Applied Mathematics and Computation 217 (2010) 1463–1469

Contents lists available at ScienceDirect

Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Positons, negatons and complexitons of the mKdV equationwith non-uniformity terms

Yi Zhang a,*, Li-Jin Chu a, Bo-Ling Guo b

a Department of Mathematics, Zhejiang Normal University, Jinhua 321004, PR Chinab Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China

a r t i c l e i n f o a b s t r a c t

Keywords:ComplexitonsHirota methodWronskian techniqueThe mKdV equation with non-uniformity

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.05.064

* Corresponding author.E-mail address: zy2836@163.com (Y. Zhang).

The N-soliton solution of the mKdV equation with non-uniformity terms is obtainedthrough Hirota method and Wronskian technique. We can also derive its positons, nega-tons and complexitons by a matrix extension of the Wronskian formulation.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

In 1971, Hirota first proposed the formal perturbation technique, called the Hirota method [1], to obtain N-soliton solu-tion of the KdV equation. Equivalently at the same time a more compact form of obtaining N-soliton solution is derivedthrough Wronskian technique [2]. Satsuma gave the Wronskian representation of the multisoliton solution to the KdV equa-tion [3]. Then the Wronskian technique was developed by Freeman and Nimmo [4,5]. And this technique together with theHirota method is considered as one of the most efficient and direct approaches in deriving soliton solutions for nonlinearevolution equations with bilinear forms. A meaningful generalization came from Siriaunpiboon and co-workers [6]. It isnoted that Ma gave a systematic analyze the solution structures in detail to the KdV equation, and first introduced a newkind of exact solution, complexitons [7]. Meanwhile he also studied interaction solutions among rational solutions, negatons,positons and complexitons [8–11]. Following the substantial extensions above, one of the authors [12] proposed a matrixform to the KdV equation in using the Wronskian method, and derived solitons, rational solutions [13], limit solutionsand complexitons [14].

In this letter, we consider the mKdV equation with non-uniformity terms

ut þ 6u2ux þ uxxx þ buþ ðaþ bxÞux ¼ 0; ð1:1Þ

where a and b are arbitrary constants, and derive its solitons, positons, negatons and complexitons by using the matrixextension of the Wronskian formulation. Zhang and Chen have ever investigated the KdV equation with loss and non-uni-formity terms [15,16].

This paper is organized as follows. In Section 2, the bilinear form of the mKdV equation with non-uniformity terms is gi-ven and solved by means of Hirota method. In Section 3, the Wronskian solutions are constructed and verified. In Section 4,solitons, positons, negatons and complexitons to Eq. (1.1) are constructed. At last, a few concluding and remarks will begiven in Section 5.

. All rights reserved.

1464 Y. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1463–1469

2. N-soliton solution in terms of the Hirota method

The bilinear form of Eq. (1.1) is given by

Dt þ ðaþ bxÞDx þ D3x

h i�f � f ¼ 0; ð2:1aÞ

D2x�f � f ¼ 0; ð2:1bÞ

where D is the well-known Hirota bilinear operator defined by

Dmx Dn

t a � b ¼ @x � @x0ð Þm @t � @t0ð Þnaðx; tÞbðx0; t0Þjx0¼x;t0¼t

with u ¼ i ln�ff

� �x, and �f being the complex conjugate f.

Expanding f as a power series in a parameter e

fN ¼ 1þ RNi¼1e

if ðiÞ: ð2:2Þ

Substituting (2.2) into (2.1) and collecting terms with the same powers of e, we get

f ð1Þt � �f ð1Þt þ ðaþ bxÞ f ð1Þx � �f ð1Þx

� �þ f ð1Þxxx � �f ð1Þxxx ¼ 0; ð2:3aÞ

f ð2Þt � �f ð2Þt þ ðaþ bxÞ f ð2Þx � �f ð2Þx

� �þ f ð2Þxxx � �f ð2Þxxx ¼ � Dt þ ðaþ bxÞDx þ D3

x

h if ð1Þ � �f ð1Þ; ð2:3bÞ

�f ð1Þxx þ f ð1Þxx ¼ 0; ð2:3cÞ�f ð2Þxx þ f ð2Þxx ¼ �D2

x f ð1Þ � �f ð1Þ ð2:3dÞ

and so on.First we consider the one-soliton solution. Let f ð1Þ ¼ x1ðtÞen1þp

2i; n1 ¼ k1ðtÞxþx1ðtÞ þ nð0Þ1 , and nð0Þ1 is an arbitrary constant,where x1ðtÞ; k1ðtÞ is real function with respect to t. One finds from (2.3)

k1ðtÞ ¼ k1e�bt ; ð2:4aÞ

x1ðtÞ ¼abðk1ðtÞ � k1Þ þ

13bðk3

1ðtÞ � k31Þ; ð2:4bÞ

where k1 ¼ k1ð0Þ and x1ð0Þ ¼ 0.The substituting (2.4) into (2.3), we can find that the r.h.s of Eq. (2.3) vanishes and all higher order terms can be taken to

be zero. Consequently, an exact one-soliton solution can be derived as follows

u ¼ k1ðtÞcos h k1ðtÞxþx1ðtÞ þ nð0Þ1

� � : ð2:5Þ

And from this way, the general N-soliton solutions can be denoted by

fN ¼X

lj¼0;1

expXN

j¼1

lj nj þp2

i� �

þXN

16j6l

ljllAjl

" #; ð2:6Þ

where nj ¼ kjðtÞxþxjðtÞ þ nð0Þj ; eAjl ¼ kj�kl

kjþkl

� �2and xjðtÞ; kjðtÞ are given by Eq. (2.4) with k1 be replaced by kj, the sum is taken

over all possible combinations of lj ¼ 0;1ðj ¼ 1;2; . . . ;NÞ.

3. Wronskian form

In this section, we would derive the N-soliton solutions in the Wronskian form for the mKdV equation with non-uniformity terms. At first, we shall use the abbreviated notion of Freeman and Nimmo for the Wronskian and its derivativesin the following [4,5]. The Wronskian solution for Eq. (2.1) is given by

f ¼Wð/1;/2; . . . ;/NÞ ¼

/1 @/1 � � � @N�1/1

/2 @/2 � � � @N�1/2

� � � � � � � � � � � �/N @/N � � � @N�1/N

���������

��������� ¼ j0;1; . . . ;N � 1j ¼ j dN� 1j; ð3:1Þ

where the entries /jðj ¼ 1;2; . . . ;NÞ satisfies

/j;x ¼ kjðtÞ/j; ð3:2aÞ

/j;t ¼ �4/j;xxx � ðaþ bxÞ/j;x þN � 1

2ð1þ 4bÞ/j; ð3:2bÞ

with kj;tðtÞ ¼ �bkjðtÞ. From Eq. (3.2a), it is easy to induce

Y. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1463–1469 1465

/j ¼ kjðtÞ@�1/jðtÞ: ð3:3Þ

Then, in order to verify this solution, we shall compute the various derivatives of f and �f , which appear in Eq. (2.1). Here,we present the derivative of f as follows

fx ¼ j dN� 2;Nj; f xx ¼ j dN� 3;N � 1;Nj þ j dN� 2;N þ 1j� �

;

fxxx ¼ j dN� 4;N � 2;N � 1;Nj þ 2j dN� 3;N � 1;N þ 1j þ j dN� 2;N þ 2j� �

:ð3:4Þ

Noting that the /jðtÞ satisfy Eq. (3.3), thus we can get

�f x ¼YNj¼1

kjðtÞj � 1; dN� 3;N � 1j;

�f xx ¼YN

j¼1

kjðtÞ j � 1; dN� 4;N � 2;N � 1j þ j � 1; dN� 3;Nj� �

;

�f xxx ¼YNj¼1

kjðtÞ j � 1; dN� 5;N � 3;N � 2;N � 1j þ 2j � 1; dN� 4;N � 2;Nj þ j � 1; dN� 3;N þ 1j� �

;

In meantime, from Eq. (3.2) we have the time derivatives for f

ft ¼ �4 j dN� 4;N � 2;N � 1;Nj � j dN� 3;N � 1;N þ 1j þ j dN� 2;N þ 2j� �

� ðaþ bxÞfx þNðN � 1Þ

2ð1þ 3bÞf : ð3:5Þ

In addition, Eq. (3.2b) can be rewritten in the form

/j;t ¼ kjðtÞ �4@�1/j;xxx � ðaþ bxÞ@�1/j;x þN � 1

2ð1þ 4bÞ@�1/j

� �; ð3:6Þ

then we obtain the time derivatives of �f

�f t ¼ A �4j � 1; dN� 5;N � 3;N � 2;N � 1j � j � 1; dN� 4;N � 2;Nj�h

þj dN� 3;N þ 1j�� ðaþ bxÞ�f x þ

ðN � 1ÞNð1þ 3bÞ2

�f�: ð3:7Þ

Substitution of these expressions into (2.2), now yields

� 6YNj¼1

kjðtÞð�1ÞN�2 �j dN� 4;N � 2;N � 1jj dN� 4;N � 2;�1;N � 1j�h

þ j dN� 4;N � 2;�1;N � 3jj dN� 4;N � 2;N � 1;Nj

þ j dN� 4;N � 2;N � 3;N � 1jj dN� 4;N � 2;�1;Nj�þ j dN� 3;N � 2;N � 1jj dN� 3;�1;N þ 1j�

�j dN� 3;�1;N � 1jj dN� 3;N � 2;N þ 1j þ j dN� 3;�1;N � 2jj dN� 3;N � 1;N þ 1j�i; ð3:8Þ

where having made use of the two identities

XN

j¼1

k2j ðtÞj dN� 2;Nj

!j � 1; dN� 2j ¼ j dN� 2;Nj

XN

j¼1

k2j ðtÞj � 1; dN� 2j

!; ð3:9Þ

XN

j¼1

k2j ðtÞj � 1; dN� 3;N � 1j

!j dN� 1j ¼ j � 1; dN� 3;N � 1j

XN

j¼1

k2j ðtÞj dN� 1j

!: ð3:10Þ

In a very similar way, (2.1b) may be shown to identically zero. So, by induction, we have proved that the Wronskian solu-tion in the Eq. (2.2) gives the solution for the Eq. (2.1).

4. The solutions with matrix form

Define U ¼ ð/1;/2; . . . ;/NÞT , then the Wronskian solution (3.2), which is mentioned in the preceding section, can be ex-

tended as follows

Ux ¼ KðtÞUk; ð4:1aÞ

Ut ¼ �4Uj;xxx � ðaþ bxÞUx þN � 1

2ð1þ 4bÞUj; ð4:1bÞ

1466 Y. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1463–1469

where kij are arbitrary real constants, KðtÞ ¼ ðkijðtÞÞN�N and KðtÞ ¼ Kð0Þe�bt . This extension constructed a very broad class ofexact solutions to the Eq. (2.1), which are solitons, positons and complexitons. The prove of this extended Wronskian solu-tion (4.1) is not difficult, and the detail is omitted.

Then we can easily get the solution formula of (4.1), which can be expressed as

U ¼ eKðtÞxþXðtÞlþ e�KðtÞx�XðtÞþp2im

� �e

N�12 ð1þ4bÞt ; ð4:2Þ

where l; m are arbitrary initial vectors, XðtÞ ¼ 43b K3ðtÞ þ a

3b KðtÞ and the matrix KðtÞ ¼ ðkijÞN�N . Then we would like to considerthe construction of solutions. By linear algebra, we only need to consider some types of the coefficient matrix, then the sol-itons, positons, negatons and complexitons to the Eq. (2.1) can be derived.

4.1. Solitons

First, eKðtÞxþXðtÞ can be expended as

eKðtÞxþXðtÞ ¼X1i¼0

1i!

KiðtÞxi

" # X1j¼0

4j

j!3jbjK3jðtÞ

" # X1k¼0

ak

k!bkKkðtÞ

" #¼X1l¼0

Xl

h¼0

X½h3�j¼0

4jah�3j

ðl� hÞ!j!ðh� 3jÞ!3jbh�2jKlðtÞxl�h

" #

¼X1k¼0

X2k

h¼0

X½h3�j¼0

4ja2k�3j

ð2k� hÞ!j!ðh� 3jÞ!3jbh�2jK2kðtÞx2k�h

" #

þX1k¼0

X2kþ1

h¼0

X½h3�j¼0

4ja2k�3jþ1

ð2k� hþ 1Þ!j!ðh� 3jÞ!3jbh�2jK2kþ1ðtÞx2k�hþ1

" #: ð4:3Þ

Then, similarly have

e�KðtÞx�XðtÞ ¼X1k¼0

X2k

h¼0

X½h3�j¼0

4ja2k�3j

ð2k� hÞ!j!ðh� 3jÞ!3jbh�2jK2kðtÞx2k�h

" #

�X1k¼0

X2kþ1

h¼0

X½h3�j¼0

4ja2k�3jþ1

ð2k� hþ 1Þ!j!ðh� 3jÞ!3jbh�2jK2kþ1ðtÞx2k�hþ1

" #: ð4:4Þ

Therefore, in Eq. (4.2) we have

U ¼X1l¼0

Xl

h¼0

X½h3�j¼0

4jah�3j½lþ ð�1Þhim�ðl� hÞ!j!ðh� 3jÞ!3jbh�2j

KlðtÞxl�h

" #e

N�12 ð1þ4bÞt : ð4:5Þ

a 00 1

If the coefficient matrix KðtÞ ¼ Ae�bt and the matrix A ¼1

. ..

0 aN

B@ CA, where aiðtÞ ¼ aie�bt and aiðtÞ–0, then the Eq.(4.5) is equal to

U ¼ eN�1

2 ð1þ4bÞt eBjlþ e�Bjþp2im

; ð4:6Þ

b 00 1

where bj ¼ aje�btxþ 43b a3

j e�3bt þ ab aje�bt and the matrix B ¼

1

. ..

0 bN

B@ CA. Therefor the associated Wronskian solution to Eq.(1.1) can be immediately obtained by !

f ¼Wð/0;/1; . . . ;/k�1Þ;u ¼ i ln�ff

x

: ð4:7Þ

Without losing generality, let l ¼ m ¼ ð1;1; . . . ;1ÞTN�1 and the first two soliton solutions of lower order are

f ¼ eb1 1þ e�2b1þp2i

� �; ð4:8Þ

f ¼ ða1 � a2Þeð1þ4bÞt�bt�ðb1þb2Þþpi 1þ e2b1þ2b2 þ e2b1�lna1�a2a1þa2

þp2i þ e2b2�ln

a1�a2a1þa2

�p2i

h i; ð4:9Þ

which are all consistent to the solutions generated from Hirota method, and more general soliton solutions can also be gen-erated from this way.

4.2. Positons and negatons

If the coefficient matrix KðtÞ has the following type of Jordan blocks

JðkiðtÞÞ ¼

kiðtÞ 01 kiðtÞ... . .

. . ..

0 � � � 1 kiðtÞ

0BBBB@1CCCCA

s�s

¼ kiðtÞIkiþ Eki

; 1 6 ki 6 s; ð4:10Þ

Y. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1463–1469 1467

which kiðtÞ ¼ kie�bt , kiðtÞ–0, ki is positive integers, Ikirepresents l� l matrix, and

Eki¼

0 01 0... . .

. . ..

0 � � � 1 0

0BBBB@1CCCCA

ki�ki

: ð4:11Þ

Without losing generality, consider the first one Jðk1ðtÞÞ. Let l ¼ m ¼ 1 we have

Uk1ðtÞ ¼X1l¼0

Xl

h¼0

X½h3�j¼0

4jah�3j 1þ ð�1Þhih i

ðl� hÞ!j!ðh� 3jÞ!3jbh�2jJlðk1ðtÞÞxl�h

24 35eN�1

2 ð1þ4bÞtlk1ðtÞ; ð4:12Þ

where lk1ðtÞ is arbitrary real vector. Noticing that

Jl½k1ðtÞ� ¼ ðk1e�bt Ik1 þ Ek1 Þl ¼ kl

1e�lbt Ik1 þ lkl�11 e�ðl�1ÞbtEk1 þ C2

l kl�21 e�ðl�2ÞbtE2

k1þ � � � þ Ck1�1

n kl�k1þ11 e�ðl�k1þ1ÞbtEk1�1

k1

¼

1 0ebt@k1 1 0

12!

e2bt@k1 ebt@k1 1 0

..

. . .. . .

. . .. . .

.

..

.12!

e2bt@2k1

ebt@k1 1 01

ðk1�1Þ! eðk1�1Þbt@

ðk1�1Þk1

� � � � � � 12!

e2bt@2k1

ebt@k1 1

0BBBBBBBBB@

1CCCCCCCCCAkl

1ðtÞ ¼ Jk11 ðtÞk

l1ðtÞ: ð4:13Þ

So

Uk1 ¼ Jk11 ed1 ðec1 þ e�c1þp

2iÞlk1; ð4:14Þ

where d1 ¼ k1�12 ð1þ 4bÞt, c1 ¼ k1e�btxþ 4

3b k31e�3bt þ a

b k1e�bt . When k1 ¼ 2, then one of the solution is generated from that

f ¼ 4k1 xþ 4b

k21e�2bt þ a

b

� �eð1þ3bÞtþp

2i þ eð1þ4bÞt e2c1 þ e�2c1þp2i

� �: ð4:15Þ

For a nonzero real eigenvalue ks, we have

UksðtÞ ¼ Jkss ðtÞeds ðecs þ e�csþp

2iÞlk1; ð4:16Þ

where ds ¼ ks�12 ð1þ 4bÞt, cs ¼ ksðtÞxþ 4

3b k3s ðtÞ þ a

b ksðtÞ;1 6 s 6 n.In addition, for KðtÞ ¼ Ae�bt , and

A ¼

Jðk1Þ 0Jðk2Þ

. ..

0 JðksÞ

0BBBB@1CCCCA

s�s

; JðkiÞ ¼

ki 01 ki

..

. . .. . .

.

0 � � � 1 ki

0BBBB@1CCCCA

ki�ki

:

When ki > 0, we get positons, and when ki < 0, we get negatons. A more general positons and negatons can be obtainedby combining n set of eigenfunctions associated with different ki > 0 or different ki < 0,

f ¼W /1ðk1Þ;/2ðk1Þ; . . . ;/l1 ðk1Þ; /1ðk2Þ;/2ðk2Þ; . . . ;/l2 ðk2Þ; . . . ; /1ðksÞ;/2ðksÞ; . . . ;/ls ðksÞ� �

; ð4:17Þ

and u ¼ i ln�ff

� �x.

4.3. Complexitons

If the coefficient matrix KðtÞ has the following type of Jordan blocks with the complex coefficient as follows

KðtÞ ¼

J1ðtÞ 0J2ðtÞ

. ..

0 JsðtÞ

0BBBB@1CCCCA

s�s

; JiðtÞ ¼

AiðtÞ 0I2 AiðtÞ... . .

. . ..

0 � � � I2 AiðtÞ

0BBBB@1CCCCA

ki�ki

;

with

AiðtÞ ¼ai �bi

bi ai

� �e�bt ¼ aie�bt I2 þ bir2;1 6 i 6 s;

1468 Y. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1463–1469

where ai, bi are real vectors, I2 ¼1 00 1

� �, r2 ¼

0 �11 0

� �. By the similar way, we can get0 1

Jl½AiðtÞ� ¼

I2 0I2ebt@k1 I2 0

12!

I2e2bt@2k1

I2ebt@k1 I2 0

..

. . .. . .

. . .. . .

.

..

.12!

I2e2bt@2k1

I2ebt@k1 I2 01

ðk1�1Þ! I2eðk1�1Þbt@ðk1�1Þk1

� � � � � � 12!

I2e2bt@2k1

I2ebt@k1 I2

BBBBBBBBBBBB@

CCCCCCCCCCCCAA0iðtÞ ¼ Tki ð@ai

ÞA0iðtÞ;

where A0i ¼Ai

. ..

Ai

0B@1CA

ki�ki

.

Take the simplest one to consider

UA1 ¼ Tk1 ð@a1 Þed1 ec1 þ e�c1þp2i

� �lA1

; ð4:18Þ

where d1 ¼ k1�12 ð1þ 4bÞt, c1 ¼ A1e�btxþ 4

3b A31e�3bt þ a

b A1e�bt . Thus, when k1 ¼ 1, one of the complexitons is generated fromthat

f ¼ 2b1e�bt ex1þx2 þ e�ðx1þx2Þ� �

� 2ia1e�bt ex1�x2 � e�ðx1�x2Þ� �

; ð4:19aÞ

where

x1 ¼ ða1 � b1Þe�btxþ 43b

a31 � 3a2

1b1 � 3a1b21 � b3

1

� �e�3bt þ a

bða1 � b1Þe�bt; ð4:19bÞ

x2 ¼ ða1 þ b1Þe�btxþ 43bða3

1 þ 3a21b1 � 3a1b

21 � b3

1Þe�3bt þ abða1 þ b1Þe�bt : ð4:19cÞ

When k1 ¼ 2, then another one of the complexitons is generated from

f ¼ 16ða21 � b2

1Þ �i ða21 � b2

1Þe1

6bðFþGþ12e�bt xbb1�24b2tÞ þ ða21 þ b2

1Þe1

6bðF�G�12e�bt xbb1�24b2tbÞh in

1

þ iða21 þ b2

1Þ e1

6bð�FþGþ12e�bt xbb1�24b2tbÞ � e1

6bð�F�Gþ12e�bt xbb1�24b2tbÞh i

þ a1b1e�4btðe 23bH þ e�

23bH�a1b1e

Mb ðeN þ e�NÞ þ 4a1b1e�4bt

o; ð4:20aÞ

where

F ¼ 2H ¼ �12e�btxba1 � 16e�3bta31 þ 48e�3bta1b

21 � 12ae�bta1 þ 16e�3btb3

1; ð4:20bÞG ¼ 48e�3bta2

1b1 þ 12ae�btb1; M ¼ 4e�btxbb1 � 4b2t þ 4ae�btb1; N ¼ 16e�3bta21b1: ð4:20cÞ

From this way, much more other comlexitons to the Eq. (2.1) can also be derived.

5. Concluding and remarks

In this article, we have used a matrix technique to get a broad class of explicit exact solutions to the mKdV equation withnon-uniformity terms. It is proved that the method mentioned can be used in deriving solitons, negatons, positons and com-plexitons. The key technique is to apply the variation of parameters in solving the matrix equation which involved partialdifferential equations of second and three order and to analyze solution structures in detail. One can easily get the mKdVequation and its exact solutions with a ¼ 0 and b ¼ 0. In addition, our scheme enables us to construct more general solutionsother than the already known solutions. We hoped that this study could further assist in understanding, identifying and clas-sifying the non-uniformity equations and their exact solutions.

Acknowledgements

The authors would like to thank Dr. Yan K.H. and Yan J.J. for their stimulating discussions. This work is supported by theNational Natural Science Foundation of China (No. 10771196, 10831003).

References

[1] R. Hirota, Exact solution of the Korteweg-de Vries equation for multiple collisions of solitons, Phys. Rev. Lett. 27 (1971) 1192–1194.[2] W.X. Ma, Wronskians, generalized Wronskians and solutions to the Korteweg-de Vries equation, Chaos Soliton Fract. 19 (2003) 163–170.[3] J. Satsuma, A Wronskian representation of N-soliton solutions of nonlinear evolution equations, J. Phys. Soc. Jpn. 40 (1979) 359–360.

Y. Zhang et al. / Applied Mathematics and Computation 217 (2010) 1463–1469 1469

[4] N.C. Freeman, J.J.C. Nimmo, Soliton solutions of the KdV and KP equations: the wronskian technique, Phys. Lett. A 95 (1983) 1–3.[5] J.J.C. Nimmo, N.C. Freeman, A method of obtaining the N-soliton solution of the Boussinesq equation in terms of a Wronskian, Phys. Lett. A 95 (1983) 4–

6.[6] S. Sirianunpiboon, S.D. Howard, S.K. Roy, A note on the Wronskian form of solutions of the KdV equation, Phys. Lett. A 134 (1988) 31–33.[7] W.X. Ma, Complexiton solutions to the Korteweg-de Vries equation, Phys. Lett. A 301 (2002) 35–44.[8] W.X. Ma, K. Maruno, Complexiton solutions of the Toda lattice equation, Physica A 343 (2004) 219–237.[9] W.X. Ma, Y. You, Solving the Korteweg-de Vries equation by its bilinear form: Wronskian solutions, Trans. Am. Math. Soc. 357 (2005) 1753–1778.

[10] W.X. Ma, Soliton, positon and negaton solutions to a Schrodinger self-consistent source equation, J. Phys, Soc. Jpn. 72 (2003) 3017–3019.[11] W.X. Ma, Complexiton solutions to integrable equations, Nonlinear Anal. 63 (2005) e2461–e2471.[12] J.Y. Ge, Y. Zhang, Extended Wronskian formula for solutions to the Korteweg-de Vries equation, J. Phy.: Conf. Ser. 96 (2008) 012071–012073.[13] J.J.C. Nimmo, Rational solutions of the Korteweg-de Vries equation in Wronskian form, Phys. Lett. A 96 (1983) 443–446.[14] C.X. Li, W.X. Ma, X.J. Liu, Y.B. Zeng, Wronskian solutions of the Boussinesq equation–solitons, negatons, positons and complexitons, Inv. Probl. 23

(2007) 279–296.[15] D.J. Zhang, The N-soliton solutions for the modified KdV equation with self-consistent sources, J. Phys, Soc. Jpn. 71 (2002) 2649–2656.[16] D.J. Zhang, Negatons, positons, rational-like solutions and conservation laws of the KortewegCde Vries equation with loss and non-uniformity terms, J.

Phys, A: Math. Gen. 37 (2004) 851–865.