new exact solutions to the -dimensional konopelchenko–dubrovsky equation

Commun Nonlinear Sci Numer Simulat 15 (2010) 216–224

Contents lists available at ScienceDirect

Commun Nonlinear Sci Numer Simulat

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

New exact solutions to the ð2þ 1Þ-dimensional Konopelchenko–Dubrovskyequation

Yang Wang, Long Wei *

Institute of Mathematics, Hangzhou Dianzi University, Xiasha Hangzhou, Zhejiang 310018, China

a r t i c l e i n f o

Article history:Received 3 September 2008Received in revised form 26 February 2009Accepted 17 March 2009Available online 26 March 2009

PACS:02.30.Ik02.30.Jr04.20.Jb

Keywords:ð2þ 1Þ-Dimensional KD equationTravelling wave solutionsComplexitons

1007-5704/$ - see front matter � 2009 Elsevier B.Vdoi:10.1016/j.cnsns.2009.03.013

* Corresponding author.E-mail addresses: [email protected] (Y. Wan

a b s t r a c t

In this paper, the extended tanh method, the sech–csch ansatz, the Hirota’s bilinear formal-ism combined with the simplified Hereman form and the Darboux transformation methodare applied to determine the traveling wave solutions and other kinds of exact solutions forthe ð2þ 1Þ-dimensional Konopelchenko–Dubrovsky equation and abundant new solitonsolutions, kink solutions, periodic wave solutions and complexiton solutions are formallyderived. The work confirms the significant features of the employed methods and showsthe variety of the obtained solutions.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

It is an interesting topic to search for new solutions of partial differential equations, especially for those which appear inmany branches of nonlinear science. Many powerful methods have been presented such as Bäcklund transformation [1], Dar-boux transformation [2], the extended Jacobian elliptic function expansion method [3], the tanh method [4–8], the sine–co-sine method [9–11], the homogeneous balance method [12], the direct reduction method [13] and many other techniques.Practically, there is no unified method that can be used to handle all types of nonlinear problems.

This paper is concerned with the ð2þ 1Þ-dimensional Konopelchenko–Dubrovsky (KD) equation presented by Kon-opelchenko and Dubrovsky in [14]

ut � uxxx � 6buux þ32

a2u2ux � 3vy þ 3auxv ¼ 0;

uy ¼ vx;

ð1Þ

where u ¼ uðx; y; tÞ is a sufficiently-often differentiable function, a and b are arbitrary real parameters.Many methods were used to find the exact solutions of Eq. (1) such as the inverse scattering transform method [14], the

extended F-expansion method [15], the improved tanh function method [16–18], homogenous balance method [22], theExp-function method [23], the cosh–sinh method, the tanh–sech method and exponential method [24] and singular

. All rights reserved.

g), [email protected], [email protected] (L. Wei).

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Y. Wang, L. Wei / Commun Nonlinear Sci Numer Simulat 15 (2010) 216–224 217

manifold method [19]. Recently, in [20] the variable separation approach was applied to study the Eq. (1) with a = 0, that is,the well-known Kadomtsev–Petviashvili (KP) equation [21].

The objectives of this work are twofold. First, we seek to establish exact solutions of distinct physical structures, solitons,kink, periodic wave solutions and complexiton solutions for the nonlinear Eq. (1). Second, we aim to implement four strat-egies to achieve our goal, namely, the extended tanh [28,29], the sech–csch method, the Hirota’s bilinear method [30] andthe Darboux transformation method [31] to obtain new kinds of solutions. Here, the so-called complexiton [25–27] is ex-pressed by combinations of trigonometric functions and hyperbolic functions. Recently, complexiton solutions for some inte-grable systems are presented by means of different approaches. For example, Prof. Ma provided the complexiton solutions ofthe KdV equation and the Toda lattice equation through the Wronskian and Casoratian techniques [26,27] and presented thepositons, negatons and complexitons and their interaction solutions for the Boussinesq equation through its Wronskiandeterminant [32]. In [33], the authors proposed some complexiton solutions for a special kind of the coupled KdV systemby Darboux transformation. In this work, we will apply Darboux transformation to obtain complexiton solutions for (1).As will be shown later, these distinct approaches, that to be used, provide distinct solutions of different physical structuresand the power of the methods is in its ease of use to determine shock or solitary type of solutions. In what follows, the meth-ods will be reviewed briefly.

2. Analysis of the methods

Now we highlight briefly the main features of the extended tanh method, the sech–csch method and the Hirota’s bilinearmethod which will be used in this work.

2.1. The extended tanh method and the sech–csch method

A PDE

Pðu;ut ;ux;uxx; � � �Þ ¼ 0 ð2Þ

can be converted to on ODE

Qðu; u0;u00;u000; � � �Þ ¼ 0 ð3Þ

upon using a wave variable n ¼ x� ct. Eq. (3) is then integrated as long as all terms contain derivatives where integrationconstants are considered zeros.

2.1.1. The extended tanh methodWazwaz has summarized for using extended tanh method [34–36]. Introducing a new independent variable

Y ¼ tanhðlnÞ; n ¼ x� ct; ð4Þ

leads to change of derivatives:

ddn¼ lð1� Y2Þ d

dY;

d2

dn2 ¼ �2l2Yð1� Y2Þ ddYþ l2ð1� Y2Þ2 d2

dY2 :

ð5Þ

The extended tanh method admits the use of a finite expansion of tanh function

uðnÞ ¼ SðYÞ ¼XM

k¼0

akYk þXM

k¼1

bkY�k; ð6Þ

where M is a positive integer, for this method, that will be determined. The parameter M is usually obtained by balancing thelinear terms of the highest order in the resulting equation with the highest order nonlinear terms. If M is not an integer, thena transformation formula should be used to overcome this difficulty. Substituting (6) into the ODE results in an algebraicsystem of equations in powers of Y which will lead to the determination of the parameters ak ðk ¼ 0; � � � ;MÞ; bk

ðk ¼ 1; � � � ;MÞ; c and l.

2.1.2. The sech–csch methodIt is appropriate to introduce two schemes that involve cosh and sinh functions. Only direct substitution can show the

existence of solutions for some ansatzs. Specific schemes of the form:

u ¼ a0 þ a1Y; ð7Þ

where Y ¼ sechðlðxþ y� ctÞÞ or Y ¼ cschðlðxþ y� ctÞÞ is introduced to obtain more travelling wave solutions, where a0, a1

and l are parameters that will be determined. The two schemes are simply applied to Eq. (1), collecting the coefficients of

218 Y. Wang, L. Wei / Commun Nonlinear Sci Numer Simulat 15 (2010) 216–224

the resulting hyperbolic functions and setting them to zero, and solving the resulting equations to determine the parametersa0; a1 and l.

2.2. The Hirota’s method

To formally derive N-soliton solutions of any completely integrable equation, we will mainly use the Hirota’s direct meth-od combined with the simplified version of Hereman and Nuseir [37] where it was shown that soliton solutions are just poly-nomials of exponentials.

We first substitute

uðx; y; tÞ ¼ ekiðxþyÞ�cit ; i ¼ 1;2; � � � ;N ð8Þ

into the linear terms of the equation under discussion to determine the dispersion relation between ki and ci. We then sub-stitute the single-soliton solution

uðx; y; tÞ ¼ Ro2 ln f ðx; y; tÞ

ox2 ¼ Rffxx � ðfxÞ2

f 2 ; ð9Þ

into the equation under discussion, where the auxiliary function f is given by

f ðx; y; tÞ ¼ 1þ eh1 ; ð10Þ

and h1 ¼ k1ðxþ yÞ � c1t, and solve the resulting equation to determine the numerical value for R. The N-soliton solutions canbe obtained by the following main steps:

(i) For dispersion relation, we use

uðx; y; tÞ ¼ ehi ; hi ¼ kiðxþ yÞ � cit ði ¼ 1;2; � � � ;NÞ; ð11Þ

(ii) For single-soliton, we use

f ¼ 1þ eh1 ; ð12Þ

(iii) For two-soliton solutions, we use

f ¼ 1þ eh1 þ eh2 þ a12eh1þh2 ; ð13Þ

(iv) For three-soliton solutions, we use

f ¼ 1þ eh1 þ eh2 þ eh3 þ a12eh1þh2 þ a13eh1þh3 þ a23eh2þh3 þ b123eh1þh2þh3 : ð14Þ

Notice that we use (11) to determine the dispersion relation, (13) to determine the factor a12 to generalize for the otherfactors aij in f, and finally we use (14) to determine b123, which is mostly given by b123 ¼ a12a13a23. The determination ofthree-soliton solutions confirms the fact that N-soliton solutions exist for any order.

2.3. The Darboux transformation method

The Darboux transformation method is a powerful tool to get the analytical solutions for the integrable nonlinear PDEs[31,38]. The most obvious advantage of this method lies in its iterative algorithm, which is purely algebraic and can be easilyachieved on the symbolic computation system. Based on the corresponding Lax pair(s), one can construct the binary Darbouxtransformation of the integrable nonlinear PDEs. By virtue of the Darboux transformation, starting from the seed solution,solving the corresponding linear equation or system and iterating step by step, one can obtain wide classes of exact analyt-ical solutions for an nonlinear PDE, such as the soliton solutions, periodic solutions, rational solutions [31,38,39] and com-plexiton solutions.

In the following, we will apply the aforementioned methods to the ð2þ 1Þ-dimensional KD equation.

3. Using the extended tanh method

In this section, the extended tanh method will be applied to study the ð2þ 1Þ-dimensional KD Eq. (1). First, we introducethe wave variable n ¼ xþ y� ct into (1) to transform the equation into a ODE given by

�cu0 � u000 � 6buu0 þ 32 a2u2u0 � 3v 0 þ 3au0v ¼ 0;

u0 ¼ v 0:

(ð15Þ

Integrating the last equation gives u ¼ v where the constants of integration are considered zeros. The first equation in (15)becomes


�ðc þ 3Þu0 � u000 þ 3ða� 2bÞuu0 þ 32

a2u2u0 ¼ 0; ð16Þ

now by integrating both sides once we obtain

�ðc þ 3Þu� u00 þ 32ða� 2bÞu2 þ a2

2u3 ¼ 0: ð17Þ

Balancing u00 with u3 gives M ¼ 1. The extended tanh method admits the use of

uðnÞ ¼ a0 þ a1Y þ b1Y�1; Y ¼ tanhðlnÞ: ð18Þ

Substituting this transformation formula into the reduced ODE (16), collecting the coefficients of Y, and solving the resultingsystem with the help of Maple we find the following sets of solutions

(i) The first set:

a0 ¼2b� a

a2 ; a1 ¼ �a� 2b

a2 ; b1 ¼ 0; l ¼ � a� 2b2a

; c ¼ 4ðab� a2 � b2Þa2 ; ð19Þ

(ii) The second set:

a0 ¼2b� a

a2 ; a1 ¼ 0; b1 ¼ �a� 2b

a2 ; l ¼ � a� 2b2a

; c ¼ 4ðab� a2 � b2Þa2 ; ð20Þ

(iii) The third set:

a0 ¼2b� a

a2 ; a1 ¼ b1 ¼ �a� 2b

2a2 ; l ¼ � a� 2b4a

; c ¼ 4ðab� a2 � b2Þa2 ; ð21Þ

(iv) The fourth set:

a0 ¼2b� a

a2 ; a1 ¼ �2la; b1 ¼ 0; c ¼ �9a2 � 12b2 þ 12abþ 4a2l2

2a2 ; ð22Þ

(v) The fifth set:

a0 ¼2b� a

a2 ; a1 ¼ 0; b1 ¼ �2la; c ¼ �9a2 � 12b2 þ 12abþ 4a2l2

2a2 ; ð23Þ

(vi) The sixth set:

a0 ¼2b� a

a2 ; a1 ¼ b1 ¼ �2la; c ¼ �9a2 � 12b2 þ 12abþ 16a2l2

2a2 ; ð24Þ

(vii) The seventh set:

a0 ¼2b� a

a2 ; a1 ¼ �b1 ¼ �2la; c ¼ �9a2 � 12b2 þ 12ab� 8a2l2

2a2 ; ð25Þ

(viii) The eighth set:

a0 ¼2b� a

a2 ; a1 ¼ �b1 ¼ �ffiffiffi2pða� 2bÞ2a2 i; l ¼ �

ffiffiffi2pða� 2bÞ

4ai; c ¼ 4ðab� a2 � b2Þ

a2 : ð26Þ

where l is a arbitrary constant. In view of this we obtain the following kinks solutions:

u1;2 ¼2b� a

a2 1� tanha� 2b

2axþ yþ 4ða2 � abþ b2Þ

a2 t

!" # !; ð27Þ

u3;4 ¼2b� a

a2 1� cotha� 2b


a2 t

!" # !; ð28Þ

u5;6 ¼2b� a

a2 1� 12

tanha� 2b


a2 t

!" # ð29Þ

�12

cotha� 2b


a2 t

!" #!;


u7;8 ¼2b� a

a2 � 2la

tanh l xþ yþ 9a2 þ 12b2 � 12ab� 4a2l2

2a2 t

!" #; ð30Þ

u9;10 ¼2b� a

a2 � 2la

coth l xþ yþ 9a2 þ 12b2 � 12ab� 4a2l2

2a2 t

!" #; ð31Þ

u11;12 ¼2b� a

a2 � 2la

tanh l xþ yþ 9a2 þ 12b2 � 12ab� 16a2l2

2a2 t

!" #

� 2la

coth l xþ yþ 9a2 þ 12b2 � 12ab� 16a2l2

2a2 t

!" #; ð32Þ

u13;14 ¼2b� a

a2 � 2la

tanh l xþ yþ 9a2 þ 12b2 � 12abþ 8a2l2

2a2 t

!" #

� 2la

coth l xþ yþ 9a2 þ 12b2 � 12abþ 8a2l2

2a2 t

!" #; ð33Þ

and periodic solutions:

u15;16¼2b�a

a2 1�ffiffiffi2p

2tan

ffiffiffi2pða�2bÞ4a

xþyþ4ða2�abþb2Þa2 t

!" # �

ffiffiffi2p

2cot

ffiffiffi2pða�2bÞ4a

xþyþ4ða2�abþb2Þa2 t

!" #!;

ð34Þ

where l is an arbitrary constant. These obtained solutions contain four pairs of solutions obtained in [24].

4. Using the sech–csch method

In a manner parallel to our discussion presented above, we can assume that the solution is of the form:

uðnÞ ¼ a0 þ a1Y; Y ¼ sechðlnÞ or cschðlnÞ: ð35Þ

Substituting (35) into (17), collecting the coefficients of Y we obtain the system of algebraic equations for a0; a1 and l, thenwe solve the system with the aid of shape Maple to obtain the periodic solutions:

u17;18 ¼2b� a

a2 1�ffiffiffi2p

sec

ffiffiffi2pð2b� aÞ


a2 t

!" # !; ð36Þ

u19;20 ¼2b� a

a2 1�ffiffiffi2p

csc

ffiffiffi2pð2b� aÞ


a2 t

!" # !; ð37Þ

the kink solutions:

u21;22 ¼2b� a

a2 � 2la

csch l xþ yþ 18a2 � 24abþ 24b2 þ 4a2l2

a2 t

!" #ð38Þ

and the complex solution

u23;24 ¼2b� a

a2 � 2lia

sech l xþ yþ 9a2 � 12abþ 12b2 þ 2a2l2

a2 t

!" #; ð39Þ

where i2 ¼ �1 and l is an arbitrary constant. The solutions u17;18 and u19;20 were obtained in [24].

5. Using the Hirota’s bilinear method

Substituting

uðx; y; tÞ ¼ vðx; y; tÞ ¼ ehi ; hi ¼ kiðxþ yÞ � cit ði ¼ 1;2; � � � ;NÞ; ð40Þ

into the linear terms of the first equation of (1), one can get easily the dispersion relation

ci ¼ �ðk3i þ 3kiÞ; i ¼ 1;2; � � � ;N ð41Þ

and hence hi becomes

hi ¼ kiðxþ yÞ þ ðk3i þ 3kiÞt: ð42Þ


Using the simplified version of Hereman and Nuseir [37], the multi-soliton solutions of Eq. (1) can be assumed to have thefollowing form

uðx; y; tÞ ¼ Ro2 ln f ðx; y; tÞ

ox2 ¼ Rffxx � ðfxÞ2

f 2 ;

vðx; y; tÞ ¼ Ro2 ln f ðx; y; tÞ

oxoy¼ R

ffxy � fxfy

f 2 ;

ð43Þ

where f ðx; y; tÞ, for the single-soliton solution, is given by

f ðx; y; tÞ ¼ 1þ ek1ðxþyÞþðk31þ3k1Þt : ð44Þ

To determine the numerical value for R, we substitute (43) into Eq. (1) and find that

R ¼ 2b

and a ¼ 0: ð45Þ

This means that the single-soliton solution is given by

uðx; y; tÞ ¼ 2k21ek1ðxþyÞþðk3

1þ3k1Þt

bð1þ ek1ðxþyÞþðk31þ3k1ÞtÞ2

;

vðx; y; tÞ ¼ 2k21ek1ðxþyÞþðk3

1þ3k1Þt

bð1þ ek1ðxþyÞþðk31þ3k1ÞtÞ2

;

ð46Þ

provided that a ¼ 0 and b–0.For two-soliton solutions, we assume that

f ¼ 1þ eh1 þ eh2 þ a12eh1þh2 ; ð47Þ

where h1 and h2 are given by (42). To determine the coefficient a12, we substitute (43) with f given by (47) into the KD Eq. (1)with constrain a ¼ 0 (i.e. KP equation). With the aid of the Maple, we obtain that

a12 ¼ðk1 � k2Þ2

ðk1 þ k2Þ2ð48Þ

and hence we can obtain that for 1 6 i < j 6 N

aij ¼ðki � kjÞ2

ðki þ kjÞ2: ð49Þ

Thus we get

f ¼ 1þ exp k1ðxþ yÞ þ ðk31 þ 3k1Þt

� �þ exp k2ðxþ yÞ þ ðk3

2 þ 3k2Þt� �

þ ðk1 � k2Þ2

ðk1 þ k2Þ2exp ðk1 þ k2Þðxþ yÞ þ ðk3

1 þ k32 þ 3k1 þ 3k2Þt

� �: ð50Þ

Now, substituting the last result for f ðx; y; tÞ into the following expressions

uðx; y; tÞ ¼ 2ðffxx � ðfxÞ2Þbf 2 ;

vðx; y; tÞ ¼ 2ðffxy � fxfyÞbf 2 ;

we can obtain the two-soliton solutions explicitly to Eq. (1) provided that a ¼ 0 and b–0.Similarly, to determine the three-soliton solutions, we set

f ¼ 1þ eh1 þ eh2 þ eh3 þ a12eh1þh2 þ a13eh1þh3 þ a23eh2þh3 þ b123eh1þh2þh3 ð51Þ

in (43) and substitute it in the KD Eq. (1) with the constrain a ¼ 0 (that is, KP equation) to find that

b123 ¼ a12a13a23 ¼ðk1 � k2Þ2ðk1 � k3Þ2ðk2 � k3Þ2

ðk1 þ k2Þ2ðk1 þ k3Þ2ðk2 þ k3Þ2: ð52Þ

To find the three-soliton solutions explicitly, we substitute the last result for f ðx; y; tÞ into (43) with R ¼ 2b. The higher level

soliton solutions, for N P 4 can be obtained in a parallel manner. This confirms that the KP equation provides multiple-sol-iton solutions of any order.


Recall that the multiple soliton solutions require the restriction a ¼ 0. However, the results obtained by using theextended tanh method and the sech–csch method work effectively for any arbitrary constants a and b.

6. Using the Darboux transformation method

In [19], the authors obtained the Lax pairs of Eq. (1) in view of the singular manifold method [40]. Starting from the twoLax pairs, they had given the Darboux transformation for Eq. (1) expressed by the two singular manifolds, f and g, in relationwith the two different branches in the expansion of Painlevé analysis. The Lax pairs of Eq. (1) read

wy ¼ �wxx � au� 2ba

� �wx;

wt ¼ 4wxxx þ 6au� 12ba

� �wxx þ

12b2

a2 � 6buþ 32

a2u2 � 3av þ 3aux

!wx

ð53Þ

and

/y ¼ /xx � au� 2ba

� �/x;

/t ¼ 4/xxx � 6au� 12ba

� �/xx þ

12b2

a2 � 6buþ 32

a2u2 � 3av � 3aux

!/x;

ð54Þ

where fu;vg is a solution of Eq. (1). Using the above Lax pairs, one can define the two singular manifolds f and g in an abbre-viated form as

f ¼ Dðw;/Þ; g ¼ Xðw;/Þ; ð55Þ

where Dðw;/Þ and Xðw;/Þ have been defined as the solutions of the over-determined system

½Dðw;/Þ�x ¼ wx/; ½Xðw;/Þ�x ¼ w/x;

½Dðw;/Þ�y ¼ wy/þ wx/x; ½Xðw;/Þ�y ¼ w/y � wx/x;

½Dðw;/Þ�t ¼ wt/� 2wxx/x þ 2wy/x þ 4wx/y;

½Xðw;/Þ�t ¼ w/t þ 2wxx/x � 2wy/x � 4wx/y:

ð56Þ

Then we arrive the Darboux transformation of Eq. (1)

u0 ¼ uþ 2a

lnfg

� �x

; v 0 ¼ v þ 2a

lnfg

� �y

;

w0 ¼ w1 � wDðw1;/ÞDðw;/Þ ; /0 ¼ /1 � /

Xðw1;/ÞXðw;/Þ ;

where w;/ satisfy the Lax pairs (53) and (54) respectively, and w0;/0 also satisfy (53) and (54) except that fu;vg is replaced bynew solution fu0;v 0g.

Usually, Darboux transformation can be iterated step by step. Iterating DT N times, we obtain the following Grammiansolutions to Eq. (1)

u½N� ¼ uþ 2a

lnF½N�G½N�

� �x; v ½N� ¼ v þ 2

aln

F½N�G½N�

� �y

ð57Þ

with

F½N� ¼

Dðw1;/1Þ Dðw1;/2Þ � � � Dðw1;/NÞ

Dðw2;/1Þ Dðw2;/2Þ � � � Dðw2;/NÞ

� � � � � � � � � � � �

DðwN;/1Þ DðwN ;/2Þ � � � DðwN ;/NÞ

;

G½N� ¼

Xðw1;/1Þ Xðw1;/2Þ � � � Xðw1;/NÞ

Xðw2;/1Þ Xðw2;/2Þ � � � Xðw2;/NÞ

� � � � � � � � � � � �

XðwN;/1Þ XðwN;/2Þ � � � XðwN;/NÞ

;

where wi and /i ði ¼ 1;2; � � � ;NÞ satisfy Lax pairs (53) and (54).


In order to show some new solutions, in particular the complexiton solutions, of Eq. (1) under seed solution fu ¼ 0;v ¼ 0gfrom Grammian solutions (57), we find first that the Lax pairs (53) and (54) have general solutions as follows:

w ¼ ek2 ðayþ12btÞ

a C1 coskða2xþ 2abyþ 12b2t � 4k2a2tÞ

a2

!þ C2 sin

kða2xþ 2abyþ 12b2t � 4k2a2tÞa2

!" #

þ e�l2ðayþ12btÞ

a C3 coshlða2xþ 2abyþ 12b2t þ 4l2a2tÞ

a2

!þ C4 sinh

lða2xþ 2abyþ 12b2t þ 4l2a2tÞa2

!" #ð58Þ

and

/ ¼ e�m2 ðayþ12btÞ

a D1 cosmða2xþ 2abyþ 12b2t � 4m2a2tÞ

a2

!þ D2 sin

mða2xþ 2abyþ 12b2t � 4m2a2tÞa2

!" #

þ en2 ðayþ12btÞ

a D3 coshnða2xþ 2abyþ 12b2t þ 4n2a2tÞ

a2

!þ D4 sinh

nða2xþ 2abyþ 12b2t þ 4n2a2tÞa2

!" #; ð59Þ

where k; l;m;n 2 R and Ci;Di 2 C; i ¼ 1;2;3;4. Next, by selecting k; l;m;n;Ci;Di properly one can obtain the desired solutionsto Eq. (1) from the general representation (57). Here, we should point out that integration of system (56) with respect to x; yand t results in the opposite integral constants in f and g. For instance, taking N ¼ 1; C1 ¼ D3 ¼ 1 andC2 ¼ C3 ¼ C4 ¼ D1 ¼ D2 ¼ D4 ¼ 0 and integrating (56) we get that

f1 ¼k

ðn2 þ k2Þerðk cos n cosh g� n sin n sinh gÞ þ c ð60Þ

and

g1 ¼n

ðn2 þ k2Þerðn cos n cosh gþ k sin n sinh gÞ � c; ð61Þ

If C2 ¼ D3 ¼ 1 and C1 ¼ C3 ¼ C4 ¼ D1 ¼ D2 ¼ D4 ¼ 0, we have

f2 ¼k

ðn2 þ k2Þerðk sin n cosh gþ n cos n sinh gÞ þ d ð62Þ

and

g2 ¼n

ðn2 þ k2Þerðn sin n cosh g� k cos n sinhgÞ � d; ð63Þ

where c; d are two arbitrary constants and

r ¼ ðn2 þ k2Þðayþ 12btÞ

a;

n ¼ kða2xþ 2abyþ 12b2t � 4k2a2tÞa2 ;

g ¼ nða2xþ 2abyþ 12b2t þ 4n2a2tÞa2 :

ð64Þ

Substituting (60), (61) and (62), (63) into (57) respectively, we can obtain two complexiton solutions of Eq. (1). Similarly, onecan obtain abundant new solutions including complexiton solutions of Eq. (1) by above steps. Here, we should note that oursolutions contain the solutions obtained in [19] if we take Ci;Di 2 C properly.

7. Discussion

The ð2þ 1Þ-dimensional KD equation is investigated. The extended tanh method, the sech–csch method, Hereman’smethod combined with the Hirota’s bilinear sense and the Darboux transformation method were employed to achievethe goals set for this work. Many new soliton solutions, kink solutions, periodic wave solutions and complexiton solutionsare formally derived. The Hirota’s bilinear method imposed restrictions of a ¼ 0. However, the extended tanh method, thesech–csch method and the Darboux transformation method were implemented to obtain kink, periodic wave solutionsand complexiton solutions without any need to this restriction.


Acknowledgements

The authors express their sincere gratitude to the referees for introducing the Refs. [25–28], many valuable commentsand suggestions which served to improve the article. The research of the authors is partly supported by Scientific ResearchFoundation of Hangzhou Dianzi University.

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new exact solutions to the -dimensional konopelchenko–dubrovsky equation

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