Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method

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Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method Abdullahi Rashid Adem Department of Mathematical Sciences, Material Science Innovation and Modelling Focus Area, North-West University, Mafikeng Campus, Private Bag X 2046, Mmabatho 2735, South Africa

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Article history: Received 1 March 2017 Received in revised form 17 May 2017 Accepted 29 June 2017 Available online xxxx

A coupled Kadomtsev–Petviashvili equation is investigated by using Lie symmetry analysis. The similarity reductions and new exact solutions are obtained via the extended tanh method with symbolic computation. Exact solutions including solitons are shown. The solutions derived have dissimilar physical structures and depend on the real parameters. © 2017 Elsevier Ltd. All rights reserved.

Keywords: Coupled Kadomtsev–Petviashvili equation Lie symmetry method Extended tanh method

1. Introduction Nonlinear evolution equations (NLEEs) model a plethora of physical phenomena that appear in many branches of physics, engineering and applied mathematics. Therefore it is central to investigate the exact explicit solutions of NLEEs. To obtain the exact solutions of NLEEs, a number of methods have been suggested in the literature. Some of the renowned approaches include the inverse scattering transform method, Hirota’s bilinear method, tanh and sine–cosine methods, etc. [1–18]. The illustrious Korteweg–de Vries (KdV) equation [19] ut + 6uux + uxxx = 0

(1.1)

is an example of a NLEE. It describes the dynamics of solitary waves. Initially it was derived to describe shallow water waves of long wavelength and small amplitude. It is a significant equation in the theory of integrable systems since it has an infinite number of conservation laws, multiple-soliton solutions, and many other physical properties. A natural generalization of the KdV equation is the Kadomtsev–Petviashvili (KP) equation [20] (ut + 6uux + uxxx )x + uyy = 0

(1.2)

which is a model for shallow long waves in the x-direction with some mild dispersion in the y-direction. It is entirely integrable by the inverse scattering transform method and provides multiple-soliton solutions. In recent years, the coupled Korteweg–de Vries equations and the coupled Kadomtsev–Petviashvili equations have been the focus of attraction and some studies have been conducted by many authors [21–23]. In this paper, we will investigate a new coupled KP equation

( ut + uxxx −

7 4

uux − vvx +

5 4

(uv )x

)

+ uyy = 0,

(1.3a)

x

E-mail addresses: [email protected], [email protected]. http://dx.doi.org/10.1016/j.camwa.2017.06.049 0898-1221/© 2017 Elsevier Ltd. All rights reserved.

Please cite this article in press as: A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.049.

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) ( 7 5 vt + vxxx − uux − vvx + 2(uv )x + vyy = 0, 4

4

(1.3b)

x

by using the Lie symmetry method and extended tanh function method. Eq. (1.3) was firstly given in [10], and it was shown that Eq. (1.3) is completely integrable by the simplified form of Hirota’s bilinear method. Further, multiple soliton solutions and multiple singular soliton solutions for Eq. (1.3) have been derived. To our knowledge, Eq. (1.3) is a new coupled KP equation, and little work has been given. We will pay our attention to the study on Eq. (1.3). 2. Symmetry reductions and exact solutions of (1.3) The vector field

∂ ∂ ∂ ∂ + ξ 2 (t , x, y, u, v ) + ξ 3 (t , x, y, u, v ) + η1 (t , x, y, u, v ) ∂t ∂x ∂y ∂u ∂ + η2 (t , x, y, u, v ) ∂v

X = ξ 1 (t , x, y, u, v )

(2.4)

is a Lie point symmetry of (1.3) if X

[4]

{( ut + uxxx −

7 4

uux − vvx +

5 4

(uv )x

)

(

5

7

4

4

+ uyy , vt + vxxx − uux − vvx + 2(uv )x

x

) x

}⏐ ⏐ + vyy ⏐⏐

= 0, (1.3)

where X[4] is the fourth prolongation of (2.4). Expanding the above equation and splitting on the derivatives of u and v leads to the following overdetermined system of linear partial differential equations:

ξu3 = 0,

ξv3 = 0,

ξv = 0,

ξ

1

η

= 0,

η

= 0,

1 x, u 2 x, u 1

ξy1 = 0,

= ,

2 0 x, x 2 0 y,u 2 0 u ,u 1 3 2 y ,u y ,y

ηv,v = 0,

η

= ,

η

= ,

η

−ξ

η

ξx3 = 0,

ξu2 = 0,

ξv2 = 0,

ξx1 = 0,

ξu1 = 0,

= 0,

2 x,v 1 y,v 1 x,v

η

= 0,

η

= 0,

= 0,

ηu1,u = 0,

2ηy2,v − ξy3,y = 0,

2 ηv,v = 0,

ηu2,v = 0,

−ξt1 + 3ξx2 = 0,

ηu1,v = 0,

−ξy3 + 2ξx2 = 0,

−ξ − 2ξ = 0, 3 t

2 y

−5ηx1 + 8ηx2 + 2ηt2,u = 0,

5ηx1 − 4ηx2 + 2ηt1,v = 0,

−14ξx2 − 7ηu1 + 10ηu2 + 5ηv1 = 0,

−14ξx2 + 4ηu2 + 16ηv1 − 7ηv2 = 0, −10ξx2 − 10ηu1 + 23ηu2 + 5ηv2 = 0, 2 1 1 2 −8ξx + 4ηu + 17ηv − 8ηv = 0, 10ξx2 − 4ηu2 − 15ηv1 + 5ηv2 = 0, 16ξx2 + 8ηu1 − 12ηu2 − 5ηv1 = 0, −7ηx1 + 5ηx2 − 2ξt2,x + 2ηt1,u − 2ξy2,y = 0, 8ηx1 − 7ηx2 − 2ξt2,x + 2ηt2,v − 2ξy2,y = 0, −7ηx1,x u + 5ηx2,x u + 5ηx1,x v − 4ηx2,x v + 4ηt1,x + 4ηy1,y + 4ηx1,x,x,x = 0, −5ηx1,x u + 8ηx2,x u + 8ηx1,x v − 7ηx2,x v + 4ηt2,x + 4ηy2,y + 4ηx2,x,x,x = 0, 5ηv1 u + 5ηu2 u − 14ξx2 u − 8ηv1 v − 4ηu2 v + 10ξx2 v − 7η1 + 5η2 − 4ξt2 = 0,

−5ηv1 u − 5ηu2 u + 16ξx2 u + 8ηv1 v + 4ηu2 v − 14ξx2 v + 8η1 − 7η2 − 4ξt2 = 0, −5ηu1 u + 15ηu2 u + 5ηv2 u − 10ξx2 u + 8ηu1 v − 12ηu2 v − 8ηv2 v + 16ξx2 v − 5η1 + 8η2 = 0, −5ηu1 u − 15ηv1 u + 5ηv2 u + 10ξx2 u + 4ηu1 v + 12ηv1 v − 4ηv2 v − 8ξx2 v + 5η1 − 4η2 = 0. Solving the above equations we obtain the values of ξ 1 , ξ 2 , ξ 3 , η1 and η2 with the aid of Maple. We now state the result in the following theorem. Theorem. The infinitesimal symmetries of (1.3) constitute the five dimensional Lie algebra spanned by the following linearly independent operators:

∂ , ∂t ∂ X2 = , ∂x ∂ X3 = , ∂y

X1 =

∂ ∂ + 2t , ∂x ∂y ∂ ∂ ∂ ∂ ∂ X5 = x + 2y + 3t − 2u − 2v . ∂x ∂y ∂t ∂u ∂v

X4 = −y

Please cite this article in press as: A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.049.

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2.1. Symmetry reductions of (1.3) Linear combination of the translation symmetries (so as to obtain a travelling wave transformation), Γ = X1 + X2 + ωX3 , where ω is a constant diminishes (1.3) to a partial differential equation (PDE) in two independent variables. The symmetry Γ produces the following three invariants: f = y − ωt ,

g = t − x,

φ = u,

ψ = v.

By the above invariants we can compose the altered equation as 5

5

7

4

4

4 5

5

7φg2

2

4

−ψgg ψ + φgg ψ + ψgg φ − φgg φ + ωφfg + φff − ψg2 + ψg φg − 7

− ψgg ψ + 2φgg ψ + 2ψgg φ − φgg φ + ωψfg + ψff −

7ψ

2 g

− φgg + φgggg = 0,

− ψgg + ψgggg + 4ψg φg −

5φg2

=0 4 4 4 4 which is a system in two independent variables f and g. The above equation has the two translation symmetries, namely Υ1 =

∂ , ∂f

Υ2 =

∂ . ∂g

By taking a linear combination Υ1 + Υ2 (so as to end up with a travelling wave transformation of the form z = x + y − (ω + 1) t) of the above symmetries, we see that it yields the invariants z = f − g = x + y − (ω + 1) t ,

φ = F,

ψ = G.

Treating F , G as new dependent variables and z as the new independent variable the above system reduces to the following system: ′

′

5F G 2 ′

5

7

+ F G(z) − ωF − F (z)F − ′′

′′

4

′′

7F

4

′

4F G + 2F ′′ G(z) −

5 4

F (z)F ′′ −

5F

′2

4

′2

4

5

′

+ F ′′′′ + F (z)G′′ − G(z)G′′ − G 2 = 0, 4

′2

7

7G

4

4

+ 2F (z)G′′ − ωG′′ − G(z)G′′ −

+ G′′′′ = 0.

(2.5a) (2.5b)

2.2. Exact solutions using the extended tan h method We use the extended tanh function method which was presented by Wazwaz [7]. The elementary idea in this technique is to assume that the solution of (2.5) can be written in the form F (z) =

M ∑

Ai H(z)i ,

G(z) =

i=−M

M ∑

Bi H(z)i ,

(2.6)

i=−M

where H(z) satisfies the Riccati equation H ′ (z) = 1 − H 2 (z),

(2.7)

whose solution is H(z) = tan h(z). The positive integer M will be computed by the homogeneous balance method between the highest order derivative and highest order nonlinear term appearing in (2.5). Ai , Bi are parameters to be computed. Here, the balancing procedure gives M = 2 and so the solutions of (2.5) are of the form F (z) = A−2 H −2 + A−1 H −1 + A0 + A1 H + A2 H 2 ,

(2.8a)

G(z) = B−2 H −2 + B−1 H −1 + B0 + B1 H + B2 H 2 .

(2.8b)

We have the following four cases with the aid of Maple: Case 1

ω=− A−2

469 A0 k2

16 = 48 k,

+

13 2

A0 k −

19 A0 16

− 8,

A−1 = 0, A1 = 0, A2 = 0, Please cite this article in press as: A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.049.

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A.R. Adem / Computers and Mathematics with Applications (

B−2 = −

11256 k2 37

B−1 = 0,

+

3576 k 37

−

144 37

(

B0 = −

)

–

,

)

A0 1407 k2 + 626 k − 241

,

148

B1 = 0, B2 = 0; Case 2

ω=−

81137 k2 B0 1480

A−2 = 0,

+

4391 B0 k 1480

1777 B0

−

740

− 8,

A−1 = 0,

(

A0 =

B0 13601 k2 + 267 k + 692

) ,

370

A1 = 0, A2 = 48 k, B−2 = 0, B−1 = 0, B0 = B0 , B1 = 0, B2 = −

11256 k2 37

+

3576 k 37

−

144 37

;

Case 3

ω=− A−2

81137 k2 B0

1480 = 48 k,

+

4391 B0 k 1480

1777 B0

−

740

− 8,

A−1 = 0,

(

A0 =

B0 13601 k2 + 267 k + 692

)

370

A1 = 0,

,

A2 = 48 k, 11256 k2 3576 k 144 B−2 = − + − , 37 37 37 B−1 = 0, B1 = 0, B2 = −

11256 k2 37

+

3576 k 37

−

144 37

;

Case 4

ω = −8, A−2 = 48 k, A−1 = 0, A0 = 0, A1 = 0, A2 = 48 k, B−2 = − B−1 = 0,

11256 k2 37

+

3576 k 37

−

144 37

,

B0 = 0, B1 = 0, B2 = −

11256 k2 37

+

3576 k 37

−

144 37

.

Please cite this article in press as: A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.049.

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5

2.86

10.7

u

)

2.84 2.82

10.6 v

10

10.5

10

2.80 2.78

10.4

0 y

0 y

–10

–10 0

0

–10

x

–10

x

10

10

Fig. 1. Profiles of solutions (2.9) for Case 1 with k = 0.216547, A0 = 0, t = 0.

0

0.0 0.5

u

v

1.0

1 2

1.5

5

5 0

0

y

0

y

x

0

x

5

5

Fig. 2. Profiles of solitonic solutions (2.9) for Case 2 with k = 0.216547, B0 = 0, t = 0.

5

v

26

u

7.0

24

0

22

5

6.5 6.0

y 0

x

y

0

0 5

x

5

Fig. 3. Profiles of solutions (2.9) for Case 3 with k = 0.216547, B0 = 0, t = 0.

Note that k is any root of 469k3 − 104k2 + 19 k − 4 = 0. Consequently, a solution of (1.3) is u(x, y, t) = A−2 cot h2 (z) + A−1 cot h(z) + A0 + A1 tan h(z) + A2 tan h2 (z),

(2.9a)

v (x, y, t) = B−2 cot h (z) + B−1 coth(z) + B0 + B1 tan h(z) + B2 tan h (z),

(2.9b)

2

2

where z = x + y − (ω + 1) t (see Figs. 1–3). Please cite this article in press as: A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.049.

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Hereby, we give some figures to describe our solutions. Note that our solutions are different from those derived in [21]. We have obtained four families of solutions with different coefficients given by Cases 1–4. We can see from these figures that solutions in case 2 can generate solitonic profiles, and the other cases give singular profiles. 3. Concluding remarks Lie symmetry method and the extended tanh method were successfully used for analytic handling of the new coupled KP equation. Exact solutions including solitons were correctly computed, and some of the obtained solutions are new ones. The solutions have dissimilar physical structures and depend on the real parameters. It is anticipated that the exact solutions shown in this paper can be used as benchmarks for numerical simulations of the underlying equation. References [1] X. Lü, M. Peng, Nonautonomous motion study on accelerated and decelerated solitons for the variable-coefficient Lenells-Fokas model, Chaos 23 (013122) (2013) 1-7. [2] X. Lü, F. Lin, Soliton excitations and shape-changing collisions in alpha helical proteins with interspine coupling at higher order, Commun. Nonlinear Sci. Numer. Simul. 32 (2016) 241–261. [3] X. Lü, F. Lin, F. Qi, Analytical study on a two-dimensional Korteweg–de Vries model with bilinear representation, Bäcklund transformation and soliton solutions, Appl. Math. Model. 39 (2015) 3221–3226. [4] X. Lü, L. Ling, Vector bright solitons associated with positive coherent coupling via Darboux transformation, Chaos 25 (123103) (2015) 1-8 em. [5] A.M. Wazwaz, Exact solutions for the ZK-MEW equation by using the tanh and sine-cosine methods, Int. J. Comput. Math. 82 (2005) 699–708. [6] A.M. Wazwaz, A study on KdV and Gardner equations with time-dependent coefficients and forcing terms, Appl. Math. Comput. 217 (2010) 2277–2281. [7] A.M. Wazwaz, Completely integrable coupled KdV and coupled KP systems, Commun. Nonlinear Sci. Numer. Simul. 15 (2010) 2828–2835. [8] X. Lü, S.T. Chen, W.X. Ma, Constructing lump solutions to a generalized Kadomtsev–Petviashvili-Boussinesq equation, Nonlinear Dynam. 86 (2016) 523–534 em. [9] X. Lü, W.X. Ma, Study of lump dynamics based on a dimensionally reduced Hirota bilinear equation, Nonlinear Dynam. 85 (2016) 1217–1222 em. [10] X. Lü, W.X. Ma, Y. Zhou, C.M. Khalique, Rational solutions to an extended Kadomtsev–Petviashvili-like equation with symbolic computation, Comput. Math. Appl. 71 (2016) 1560–1567. [11] X. Lü, W.X. Ma, S. Chen, C.M. Khalique, A note on rational solutions to a Hirota-Satsuma-like equation, Appl. Math. Lett. 58 (2016) 13–18. [12] X. Lü, W.X. Ma, J. Yu, C.M. Khalique, Solitary waves with the Madelung fluid description: A generalized derivative nonlinear Schrödinger equation, Commun. Nonlinear Sci. Numer. Simul. 31 (2016) 40–46. [13] M.S. Mohamed, K.A. Gepreel, Reduced differential transform method for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations, J. Egyptian Math. Soc. 25 (2017) 1–7. [14] M.S. Mohamed, K.A. Gepreel, M.R. Alharthi, R.A. Alotabi, Homotopy analysis transform method for integro-differential equations, Gen. Math. Notes 32 (2016) 32–48. [15] K.A. Gepreel, T.A. Nofal, K.O. Alweil, Simple equation method for nonlinear evolution equations in mathematical physics, WSEAS Trans. Math. 15 (2016) 2224–2880. [16] K.A. Gepreel, T.A. Nofal, N.S. Al-Sayali, Direct method for solving nonlinear strain wave equation in microstructure solids, Int. J. Phys. Sci. 11 (2016) 121–131. [17] K.A. Gepreel, T.A. Nofal, N.S. Al-Sayali, Exact solutions to the generalized Hirota-Satsuma KdV equations using the extended trial equation method, Eng. Lett. 24 (2016) 274–283. [18] K.A. Gepreel, Exact solutions for nonlinear integral member of Kadomtsev–Petviashvili hierarchy differential equations using the modified (w/g)expansion method, Comput. Math. Appl. 72 (2016) 2072–2083. [19] D.J. Korteweg, G. de Vries, On the change of form of long waves advancing in a rectangular canal, and on a new type of long stationary waves, Phil. Mag. 39 (1895) 422–443. [20] B.B. Kadomtsev, V.I. Petviashvili, On the stability of solitary waves in weakly dispersive media, Sov. Phys. Dokl. 15 (1970) 539–541. [21] A.M. Wazwaz, Integrability of two coupled Kadomtsev–Petviashvili equations, Pramana J. Phys. 77 (2) (2011) 233–242. [22] A.R. Adem, X. Lü, Travelling wave solutions of a two-dimensional generalized Sawada-Kotera equation, Nonlinear Dynam. 84 (2016) 915–922. [23] A.M. Wazwaz, New solitary wave solutions to the Kuramoto-Sivashinsky and the Kawahara equations, Appl. Math. Comput. 182 (2006) 1642–1650.

Please cite this article in press as: A.R. Adem, Symbolic computation on exact solutions of a coupled Kadomtsev–Petviashvili equation: Lie symmetry analysis and extended tanh method, Computers and Mathematics with Applications (2017), http://dx.doi.org/10.1016/j.camwa.2017.06.049.