Similarity solutions of the Konopelchenko–Dubrovsky system using Lie group theory

Computers and Mathematics with Applications 71 (2016) 2051–2059

Contents lists available at ScienceDirect

Computers and Mathematics with Applications journal homepage: www.elsevier.com/locate/camwa

Similarity solutions of the Konopelchenko–Dubrovsky system using Lie group theory Mukesh Kumar a , Anshu Kumar a , Raj Kumar b,∗ a

Department of Mathematics, Motilal Nehru National Institute of Technology, Allahabad-211004, India

b

Department of Mathematics, VBS Purvanchal University, Jaunpur-222003, India

article

abstract

info

Article history: Received 20 October 2015 Received in revised form 18 February 2016 Accepted 26 March 2016 Available online 18 April 2016 Keywords: Konopelchenko–Dubrovsky equations Similarity transformations method Similarity variables Similarity solutions

This research deals with the similarity solutions of (2 + 1)-dimensional Konopelchenko– Dubrovsky (KD) system. Solutions so obtained are derived by using similarity transformations method based on Lie group theory. The method reduces the number of independent variables by one exploiting Lie symmetries and using invariance property. Thus, the KD system can further be reduced to a new system of ordinary differential equations. Under a suitable choice of functions and the arbitrary constants, these new equations yield the explicit solutions of the KD system which are discussed in the Similarity Solutions section of the article. Moreover, the physical analysis of the solutions is illustrated graphically in the Analysis and Discussions section based on numerical simulations in order to highlight the importance of the study. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction Nonlinear evolution equations (NLEEs) occur in various fields of mathematical and theoretical sciences. These equations have wide role in Engineering, Biology and the like [1–28] where the system has complex behavior. The phenomena governed by NLEE are quite demanding and complex. To understand the physical nature of the system it is highly desirable to have solutions of NLEEs. Many researchers [1–28] contributed to get exact solutions of these NLEEs in various contexts. However, authors are seeking the closed form solution of the following (2 + 1)-dimensional system of KD equations given by ut − uxxx − 6b u ux +

3 2

a2 u2 ux − 3vy + 3av ux = 0,

uy = v x ,

(1.1a) (1.1b)

where u(x, y, t ) and v(x, y, t ) are components of the velocity along x- and y-directions respectively. In the article, subscript with a dependent variable (u and v ) denotes its partial derivative with respect to the subscript variable. In Eq. (1.1a), ut describes the time evolution of the physical quantity u, and the terms u ux and u2 ux , etc. are responsible for nonlinearity. The KD system came into existence due to the remarkable contribution of B.G. Konopelcheno and V.G. Dubrovsky [6] while they were solving some new (2 + 1)-dimensional integrable NLEEs by using the inverse scattering transformation method. The KD equation is one of them. Eq. (1.1a) turns into the Gardner equation (which is usually obtained combining both KdV and mKdV) if uy = 0; while it recasts into the famous Kadomtsev–Petviashvili (KP) equation when a = 0. On the other hand, for b = 0, it provides modified KP, which has a number of applications in the soliton theory.

∗

Corresponding author. E-mail addresses: [email protected] (M. Kumar), [email protected] (A. Kumar), [email protected] (R. Kumar).

http://dx.doi.org/10.1016/j.camwa.2016.03.023 0898-1221/© 2016 Elsevier Ltd. All rights reserved.

2052

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

Recent history of some past researches enables us to expose about several effective methods [7–21] of getting exact solutions of the KD system by a diverse group of researchers across the globe, notably Wang and Zhang [7] obtained some exact solutions of (2 + 1)-KD equation in terms of Jacobi–Elliptic functions by applying the improved form of the extended F-expansion method. Lin et al. [8] obtained some multi-soliton solutions to the equation with the help of Bäcklund transformation. Applying generalized F-expansion method with Mathematica to the KD equations, Zhang and Xia obtained more general non-traveling wave solutions [9]. Furthermore, the Tanh–Sech method, the Cosh–Sinh method and exponential function method are efficiently employed on the KD equation by A.M. Wazwaz [10]. Wazwaz established the exact solutions of distinct physical structures like solitons, kink and periodic. On the other hand, an extended Riccati equation rational expansion method was presented by Song and Zhang [11] to construct exact solutions for the KD equation. The bifurcation theory method of planar dynamical systems is applied to find the bounded traveling wave solutions of the (2 + 1)-KD equations by T.L. He [12]. The exact solutions show solitary wave, kink (anti-kink) wave and periodic wave profiles which have been obtained under different parameter conditions. Soliton solutions of (2 + 1)-KD equations derived by Zeng and Yong applying new mapping method [13] which added one more contribution to the previously established works done, while soliton solutions for the same equation are attained by H.Y. Zhi by using Lie point symmetry [14]. Feng and Lin [15] used the improved mapping approach and variable separation method to investigate the (2 + 1)coupled KD equations. A.H. Salas [16] with his group employed Hirota’s bilinear approach to find one–four, and in general N-soliton solutions to (2 + 1)-KD system. H.Y. Zhi [17] found that the KD equation and its Lax pair admits the same symmetry transformations of the independent variables. S. Zhang [18] found new exact solutions of the KD system with two arbitrary functions by means of the Riccati equation and its generalized solitary wave solutions by using the Exp-function method. The first integral method had been used to construct exact traveling wave solutions of KD equation by Taghizadeh and Mirzazadeh [19]. H.W. Hu and J. Yu [20] obtained a new multi-soliton solution for the KD system by using the general direct method. They started with the formation of weak Lax pair then attained general Lie point symmetry group of the KD. Their work was supplemented proving a Kac–Moody–Virasoro type to the corresponding Lie algebra structure. S. Kumar et al. [21] applied the traveling wave hypothesis to retrieve a solitary wave solution. Subsequently, optimal sub algebra is generated using Lie group theory to achieve more solutions like solitons, periodic singular waves as well as cnoidal and snoidal waves. The researches [6–21] motivate the authors to solve Eq. (1.1) analytically by using the similarity transformations method (STM). The fundamental idea behind the method is to reduce the number of independent variables by one in the system of PDEs. The reduction of variables occurs exploiting the symmetries and using invariance property of the system of PDEs under Lie group theory. The theory and its applications can be grasped by the researches and the text books [3–5,14,17, 20–27]. In the present article, once applying the STM, the system of KD equations is reduced to another form of PDEs with two dependent and two independent variables. Again repeating the similarity reduction process, the PDEs are reduced to ordinary differential equations (ODEs) with two dependent and one independent variables. Ultimately, the solutions of resulting ODEs have been expressed in explicit forms bifurcating in different cases. Results so obtained in different cases have been analyzed physically on the basis of their graphical representations. 2. Similarity solutions In order to obtain the similarity solutions of the KD equations, the STM is applied. The detailed description of the method can be studied from the text books [22,23]. Since the system (1.1) is nonlinear, therefore using the following oneparameter (ϵ ) Lie-group of infinitesimal transformations with compliance of the invariant conditions [22,23], the solution space (x, y, t , u, v ) of KD system remains invariant even transforming into another space (x∗ , y∗ , t ∗ , u∗ , v ∗ ). x∗ → x + ϵ ξ (1) (x, y, t , u, v) + O(ϵ 2 ), y∗ → y + ϵ ξ (2) (x, y, t , u, v) + O(ϵ 2 ), t ∗ → t + ϵ τ (x, y, t , u, v) + O(ϵ 2 ), u∗ → u + ϵ η(1) (x, y, t , u, v) + O(ϵ 2 ),

v ∗ → v + ϵ η(2) (x, y, t , u, v) + O(ϵ 2 ),

etc.

(2.1)

where ξ (1) , ξ (2) , τ , η(1) and η(2) are known as infinitesimals (see Bluman [22] and Olver [23]) which are in general, the functions of x, y, t, u and v . Since, to obtain the infinitesimals ξ (1) , ξ (2) , τ , η(1) and η(2) in above transformations is not an easier work in view of handling the calculations manually, so authors have obtained them by using symbolic program in Maple 16. Thus, authors get the infinitesimals ξ (1) , ξ (2) , τ , η(1) and η(2) as

ξ (1) = 18xf¯1 + 3y2 f¯¯1 + 3yf¯2 + 3f3 , ξ (2) = 36yf¯1 + 18f2 + 1, τ = 54f1 ,

(2.2)

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

η(1) =

6y ¯ 3 f¯1 + 18 2 − u f¯1 + f¯2 , a a a

η(2) =

  x  2b2  2b y2 ¯¯ 2by  y f¯3 − uy + 2 f¯¯1 + 36 3 − v f¯1 + f¯¯2 + 3 2 − u f¯2 + , f¯1 + 6 a a a a a a a

 2b

2053



where fi , 1 ≤ i ≤ 3 are the arbitrary functions of t, and bar denotes the differentiation with respect to t. Thus, to get the exact solutions of the system (1.1a)–(1.1b), the corresponding characteristic equations can be written as dx

=

18xf¯1 + 3y2 f¯¯1 + 3yf¯2 + 3f3

=

 ¯¯¯ + 6 ax − uy +

y2 f a 1

2by a2



dy

=

36yf¯1 + 18f2 + 1

dt 54f1

du

=



¯¯ + 18

6y f a 1

2b a2

dv     ¯f¯ + 36 2b2 − v f¯ + y f¯¯ + 3 2b − u f¯ + 1 2 1 a 2 a3 a2

 − u f¯1 + 3a f¯2 f¯3 a

.

(2.3)

Taking appropriate combinations from the above characteristic equation, a variety of the first order ODEs can be obtained. Hence, sufficient integral curves can be found by solving the differential equations. Ultimately, considering the suitable combination of the integral curves, the closed form of the solutions of KD equations can be predicted. These forms of the solutions contain two new functions (similarity functions) involving two new variables (similarity variables). Once these similarity functions evaluated then the solution of the KD equation can be furnished completely. In the present article, the similarity solutions are bifurcated into two cases on the basis of suitable choices of arbitrary constants. Each case is again divided into two sub-cases, hence four exact solutions have been attained. All these solutions are appearing very first time as far as authors are aware. 1 and f3 = a4 f¯¯1 , where ai ’s (0 ≤ i ≤ 4) are arbitrary constants, then Case (I): Taking f1 (t ) = a0 + a1 ea2 t , f2 = a3 f¯1 − 18 similarity reduction of the coupled KD system (1.1a)–(1.1b) provides u=

v=

a2  18a a2 9a



2y + a3 +

x+

U (X , Y )

(2y + a3 )

1 2

2b

+

a2

,

(2y + a3 ) > 0

(2.4)

A1 y + A2 a2 V (X , Y ) 1 − (2y + a3 ) 2 U (X , Y ) + , (2y + a3 ) 18 (2y + a3 )

(2.5)

while U (X , Y ) and V (X , Y ) are similarity functions of X and Y which can be expressed by x + A3

X =

(2y + a3 )

1 2

a2

−

72

3

(2y + a3 ) 2 ,

(2.6)

f1

Y =

(2.7)

3

(2y + a3 ) 2

a2 a2

2

2a2 a4

a2 (4a4 −a2 )

a2

a2 a3

1 72b 3 2 2 where A1 = 18 ( a3 − 26a3 + 3a ), A2 = 54a (a4 − 33 ) and A3 = are arbitrary constants. 24 Making use of Eqs. (2.4)–(2.7) into the KD system (1.1a)–(1.1b) yields the following system

2UXXX − (3a2 U 2 + 6aV − aA4 )UX + 2A5 UY − 6XVX − 18YVY − 12V + 2A4 = 0,

(2.8)

XUX + 3YUY + U + VX = 0,

(2.9)

where A4 =

12a3 b2 a3

a22 a33

+

108a

and A5 = a0 a2 .

To solve this system, one can find the following set of infinitesimals ξˆ1 , ξˆ2 , ηˆ1 and ηˆ2 for X , Y , U and V . 3aa6 A6 1 , ξˆ1 = a5 Y 3 + 24a5 A6 ηˆ2 = − + 2 Y

12a6 A6 ηˆ1 = ,

ξˆ2 = a6 Y ,

Y

Y

a6 A 6 

3

Y2



A5 + (24X − 12aU )Y ,

where A6 =

A5 216a

.

(2.10)

To derive the exact solutions, further process can be continued by splitting into the following cases: Case (Ia): a6 ̸= 0 yields dX a5 Y

1 3

=

+ 3aa6 A6

Y −1

dY a6 Y

=

dU 12a6 A6 Y −1

dV

=− 24a5 A6 Y

− 23

− a6 A 6

Y −2



A5 + (24X − 12aU )Y

,

(2.11)

which provides the following similarity forms U = U1 (X1 ) −

12A6 Y

,

and V = −

216aA26 Y2

− 24A6

X Y

+ 12aA6

U1 (X1 ) Y

+ V1 (X1 ),

(2.12)

2054

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059 1

where U1 (X1 ) and V1 (X1 ) are similarity functions of the variable X1 = X − 3A7 Y 3 + Substituting U and V into Eqs. (2.8)–(2.9), one can have

3aA6 Y

, A7 =

a5 . a6

¯

2U¯¯1 − (3a2 U12 + 6aV1 − aA4 )U¯1 − 6X1 V¯1 − 12V1 + 2A4 = 0,

(2.13)

X1 U¯1 + U1 + V¯1 = 0.

(2.14)

Solution of which can be obtained as U1 = −

2X1

,

a

V1 =

and

2X12

+ c1 ,

a

(2.15)

where c1 is an arbitrary constant of integration. Finally, the case terminates providing the solution of KD system as 1

6A7 f13 − 2x − 2A3

u(x, y, t ) =

+

a(2y + a3 )

a

v(x, y, t ) =

2

−

6a

 a 2 12a

18A6 

−

(2y + a3 ) +

f1

2b a2

,

 3a A a2  36A6  A1 y + A2 + c1 270 a A26 2 6 − 2 x+ + (2y + a3 )2 − 2 f1 (2y + a3 ) 2f1 864a f1 2

1

1

2x2 + 4A3 x + 18A27 f13 − 12A7 f13 x − 12A3 A7 f13 + 2A23

+

(2.16)

a(2y + a3 )2

1

+

36A6 A7 2 3

−

36A3 A6

f1

f1

−

a2 A7 f13 6a

+

a2 A 3 18a

. (2.17)

Case (Ib): a6 = 0, then Lagrange’s system for (2.8)–(2.9) is dX a5 Y

1 3

=

dY 0

=

dU 0

dV

=−

2

24a5 A6 Y − 3

.

(2.18)

Thus, similarity reduction of (2.8)–(2.9) can be obtained as U = U2 (Y ),

and V = −

24A6 X

+ V2 (Y ),

Y

(2.19)

with similarity variable Y . On inserting the values of U and V from Eq. (2.19) into (2.8)–(2.9) turns into the following system 9Y V¯2 − A5 U¯2 + 6V2 − A4 = 0,

(2.20)

3Y U¯2 + YU2 − 24A6 = 0.

(2.21)

2

On solving, it provides U2 = − V2 = −

12A6 Y A5 A6 Y2

c2

+ +

(2.22)

1

Y3 A5 c2 18Y

4 3

+

c3 Y

2 3

A4

+

6

,

(2.23)

where c2 and c3 are arbitrary constants due to the integration. Comprising Eqs. (2.4)–(2.5), (2.19) and (2.22)–(2.23), one can yield the following solution of KD system u=

a2 (f1 − a0 )(2y + a3 )

− 13

+ c2 f1

+

2b

,

18af1 a2 a2 (f1 − a0 ) 6A1 y + 6A2 + A4 a2 A 5 v= x+ + (f1 − a0 )(2y + a3 )2 − 9af1 6(2y + a3 ) 216af12

(2.24) c2 4 3

(a2 f1 − A5 )(2y + a3 ) +

18f1

c3 2 3

f1

−

A3 A5 9af1

.

(2.25) The following cases are due to the different choices of arbitrary functions which are able to provide the explicit solutions. Case (II): Let us choose f1 = b0 t, f2 = b1 t and f3 = b2 t. Then the following characteristic equation for the KD system can be asserted by recasting Eq. (2.3) dx x + B1 y + B2 t where B1 =

b1 , 6b0

=

B2 =

dy 2y + 6B1 t + B3

b2 , 6b0

B3 =

1 , 18b0

=

B4 =

dt 3t 2b a2

= +

du B4 − u

b1 , 6ab0

=

B5 =

dv B5 − B1 u − 2v 4b2 a3

+

b1 b 3a2 b0

+

,

b2 18ab0

(2.26) and b0 ̸= 0.

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

2055

(a) Multi solitons of u via (2.16).

(b) Multi solitons of v via (2.17). Fig. 1. Elastic multi solitons (shocks) profile for the physical quantity u and v .

Then the first similarity reduction of the KD system is 1

u =U (X , Y )t − 3 + B4 ,

v = − B1 U (X , Y )t

− 13

(2.27)

+ V (X , Y )t

− 32

1

+ (B5 − B1 B4 ),

(2.28)

2

where similarity variables X and Y can be expressed by





X = x − B1 y +

3B21

1





− B2 t − B1 B3 t 2

− 13

,

 and Y = y − 6B1 t +

1 2



2

B3 t − 3 .

(2.29)

Making use of Eqs. (2.27)–(2.29) into the KD system yields 6UXXX + (2X − 9a2 U 2 − 18aV )UX + 4YUY + 18VY + 2U = 0,

(2.30)

UY − VX = 0,

(2.31)

follows immediately the set of infinitesimals as

  ξˆ (1) = 9ab3 + ab4 Y + b5 6X + Y 2 , ηˆ (2) = b3 − b4

Y 9



ξˆ (2) = 9ab4 + 12b5 Y ,



+ aU − b5 2YU + 12V +

To discuss further, following cases can be raised:

Y2 3a

−

2X  a

.

ηˆ (1) = b4 + b5

 2Y a

 − 6U , (2.32)

2056

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

(a) Stationary profile of u represented by (2.24).

(b) Doubly solitons behavior of v via (2.25). Fig. 2. Doubly solitons behavior of wave represented by (2.24)–(2.25) for all t.

Case IIa: b5 ̸= 0, then the characteristic equation for the system (2.30)–(2.31) is dX Y 2 + 6X + 43 B6 Y + B7 where B6 =

3ab4 , 4b5

B7 =

9ab3 , b5

=

dY 12(Y + B6 )

B8 =

b4 , 9b5

=

dU 2Y a

=−

− 6U + 9B8

dV Y2 3a

−

2X a

+ 2UY + 12V + 43 B6 U + B8 Y + B9

,

(2.33)

b

B9 = − b3 . 5

On solving it, we get the following similarity reduction U =

Y 9a

+

U1 (X1 )

(Y + B6 )

1 2

,

where similarity variable, X1 =

and

V =

18X −Y 2 +3B7 1

18(Y +B6 ) 2

X 9a

−

Y2 54a

−

YU1 (X1 ) 9(Y + B6 )

1 2

+

V1

(Y + B6 )

,

(2.34)

.

Therefore, the reduction of PDEs (2.30)–(2.31) provides

¯

2U¯¯1 − (6aV1 + 3a2 U12 )U¯1 − 3X1 V¯1 − 6V1 = 0,

(2.35)

X1 U¯1 + 2V¯1 + U1 = 0.

(2.36)

This system of ODEs is satisfied by U1 = −

X1 a

,

and

V1 =

X12 2a

+ c4 ,

where c4 is an arbitrary constant of integration.

(2.37)

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

2057

(a) Dromian annihilation of u via Eq. (2.38) begins at t = 185.

(b) Non-linearity of v ceases via Eq. (2.39) as time passes over 497. Fig. 3. Profile shows dromian annihilation of u, v via Eq. (2.38)–(2.39) as time passes over 497.

Collectively, Eqs. (2.27)–(2.28), (2.34) and (2.37) reach to the following solutions of the KD system

(Y 2 − 18X − 3B7 ) − 1 t 3 + B4 , 9a 18a(Y + B6 ) B1 (18X − Y 2 + 3B7 ) − 1 1 1 t 3 + (B5 − B1 B4 ) v = − Yt − 3 + B1 9a 18a(Y + B6 ) 2 1  (18X − Y 2 + 3B7 )Y (18X − Y 2 + 3B7 )2 9ac4  − 2 Y2 + + + + X− t 3, 9a 6 18(Y + B6 ) 72(Y + B6 )2 (Y + B6 ) u=

1

1

Yt − 3 +

1

(2.38)

(2.39)

2

where X = [x − B1 y + (3B21 − 21 B2 )t − B1 B3 ]t − 3 , and Y = [y − 6B1 t + 12 B3 ]t − 3 . Case IIb: b4 ̸= 0, b5 = 0, then the characteristic equation for the system (2.30)–(2.31) follows dX a(Y + 9B10 )

=

dY 9a

=

dU 1

dV

=

Y 9

B10 −

− aU

,

with B10 =

b3

(2.40)

b4

results into the similarity form U =

Y 9a

+ U2 (X2 ),

and V =

B10 9a

Y−

1 81a

Y2 −

1 9

YU2 (X2 ) + V2 (X2 ),

(2.41)

2

with similarity variable X2 = X − B10 Y − Y18 . On inserting the values of U and V in Eqs. (2.30)–(2.31), one can get 9a(B10 U¯2 + V¯2 ) − 1 = 0

and

¯

6aU¯¯2 + a(2X2 − 18aV2 − 9a2 U22 )U¯2 − 18aB10 V¯2 + 2B10 = 0.

To get the particular solution of the system (2.42), let us take U2 = c5 , then V2 = constants. u(x, y, t ) =

2y + B3 18at

1

+ c5 t − 3 + B4 −

2B1 3a

,

X2 9a

(2.42)

+ c6 , where c5 and c6 are arbitrary (2.43)

2058

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

(a) Stationary profile of u via (2.43) for all t ≥ 0.

(b) Parabolic profiles of v via Eq. (2.44) losses its non-linearity as time passes over 5.25. Fig. 4. Profile shows dromian annihilation of u, v via Eqs. (2.43)–(2.44).

v(x, y, t ) =

(2x − 4B1 y − 3B1 B3 ) 18at 1

2

− B1 c5 t − 3 + c6 t − 3 +

−

(2y − 12B1 t + B3 ) 

[18aB1

4

2

c5 +

(2y − 12B1 t + B3 ) 

18t 3 − B2 + 9a(B5 − B1 B4 )] 18a

2

12at 3

.

(2.44)

3. Analysis and discussions This section deals with physical analysis of explicit solutions viz. (2.16)–(2.17), (2.24)–(2.25), (2.38)–(2.39) and (2.43)–(2.44) of the KD system. All the solutions are verified by Maple16. The solutions so obtained are absolutely different from previous findings [7–21] for the equations of KD system. The basic aim of the analysis is to explain the nature of solutions on the basis of their graphical representations. The remarkable research by Tang [28] proved that the shapes, former heights, velocities i.e. u(x, y, t ), v(x, y, t ) of two solitary pulses remain unchanged after their collision/interaction. This fact helped us to arrive at some conclusions for describing the physical nature of the solutions. The graphical behavior of the solutions is analyzed in the following manner: Fig. 1: Eqs. (2.16)–(2.17) are explicit solutions of the problem (1.1). Since the solutions involve arbitrary constants a, ai (0 ≤ i ≤ 6), b, A1 , A2 , etc., as well as the function f1 (t ) = a0 + a1 ea2 t , therefore multisolitons (shocks) evolutionary profile can be observed via Fig. 1 at t = 0.05 for all −20 ≤ x, y ≤ 20. Physical structures of the quantities u and v remain the same and showing elasticity in amplitude of the solitons. Authors have set a = 1.6020, a0 = 1.3112, a1 = 1.5285, a2 = 1.1656, a3 = 0.7482, a4 = 0.4505, a5 = 0.6541, a6 = 1.2630, b = 0.6892, A1 = 0.4719, A2 = 0.0031, A3 = −0.0053, A6 = 0.0044, A7 = 0.5179, c1 = 0.0838 and f1 = 2.9315 in Fig. 1. The value of an arbitrary constant is taken as a random number in the MATLAB code to get physically meaningful profile of the solution, authors have read that specific value and put here. Thus, analytical solutions (2.16)–(2.17) are supplemented by numerical simulations. Fig. 2: Doubly solitons profile of the solution (2.24)–(2.25) is represented in Fig. 2 for t = 0.15. For all t, it can be observed that the profile remains always doubly soliton whatever choice of arbitrary constants has been taken in MATLAB coding. It

M. Kumar et al. / Computers and Mathematics with Applications 71 (2016) 2051–2059

2059

is also a consequence of the fact due to Tang et al. [28]. Figure is plotted by setting c2 = 0.0975, c3 = 0.7431, A4 = 1.0406, remaining arbitrary constants are kept same as in Fig. 1. Fig. 3: Authors have taken bi = ai (0 ≤ i ≤ 2), b3 = 0.9961, b4 = 0.0782, b5 = 1.1067, B1 = 0.0626, B2 = 0.2595, B3 = 0.0743, B4 = 0.5997, B5 = 0.6026, B6 = 0.1039, B7 = 12.9781, c4 = 0.2213, and other arbitrary constants are similar as in Fig. 1. The nonlinear profile of (2.38)–(2.39) annihilates after t = 487. This shows that external disturbances (dissipation factors a and b) dominate over the wave motion to dissipate its energy whatever choice of arbitrary parameters is taken. Fig. 4: Figure is plotted by setting B5 = 0.6026, c5 = 0.4218, c6 = 0.6324 in Eqs. (2.43)–(2.44), while other arbitrary constants are similar as in Fig. 1. The nonlinearity in the profile of (2.43)–(2.44) begins to vanish after t = 5.25 which supports the role of dissipation factors a, b to neutralize the impact of external forces. Also, solution profile remains conserved since u, v → const . as t → ∞. 4. Conclusions As discussed in the above, Lie-group theory is applied successfully to obtain four new closed form solutions of the equations of KD system and are given by Eqs. (2.16)–(2.17), (2.24)–(2.25), (2.38)–(2.39) and (2.43)–(2.44). These solutions reflect multi solitons and stationary behavior of the wave which can be used to test accuracy, comparison and analysis of numerical results in the field. This appears to be more suitable than the previous findings [7–21] as it provides physical analysis of some more exact solutions of (2 + 1)-KD equations. Again the similarity transformations method used here can be extended to obtain other exact solutions of NLEEs which are arising in theoretical and applied Physics, Engineering and the like. Acknowledgments The authors would like to thank reviewers for their fruitful comments and valuable suggestions. References [1] A.H. Bhrawy, M.A. Abdelkewy, S. Kumar, A. Biswas, Soliton and other solutions to Kadomtsev–Petviashvili equation of B-type, Romanian J. Phys. 58 (7–8) (2013) 729–748. [2] A.J.M. Jawad, M.D. Petković, A. Biswas, Soliton solutions for nonlinear Calaogero–Degasperis and potential Kadomtsev–Petviashvili equations, Comput. Math. Appl. 62 (6) (2011) 2621–2628. [3] M. Kumar, R. Kumar, On some new exact solutions of incompressible steady state Navier–Stokes equations, Meccanica 49 (2014) 335–345. [4] M. Kumar, R. Kumar, On new similarity solutions of the Boiti–Leon–Pempinelli system, Commun. Theor. Phys. (Beijing) 61 (2014) 121–126. [5] M. Kumar, Y.K. Gupta, Some invariant solutions for non conformal perfect fluid plates in 5-flat form in general relativity, Pramana 74 (2010) 883–893. [6] B.G. Konopelcheno, V.G. Dubrovsky, Some new integrable nonlinear evolution equations in (2 + 1)-dimensions, Phys. Lett. A 102 (1984) 15–17. [7] D. Wang, H.-Q. Zhang, Further improved F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equation, Chaos Solitons Fractals 25 (2005) 601–610. [8] J. Lin, S.Y. Lou, K.L. Wang, Multi-soliton solutions of the Konopelchenko–Dubrovsky equation, Chin. Phys. Lett. 18 (2001) 1173–1175. [9] S. Zhang, T.C. Xia, A generalized F-expansion method and new exact solutions of Konopelchenko–Dubrovsky equations, Appl. Math. Comput. 183 (2006) 1190–1200. [10] A.M. Wazwaz, New kinks and solitons solutions to the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation, Math. Comput. Modelling 45 (2007) 473–479. [11] L. Song, H. Zhang, New exact solutions for the Konopelchenko–Dubrovsky equation using an extended Riccati equation rational expansion method and symbolic computation, Appl. Math. Comput. 187 (2007) 1373–1388. [12] T.L. He, Bifurcation of travelling wave solutions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations, Appl. Math. Comput. 204 (2008) 773–783. [13] X. Zeng, X. Yong, A new mapping method and its applications to nonlinear partial differential equations, Phys. Lett. A 372 (2008) 6602–6607. [14] H.Y. Zhi, Lie point symmetry and some new soliton-like solutions of the Konopelchenko–Dubrovsky equations, Appl. Math. Comput. 203 (2008) 931–936. [15] W.-G. Feng, C. Lin, Explicit exact solutions for the (2 + 1)-dimensional Konopelchenko–Dubrovsky equation, Appl. Math. Comput. 210 (2009) 298–302. [16] A.H. Salas, Construction of N-soliton solutions to (2 + 1)-dimensional Konopelchenko–Dubrovsky (KD) equations, Appl. Math. Comput. 217 (2011) 7391–7399. [17] H.Y. Zhi, Symmetry reductions of the Lax pair for the 2 + 1-dimensional Konopelchenko–Dubrovsky equation, Appl. Math. Comput. 210 (2009) 530–535. [18] S. Zhang, Exp-function method for Riccati equation and new exact solutions with two arbitrary functions of (2 + 1)-dimensional Konopelchenko–Dubrovsky equations, Appl. Math. Comput. 216 (2010) 1546–1552. [19] N. Taghizadeh, M. Mirzazadeh, Exact travelling wave solutions for Konopelchenko–Dubrovsky equation by the first integral method, Appl. Appl. Math. 6 (2011) 1893–1901. [20] H.W. Hu, J. Yu, Finite symmetry transformation group of the Konopelchenko–Dubrovsky equation from its Lax pair, Chin. Phys. B 21 (2) (2012) 020202. [21] S. Kumar, A. Hama, A. Biswas, Solutions of Konopelchenko–Dubrovsky equation by traveling wave hypothesis and Lie Symmetry Approach, Appl. Math. Inf. Sci. 8 (4) (2014) 1533–1539. [22] G.W. Bluman, J.D. Cole, Similarity Methods for Differential Equations, Springer-Verlag, New York, 1974, pp. 143–166. [23] P.J. Olver, Applications of Lie Groups to Differential Equations, Springer-Verlag, New York, 1993, pp. 30–130. [24] M. Kumar, R. Kumar, A. Kumar, Some more similarity solutions of the (2 + 1)-dimensional BLP system, Comput. Math. Appl. 70 (2015) 212–221. [25] M. Kumar, R. Kumar, A. Kumar, On similarity solutions of Zabolotskaya–Khokhlov equation, Comput. Math. Appl. 68 (2014) 454–463. [26] T. Raja Sekhar, V.D. Sharma, Similarity solutions for three dimensional Euler equations using Lie group analysis, Appl. Math. Comput. 196 (1) (2008) 147–157. [27] T. Raja Sekhar, V.D. Sharma, Similarity analysis of modified shallow water equations and evolution of weak waves, Commun. Nonlinear Sci. Numer. Simul. 17 (2) (2012) 630–636. [28] X.Y. Tang, S.Y. Lou, Y. Zhang, Localized excitations in (2 + 1)-dimensional systems, Phys. Rev. E 66 (2002) 046601.