

implied from (36) must be imposed. In other words, for all defocusing cases, the asymptoticvalue ρ| |1 for the dark soliton q| |3 as → ±∞x must be larger than β−p| i |i 1 . It is also notedthat the smaller the difference between ρ| |1 and β−p| i |i 1 , the larger the amplitude of the twobright solitons.

Remark 2.2. For the defocusing-defocusing-focusing case (σ σ= = − 11 2 , σ = 13 ), the two-bright-one-dark soliton solution does not exist since the condition (36) cannot be satisfied,whereas the three-coupled NLS equation with other mixed-type nonlinearities supports thetwo-bright-one-dark soliton solution. For example, if

α α ρ β> < −p, i , (38)i i i(1) (2)

1 1

or

α α ρ β< > −p, i , (39)i i i(1) (2)

1 1

there exists a two-bright-one-dark soliton solution for the focusing-defocusing-defocusingCNLS equation (σ = 11 , σ σ= = −12 3 ). The condition (38)–(39) can be interpreted asfollows: if the amplitude of the bright soliton in the focusing component (σ = 11 ) is greaterthan that of the bright soliton in the defocusing component (σ = −12 ), then ρ| |1 cannot exceed

β−p| i |i 1 . On the other hand, if the amplitude of the bright soliton in the focusing component(σ = 11 ) is smaller than that of the bright soliton in the defocusing component (σ = −12 ), thenρ| |1 must be greater than β−p| i |i 1 .

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

8

Remark 2.3. Based on the tau functions of the two-soliton solution listed above, we canconduct the asymptotic analysis as in [19] and conclude that it only undertakes the elasticcollision when

α

α

α

α= . (40)

1(1)

1(2)

2(1)

2(2)

Otherwise, it undertakes the inelastic (shape-changing) collision. Furthermore, it is easy tofind from (33)–(35) that the two-soliton solution in the focusing-defocusing-defocusingCNLS equation exists under the following three conditions

α α ρ β β

α α ρ β β

α α α α β ρ β

> = < − −

< = > − −

> < − < < −

{ }{ }

i p p

i p p

p p

(i): ( 1, 2), min i , i ,

(ii): ( 1, 2), max i , i ,

(iii): , , i i .

i i

i i

(1) (2)1 1 1 2 1

(1) (2)1 1 1 2 1

1(1)

1(2)

2(1)

2(2)

2 1 1 1 1

In case (i), the amplitudes of the two bright solitons in the focusing component are largerthan those in the defocusing component, and ρ| |1 has an upper bound. In case (ii), theamplitudes of the two bright solitons in the focusing component are smaller than those in thedefocusing component, and ρ| |1 has a lower bound. In case (iii) where the amplitudes of thetwo bright solitons have a combination of case (i) and (ii), ρ| |1 has both an upper bound and alower bound. It is further noted that both elastic and inelastic collisions between two brightsolitons could occur in cases (i) and (ii) since the condition for elastic collision (40) can besatisfied. An example of elastic collision is illustrated in figure 1 for the parameters chosen asα = 0.81

(1) , α = 1.01(2) , α = 1.62

(1) , α = 2.02(2) , β = −1.01 , ρ = 4.01 , = −p 3 i1 , = +p 2 i2 .

Conversely, since the condition for elastic collision conflicts with the existence condition incase (iii), the collision between two bright solitons is always inelastic. We show a specificexample by choosing the following parameters: α = 1.01

(1) , α = 0.51(2) , α = 1.62

(1) , α = 2.02(2) ,

β = 1.01 , ρ = 3.01 , = −p 3 i1 , = +p 2 i2 . Figures 2 (a) and (b) shows the wave profiles ofall three components before and after the collision.

Figure 1.An example of elastic collision between two solitons, where the wave profilesfor q1, q2 and q3 are shown in (a) before the collision at = −t 5, and in (b) after thecollision at t = 5 for the two-bright-one-dark soliton solution in the three-coupled NLSequation (2) with σ = 11 , σ σ= = − 12 3 .

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

9

2.2. One-bright-two-dark soliton solution

Assuming q1 is of bright type and q2 and q3 are of dark type, the dependent variabletransformations

ρ= =σ ρ σ ρ β σ ρ σ ρ β+ + + −( ) ( ( )qg

fe q

h

fe, , (41)t x t

11 2i

2 11 i )2 1

23 2

21 2 1

23 2

212

ρ= β σ ρ σ ρ β+ + −( ( )qh

fe , (42)x t

3 22 i )2 2 1

23 2

222

convert the three-coupled NLS equation (4) into the following bilinear equations

∑ ∑

β

σ ρ σ σ ρ

+ =

+ + = =

+ = +=

+=

+

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

⎛⎝⎜⎜

⎞⎠⎟⎟

( )( )

D D g f

D D D h f l

D f f g h

i · 0 ,

i 2i · 0 , 1, 2

1

2· .

(43)

t x

t x l x l

xl

l ll

l l l

21

2

2

1

2

12

1 12

1

2

12 2

Similarly, we first present the Gram determinant solution satisfying the bilinear equations (43)

Φ=

−=

−= −f A I

I Bh A I

I Bg

A II B

C0

0, , ,

0

(44)ll

T

T( )

1

1

where A, A l( ) and B are N × N matrices, and Φ and C1 are N -component row vectors whoseelements are defined respectively by

ββ

=+

=+

−−+

ξ ξ ξ ξ+ +⎛⎝⎜⎜

⎞⎠⎟⎟a

p pe a

p p

p

pe

1

¯,

1

¯

i

¯ i, (45)ij

i jij

l

i j

i l

j l

¯ ( ) ¯i j i j

Figure 2. An example of inelastic collision between two solitons, where the waveprofiles for q1, q2 and q3 are shown in (a) before the collision at = −t 5, and in (b) afterthe collision at t = 5 for the two-bright-one-dark soliton solution in the three-coupledNLS equation (2) with σ = 11 , σ σ= = − 12 3 .

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

10

α σ α=

+ + ∑σ ρ

β β= + −

+⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( )( )

b

p p

¯

¯ 1

, (46)iji j

i j l p p

(1)1

(1)

12

¯ i i

l l

i l j l

12

Φ α α α= ⋯ = − ⋯ξ ξ ξ( ) ( )e e e C, , , , , , , , (47)N1 1(1)

2(1) (1)N1 2

where ξ ξ= + +p x p tii i i i2

0, pi, αi(1) and ξi0 ( = ⋯i N1, 2, , ) are complex constants.

To construct the above soliton solution, we start with the tau functions for a two-component KP hierarchy expressed in Gram determinants

τ =−A II B

, (48)k k0

,1 2

τΦ

Ψτ Ψ

Φ= −

−= −

−−

A II B

A I

I B00

0

0¯ 0

,¯ 0

, (49)k k

T

T k k

T

T1,

1,1 2 1 2

where Φ, Ψ , Φ and Ψ are the N-component row vectors defined previously, and A and B areN×N matrices whose elements are defined respectively as

=+

−−+

−−+

=+

ξ ξ η η+ +⎛⎝⎜⎜

⎞⎠⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟a

p p

p c

p c

p d

p de b

q qe

1

¯ ¯ ¯,

1

¯,ij

k k

i j

i

j

k

i

j

k

iji j

, ¯ ¯i j i j1 2

1 2

with

ξ ξ=−

+−

+ + +− −p c

xp d

x p x p x1 1

,ii i

i i i1(1)

1(2)

12

2 0

ξ ξ=+

++

+ − +− −p c

xp d

x p x p x¯ 1

¯

1

¯¯ ¯ ¯ ,j

i ij j j1

(1)1

(2)1

22 0

η η η η= + = +q y q y, ¯ ¯ ¯ .i i i j j j1(1)

0 1 0

Here pi, pj, qi, qj, ξi0, ξ j0, ηi0, η j0, c are complex constants. Based on the Sato theory, theabove tau functions satisfy the following bilinear equations

τ τ

τ τ

τ τ

τ τ τ τ

τ τ τ τ

τ τ τ τ

− =

− − =

− − =

= −

− = −

− = −

+

+

−

+ −

+ −

−

−

⎧

⎨

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

( )( )( )

( )( )

D D

D D cD

D D dD

D D

D D

D D

· 0 ,

2 · 0 ,

2 · 0 ,

· 2 ,

2 · 2 · ,

2 · 2 · .

(50)

x xk k k k

x x xk k k k

x x xk k k k

x yk k k k k k k k

x xk k k k k k k k

x xk k k k k k k k

21

,0

,

20

1,0

,

20

, 10

,

0,

0,

1,

1,

0,

0,

01,

01,

0,

0,

0, 1

0, 1

2 11 2 1 2

2 1 11 2 1 2

2 1 11 2 1 2

1 1(1) 1 2 1 2 1 2 1 2

1 1(1) 1 2 1 2 1 2 1 2

1 1(2) 1 2 1 2 1 2 1 2

Similarly, we perform reductions to the above bilinear equations. First, we apply the complexconjugate reduction by assuming − −x x x y, , ,1 1

(1)1

(2)1(1) are real, x c,2 and d are pure imaginary

and by letting pj and qj be the complex conjugates of pj and qj, respectively. Further, bydefining

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

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τ τ τ τ= = = =f g h h, , , ,0, 00

1 10, 0

1 01, 0

2 00, 1

it is easy to check that

= =− −a a b b, ,ijk k

jik k

ij ji, ,1 2 1 2

thus, f is real and

τ τ τ= − = =−− −g h h¯ , ¯ , ¯ .1 1

0, 01 0

1, 02 0

0, 1

The bilinear equations (50) can be recast into

β

− =

− − = =

=

− = − =−

⎧

⎨

⎪⎪⎪

⎩

⎪⎪⎪

( )( )

( )

D D g f

D D D h f l

D D f f g

D D f f h l

· 0 ,

2 · 0 , 1, 2 ,

· 2 ,

2 · 2 , 1, 2 .

(51)

x x

x x l x l

x y

x x l

21

2

12

2l

2 1

2 1 1

1 1(1)

1 1( )

Similar to the two-bright-one-dark soliton case, we can show that if qi, qi satisfy

σσ ρ σ ρ

= ++

++

q pp c p d

¯¯ ¯

, (52)i ii i

12 1

23 2

2

σσ ρ σ ρ

= +−

+−

q pp c p d

¯ , (53)i ii i

12 1

23 2

2

i.e.,

σ+

=+ + +

σ ρ σ ρ

+ − + −

⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( ) ( )( ) ( )

q qp p

1

¯¯ 1

, (54)i j

i jp c p c p d p d

1

¯ ¯i j i j

2 12

3 22

the following relation holds,

σ σ ρ σ ρ= − −− −

f f f f , (55)x y x x1 2 12

3 22

1 1(1)

1(1)

1(2)

which also implies

σ σ ρ σ ρ= − −− −

f f f f . (56)x x x y x x x x1 2 12

3 22

1 1 1 1(1)

1 1(1)

1 1(2)

By using the two relations above, the last two bilinear equation in (51) are reduced to

∑ ∑σ ρ σ σ ρ− + = +=

+=

+f f f g h , (57)x x xl

l ll

l l l2

1

2

12 2

1 12

1

2

12 2

1 1 1

which is exactly the last bilinear equation in (43) through the variable transformation (24):=x x1 , =x ti2 . Under the same transformation, the first two bilinear equations in (51)

become the first two in (43). In summary, we complete the reduction from the bilinearequations in (50) to the ones in (43); thus, we are able to construct the general one-bright-two-dark soliton solution (44) to the vector NLS equation (3).

Before giving some remarks, we illustrate the one- and two-soliton solution below.One-soliton solution: By taking N = 1 in (44), or from (A.1)–(A.3) with m = 1, M = 3,

we get the tau functions for the one-soliton solution

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

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α= + =ξ ξ ξ+f c e g e1 , ,11¯

1 1(1)1 1 1

= + =ξ ξ+h d e l1 , 1, 2ll

11( ) ¯

1 1

where

α σ α ββ

=+ + +

= −−+σ ρ

β

σ ρ

β− −

⎛⎝⎜

⎞⎠⎟( )

c

p p

dp

pc

¯

¯ 1

,i

¯ i. (58)

p p

l l

l11

1(1)

1 1(1)

1 12

i i

11( ) 1

111

2 12

1 12

3 22

1 22

The above tau functions lead to the one-soliton solution as follows

αξ θ= +ξ ( )q c e

2sech , (59)R1

1(1)

11i

1 0I1

ρ ξ θ= + + − +ζ ϕ ϕ+ ( )( ) ( )q e e e

1

21 1 tanh , (60)l R1 1

i 2i 2i1 0l l l

where ξ ξ ξ= + iR I1 1 1 , ζ β σ ρ β= + −x t(2 | | )l l l3 12 2 , =θe c2

110 , β β= − +ϕe p p(i ) (i ¯ )l l

2i1 1

l .

The amplitude of the bright soliton for q1 is α a| |1

2 1(1)

11 ; the dark soliton +ql 1 (l = 1, 2)approaches ρ| |l as → ±∞x . Moreover, the intensities of the dark solitons for +ql 1 (l = 1, 2) areρ ϕ| | cosl l, respectively.

Two-soliton solution: By taking N = 2 in (44), or from (A.4)–(A.6) with m = 1, M = 3,we get the tau functions for the one-soliton solution

= + + + + +ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ+ + + + + + +f c e c e c e c e c e1 11¯

21¯

12¯

22¯

1212¯ ¯

1 1 2 1 1 2 2 2 1 2 1 2

α α= + + +ξ ξ ξ ξ ξ ξ ξ ξ+ + + +g e e c e c e1 1(1)

2(1)

121(1) ¯

122(1) ¯

1 2 1 2 1 1 2 2

= + + + + +ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ+ + + + + + +h d e d e d e d e d e1 ,ll l l l l

11( ) ¯

21( ) ¯

12( ) ¯

22( ) ¯

1212( ) ¯ ¯

1 1 2 1 1 2 2 2 1 2 1 2

where

∑α σ α σ ρ

β β

ββ

=+

+− +

= −−+=

+

−⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟( ) ( )( )

cp p p p

dp

pc

¯

¯1

i ¯ i,

i

¯ i, (61)ij

i j

i jl

l l

i l j l

ijl i l

j lij¯

(1)1

(1)

21

21

21

¯( )

¯

= −+ +

−+ +

⎛⎝⎜⎜

⎞⎠⎟⎟( )( ) ( )( )

c p pc c

p p p p

c c

p p p p¯ ¯ ¯ ¯, (62)1212 2 1

2 11 22

1 2 2 1

12 21

1 1 2 2

α α β β

β β= −

+−

+=

− −

+ +

⎛⎝⎜⎜

⎞⎠⎟⎟

( )( )( )( )

c p pc

p p

c

p pd

p p

p pc( )

¯ ¯,

i i

¯ i ¯ i, (63)i

i

i

i

i

l l l

l l12¯(1)

2 12(1)

1¯

2

1(1)

2¯

11212( ) 1 2

1 2

1212

Remark 2.4. In the three-coupled NLS equation (4), general one-bright-two-dark solitonsolutions exist for both the focusing-focusing-focusing and defocusing-defocusing-defocus-ing cases. However, in contrast with the former case, in which there is no restriction, theconstraint below must be imposed for the latter case

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

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ρ

β

ρ

β−+

−>

p pi i1 . (64)

i i

12

12

22

22

Remark 2.5. For all possible combinations of mixed type in the three-coupled NLSequation, a one-bright-two-dark soliton solution exists with a constraint

α σ ασ ρ

β

σ ρ

β+

−+

−>

⎛

⎝⎜⎜

⎞

⎠⎟⎟( )

p p¯ 1

i i0 . (65)i i

i i

(1)1

(1) 2 12

12

3 22

22

The asymptotic analysis can be performed based on the tau functions of the two-solitonsolution listed above, whose details are omitted here. However, it should be pointed out thatthe two-soliton solution for the one-bright-two-dark soliton case always undertakes elasticcollision without shape changing. In addition, it is interesting to note that the bright solitonmay occur in the defocusing component. For example, in the defocusing-defocusing-focusingcase (σ σ= = − 11 2 , σ = 13 ), q1 can support the bright soliton if the following conditionholds

ρ

β

ρ

β−> +

−p pi1

i. (66)

i i

12

12

22

22

In figure 3, we show an example for the parameters chosen as α = 0.81(1) , α = 1.62

(1) ,β β= = − 1.01 2 , ρ = 4.01 , ρ = 2.01 , = −p 3 i1 , = +p 2 i2 .

Remark 2.6. Suppose the wave number for the i-th soliton among an N-soliton solution is= +p p pii iR iI , then ξ = −p x p t( 2 )iR iR iI . That is to say that the velocity for the i-th soliton is

p2 iI . A multiple bound state means a multiple soliton solution moving with the same velocity.As pointed out in [18], only a dark-dark bound state of order 2 exists in the two-coupled NLSequation of mixed type because the same piI can give at most two distinct piR due to theconstraint condition (43) in [18]. However, there is a less strict condition (66) in the three-coupled NLS equation. Under this condition, the same piI value can still give as many distinctpiR values as we want. Consequently, the bound states of bright-dark-dark solitons can existup to arbitrary order. To demonstrate such a bound state, we choose the parametersα = 0.81

(1) , α = 1.62(1) , β = − 1.01 , β = 1.02 , ρ = 4.01 , ρ = 2.01 , = −p 3 i1 , = −p 2 i2 .

The contour plots for q1, q2 and q3 are displayed in figures 4 (a)–(c), respectively, and theprofile is shown in figure 5.

3. General N-soliton solution to the vector NLS equation

3.1. N-bright-dark soliton solution to the vector NLS equation

Let us consider a general soliton solution consisting of m bright solitons and ( −M m) darksolitons to the vector NLS equation (3). To this end, we introduce dependent variabletransformations of the form

ρ= =∑ ∑σ ρ β σ ρ β

++ −

=

−

+=

−

+

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

qg

fe q

h

fe, ,j

j t

m l ll x t2i i 2 )

k

M m

k m k lk

M m

k m k l1

2

1

2 2

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

14

where = ⋯j m1, 2 , , = ⋯ −l M m1, 2, , , which transform equation (3) into the followingbilinear equations

Figure 3.An example of elastic collision between two solitons, where the wave profilesfor q1, q2 and q3 are shown in (a) before the collision at = −t 5, and in (b) after thecollision at t = 5 for the one-bright-two-dark soliton solution in the three-coupled NLSequation (2) with σ σ= = − 11 2 , σ = 13 .

Figure 4. The contour plots of a bound state in the three-coupled NLS equation(σ σ= = − 11 2 , σ = 13 ) for (a) q1, (b) q2 and (c) q3, respectively.

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

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∑ ∑ ∑

β

σ ρ σ σ ρ

+ = = ⋯

+ + = = ⋯ −

+ = +=

−

+= =

−

+

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

⎛⎝⎜⎜

⎞⎠⎟⎟

( )( )

D D g f j m

D D i D h f l M m

D f f g h

i · 0 , 1, 2, , ,

i 2 · 0 , 1, 2, , ,

1

2· .

(67)

t x j

t x l x l

xl

M m

l m lj

m

j jl

M m

l m l l

2

2

2

1

2

1

2

1

2 2

Similar to the bright-dark soliton in the three-coupled NLS equation, the following Gramdeterminants satisfy the above bilinear equations and thus provide a general bright-darksoliton solution

Φ=

−=

−= −f A I

I Bh A I

I Bg

A II B

C0

0, ,

0(68)l

l

j

T

T

j

( )

where A, A l( ) and B are ×N N matrices and Φ and Cj are N-component row vectors whoseelements are defined respectively by

ββ

=+

=+

−−+

ξ ξ ξ ξ+ +⎛⎝⎜⎜

⎞⎠⎟⎟a

p pe a

p p

p

pe

1

¯,

1

¯

i

¯ i, (69)ij

i jij

l

i j

i l

j l

¯ ( ) ¯i j i j

α σ α=

∑

+ + ∑σ ρ

β β

=

=−

+ −

+⎛⎝⎜⎜

⎞⎠⎟⎟( ) ( )( )

b

p p

¯

¯ 1

, (70)ijkm

ik

k jk

i j lM m

p p

1( ) ( )

1 ¯ i i

l m l

i l j l

2

Φ α α α= ⋯ = − ⋯ξ ξ ξ( ) ( )e e e C, , , , , , , , (71)jj j

Nj

1( )

2( ) ( )N1 2

where ξ ξ= + +p x p tii i i i2

0, pi, αik( ) and ξi0 ( = ⋯i N1, 2, , ) are complex constants.

In the same spirit as the three-coupled NLS equation, the general bright-dark solitonsolution to the vector NLS equation can be derived by the KP hierarchy reduction method.We consider an +m( 1)-component KP hierarchy with −M m( ) copies of shifted singularpoints in the first component. Based on the transformation group theory of solitons [27], the

Figure 5. A snapshot of the bound state in the three-coupled NLS equation(σ σ= = − 11 2 , σ = 13 ).

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

16

following bilinear equations hold

β

− =

− − =

=

− = −−

⎧

⎨⎪⎪⎪

⎩⎪⎪⎪

( )( )

D D g f

D D D h f

D D f f g g

D D f f f h h

· 0 ,

2i · 0 ,

· 2 ¯ ,

· 2 2 ¯ ,

(72)

x x j

x x l x l

x y j j

x x l l

2

2

2

j

l

2 1

2 1 1

1 1( )

1 1( )

where f g h, ,j l ( = ⋯j m1, 2, , , = ⋯ −l M m1, 2, , ) are tau functions, which can beexpressed as Gram determinants by imposing the complex conjugate condition. Furthermore,by a dimension reduction requiring

∑ ∑σ σ ρ− == =

−

+ −f f f , (73)

j

m

j yl

M m

l m l x x1 1

2j l

1( )

1( )

which can be realized by imposing constraints on wave numbers of other m-components ofthe KP hierarchy similar to (17), (18) and (52), (53), the bilinear equations (72) are reduced to(67) by applying the same variable transformations (24) as for the three-coupled NLSequation. Meanwhile, the general bright soliton solution can be reduced from the taufunctions of the KP hierarchy, which is given in (78). In the appendix, we present the explicitform of the bright-dark solution mentioned above for the N = 1 and N = 2 cases.

3.2. General bright soliton solution to the vector NLS equation

A general bright soliton solution to the vector NLS equation (3) has been found in theliterature. Here we briefly show how this solution can be constructed from the KP hierarchyreduction method. To this end, the dependent variable transformations

= = ⋯qg

fj M, 1, 2 , ,j

j

convert equation (3) into the following bilinear equations

∑σ

+ = = ⋯

==

⎧⎨⎪⎪

⎩⎪⎪

( )D D g f j M

D f f g

i · 0 , 1, 2, , ,

· .(74)

t x j

xj

M

j j

2

2

1

2

Starting from the +M( 1)-component KP hierarchy, the N-bright soliton solution inGram determinant form can be constructed in the same way. The tau functions of f, gj take thesame form as in (78)

Φ=

−= −f A I

I Bg

A II B

C0

0,

0(75)j

T

T

j

with the exception that the elements for matrices B need to be modified as

α σ α=

+=

∑

+ξ ξ+ =

( )a

p pe b

p p

1

¯,

¯

¯. (76)ij

i jij

kM

ik

k jk

i j

¯ 1( ) ( )

i j

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

17

3.3. General dark soliton solution to the vector NLS equation

To find a general dark soliton solution to the vector NLS equation (3), the dependent variabletransformations

ρ= = ⋯∑β σ ρ β+ −

=

⎛⎝⎜⎜

⎛⎝⎜⎜

⎞⎠⎟⎟

qh

fe l M, 1, 2, ,l l

l x ti 2 )lk

M

k k l1

2 2

transform equation (3) into the following bilinear equations

∑ ∑

β

σ ρ σ ρ

+ + = = ⋯

+ == =

⎧⎨⎪⎪

⎩⎪⎪

⎛⎝⎜⎜

⎞⎠⎟⎟

( )D D i D h f l M

D f f h

i 2 · 0 , 1, 2, , ,

1

2· .

(77)

t x l x l

xl

M

l ll

M

l l l

2

2

1

2

1

2 2

Starting from a single-component KP hierarchy with M copies of shifted singular points, thegeneral N-dark soliton solution can be constructed and formulated by ×N N Gramdeterminants. This has been carried out for the two-coupled NLS equation in [18]. However,it is known that the general N-dark soliton solution can alternatively be of the same form as(78)

=−

=−

f A II B

h A II B

, . (78)ll( )

with the exception that matrix B needs to be changed into an identity matrix; i.e.,

δ=+

=ξ ξ+ap p

e b1

¯, . (79)ij

i jij ij

¯i j

The reason lies in the simple fact that

−= +A I

I II A . (80)

In addition, a dimension reduction requiring

∑σ ρ ==

−f f ,

l

M

l l x x1

2l1

( )

leads to a condition for the existence of the dark soliton solution

∑σ ρ

β β+ −= −

= ( )( )p p¯ i i1 , (81)

l

Ml l

i l i l1

2

which agrees with the condition given in [18] for the two-coupled NLS equation.

4. Summary and concluding remarks

In this paper, based on the KP hierarchy reduction method, we have constructed general two-bright-one-dark and one-bright-two-dark soliton solutions in a three-coupled NLSequation (4), then we extended our analysis to a vector NLS equation (3) to obtain an m-bright- −M m( ) -dark soliton solution. This general solution exists in the vector NLSequation (3) of all types, including all-focusing, all-defocusing and mixed types. One unifiedcondition for the existence of an N-soliton solution is found as follows

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

18

∑ ∑α σ ασ ρ

β+

−> = ⋯

= =

− +⎛⎝⎜⎜

⎞⎠⎟⎟

⎛

⎝⎜⎜

⎞

⎠⎟⎟p

i N¯ 1i

0 , 1, 2, , . (82)k

mk

kk

l

M ml m l

i l11( )

1( )

1

2

2

The general bright soliton solution can be viewed as a special case for a bright-dark solitonsolution with m =M. Therefore, the N-bright soliton solution takes the same determinant formas the general bright-dark soliton solution. Usually, an N-dark soliton is reduced from thesingle KP hierarchy and its solution form is different from the ones for the bright and bright-dark solitons. In this paper, we have proposed a unified formula in the Gram determinant toinclude bright, dark and bright-dark soliton solutions.

Most recently, rogue wave solutions have been reported in vector NLS equations usingthe Darboux dressing techniques [32–34]. In addition, Ohta and Yang have derived generalrogue wave solutions for the NLS equation [35] as well as for the the Davey-Stewartson Iequation [36]. Therefore, the KP hierarchy reduction method can be applied to attain generalsoliton solutions including the general rogue wave solutions for the vector NLS equation, aswell as other coupled integrable systems such as coupled derivative NLS equations [29] andthe Yajima-Oikawa system [37].

Acknowledgements

The author is grateful to Professor Y Ohta and Professor K Maruno for their very usefuldiscussions and comments.

Appendix A

By taking N = 1 in (78), we get the tau functions for the one-soliton solution,

∑

∑

α σ α

σ ρ

β

=

+

−

+ +−

= +

ξ ξ

ξ ξ

+

=

=

− +

+

⎛

⎝⎜⎜

⎞

⎠⎟⎟( )

f

p pe

p pp

a e

1

¯1

1

¯

¯ 1i

1 , (A.1)

k

mk

kk

l

M ml m l

l

1 1

¯

11( )

1( )

1 11

2

12

11¯

1 1

1 1

∑

∑

ββ

α σ α

σ ρ

β

=

−−+ +

−

+ +−

= +

ξ ξ

ξ ξ

+

=

=

− +

+

⎛

⎝⎜⎜

⎞

⎠⎟⎟( )

h

p

p p pe

p pp

b e

i

¯ i

1

¯1

1

¯

¯ 1i

1 , (A.2)

l

l

l

k

mk

kk

l

M ml m l

l

l

1

1 1 1

¯

11( )

1( )

1 11

2

12

11( ) ¯

1 1

1 1

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

19

∑

∑

α σ α

σ ρ

β

α

α

=

+

−

+ +−

−

=

ξ ξ ξ

ξ

+

=

=

− +⎛

⎝⎜⎜

⎞

⎠⎟⎟( )

g

p pe e

p pp

e

1

¯1

1

¯

¯ 1i

0

0 0

, (A.3)

jk

mk

kk

l

M ml m l

l

j

j

1 1

¯

11( )

1( )

1 11

2

12

1( )

1( )

1 1 1

1

where

α σ α ββ

=∑

+ + ∑= −

−+σ ρ

β

=

=−

−

+⎛⎝⎜

⎞⎠⎟( )

c

p p

dp

pc

¯

¯ 1

,i

¯ i.k

m kk

k

lM m

p

111 1

( )1( )

1 12

1 i

111

111

l m l

l

2

12

From (78) with N = 2, we get the tau functions for the two-soliton solution,

=

+ +

+ +

−

−

= + + + + +

∑

∑

∑

∑

∑

∑

∑

∑

ξ ξ ξ ξ

ξ ξ ξ ξ

α σ α

σ ρ

β

α σ α

σ ρ

β β

α σ α α σ α

σ ρ

β

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

+ +

+ +

+

+−

+

++

+ −

+

+

+

+−

+ + + + + + +

σ ρ

β β

+

+

+ −+

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎛⎝⎜

⎞⎠⎟

⎤

⎦

⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

( ) ( )

( )( )

( ) ( )

f

p pe

p pe

p pe

p pe

c e c e c e c e c e

1

¯

1

¯1 0

1

¯

1

¯0 1

1 0

0 1

1 (A.4)

( )( )

p p

p

p p

m

p p

p p p p

p

1 1

¯

1 2

¯

2 1

¯

2 2

¯

¯

¯

· 1i

¯

¯

· 1¯ i i

¯

¯

· 1

¯

¯

· 1i

11¯

21¯

12¯

22¯

1212¯ ¯

kk

k

l m l

l

kk

k

l l

l l

kk

k

l m l

p l p l

kk

k

l m l

l

1 1 1 2

2 1 2 2

1( )

1( )

1 1

2

12

1( )

2( )

1 2

2

1 2

2( )

1( )

2 1

2

¯2 i 1 i

2( )

2( )

2 2

2

22

1 1 2 1 1 2 2 2 1 2 1 2

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

20

=

−

−

= + + + + +

∑

∑

∑

∑

∑

∑

∑

∑

β

β

β

β

β

β

β

β

α σ α

σ ρ

β

α σ α

σ ρ

β β

α σ α

σ ρ

β β

α σ α

σ ρ

β

ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ ξ

− −

+ +

− −

+ +

− −

+ +

− −

+ +

+

+−

+

++

+ −

+

++ −

+

+−

+ + + + + + +

ξ ξ ξ ξ

ξ ξ ξ ξ

+ +

+ +

+

+ +

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

( )( )( )

( )( )( )

( )( )( )

( )( )( )

( ) ( )

( )( )

( )

( )( )

( )

h

d e d e d e d e d e

1 0

0 1

1 0

0 1

1 , (A.5)

l

p e

p p p

p e

p p p

p e

p p p

p e

p p p

p p

p

p p

m

p p

p p

p p

p p

p

l l l l l

i

¯ i ¯

i

¯ i ¯

i

¯ i ¯

i

¯ i ¯

¯

¯

· 1i

¯

¯

· 1¯ i i

¯

¯

· 1¯ i i

¯

¯

· 1i

11( ) ¯

21( ) ¯

12( ) ¯

22( ) ¯

1212( ) ¯ ¯

l

l

l

l

l

l

l

l

kk

k

l m l

l

kk

k

l l

l l

kk

k

l m l

l l

kk

k

l m l

l

11 1

1 1 1

11 2

2 1 2

22 1

1 2 1

22 2

2 2 2

1( )

1( )

1 1

2

12

1( )

2( )

1 2

2

1 2

2( )

1( )

2 1

2

2 1

2( )

2( )

2 2

2

22

1 1 2 1 1 2 2 2 1 2 1 2

α α

α α

=

+ +

+ +

−

−

− −

= + + +

∑

∑

∑

∑

∑

∑

∑

∑

ξ ξ ξ ξξ

ξ ξ ξ ξξ

α σ α

σ ρ

β

α σ α

σ ρ

β β

α σ α

σ ρ

β β

α σ α

σ ρ

β

ξ ξ ξ ξ ξ ξ ξ ξ

+ +

+ +

+

+−

+

++

+ −

+

++ −

+

+−

+ + + +

+

+ +

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

⎡

⎣

⎢⎢⎢⎢⎢

⎛

⎝⎜⎜

⎞

⎠⎟⎟

⎤

⎦

⎥⎥⎥⎥⎥

( ) ( )

( )( )

( )

( )( )

( )

g

e

p p

e

p pe

e

p p

e

p pe

e e c e c e

¯ ¯1 0

¯ ¯0 1

1 0 0

0 1 0

0 0 0

, (A.6)

jp p

p

p p

m

p p

p p

p p

p p

p

j j

j j j j

( )

¯

1 1

¯

1 2

¯

2 1

¯

2 2

¯

¯

· 1i

¯

¯

· 1¯ i i

¯

¯

· 1¯ i i

¯

¯

· 1i

1( )

2( )

1( )

2( )

121( ) ¯

122( ) ¯

kk

k

l m l

l

kk

k

l l

l l

kk

k

l m l

l l

kk

k

l m l

l

1 1 1 21

2 1 2 22

1( )

1( )

1 1

2

12

1( )

2( )

1 2

2

1 2

2( )

1( )

2 1

2

2 1

2( )

2( )

2 2

2

22

1 2 1 2 1 1 2 2

where

∑∑

α σ ασ ρ

β β

ββ

=+

+− +

= −−+

= −+ +

−+ +

=

=

− +

−⎛

⎝⎜⎜⎜

⎞

⎠⎟⎟⎟

⎛⎝⎜⎜

⎞⎠⎟⎟

( ) ( )( )

( )( ) ( )( )

cp p p p

dp

pc

c p pc c

p p p p

c c

p p p p

¯

¯1

i ¯ i,

i

¯ i,

¯ ¯ ¯ ¯,

ijk

m

ik

k jk

i jl

M ml m l

i l j l

ijl i l

j lij¯

1

( ) ( )

21

21

¯( )

¯

1212 2 12 11 22

1 2 2 1

12 21

1 1 2 2

J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

21

α α β β

β β= −

+−

+=

− −

+ +

⎛⎝⎜⎜

⎞⎠⎟⎟

( )( )( )( )

c p pc

p p

c

p pd

p p

p pc( )

¯ ¯,

i i

¯ i ¯ i.i

jj

i

i

ji

i

l l l

l l12¯( )

2 12( )

1¯

2

1( )

2¯

11212( ) 1 2

1 2

1212

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J. Phys. A: Math. Theor 47 (2014) 355203 B-F Feng

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