Physics and Mathematics Department, Faculty of Engineering, Zagazig University, Egypt.
A modified version of the generalized Hirota-Satsuma equation is solved analytically using
Darboux transformations (DT), we start with the Lax pair of this equation and apply DT. This leads to another solvable pair containing a new eigen functions that is a solution of the equation. Several seeds solutions are tested as well as one and two solitons forms are obtained using DT. A suitable choice of the seed fields leads to new solutions.
Darboux transformation, Hirota-Satsuma equation, lax pair, solitons.
The wave observed in plasma, elastic media, optical fibers, fluid dynamics are described by nonlinear partial differential equations. In the past decades, several methods for obtaining analytic solutions of nonlinear partial differential equations (NPDEs) have been presented, such as the inverse scattering method[1], Hirota's method [2], the Backlund and Darboux transformation [3,4], Painlevé expansions [5], homogenous balance method [6], Jacobi elliptic function [7, 8], extended tanh-function methods[9-11], extended F-expansion methods [12-15], Adomain methods [16,17], Exp -function methods [18, 19] and finally the Mapping method [20-24] .
In this paper, we solve using Darboux transformation method (25- 30), the generalized
Hirota–Satsuma equation in three dimensions (3D) described as follows;
[ℎ 𝑥𝑥𝑧
−
3
4
( ℎ
2 𝑥𝑧 ℎ 𝑧
) + 3ℎ 𝑥 ℎ 𝑧
] x
= h yz
(1)
This equation describes the flow of an incompressible fluid. Using the Singular Manifold Method
(SMM), Estevez et al [31] derive its Lax pair in the form of;
−𝜓 𝑦
+ 𝜓 𝑥𝑥𝑥
+ 3ℎ 𝑥 𝜓 𝑥
+
3
2 ℎ 𝑥𝑥 𝜓 = 0 (2)
2ℎ 𝑧 𝜓 𝑥𝑧
− ℎ 𝑥𝑧 𝜓 𝑧
+ 2ℎ 2 𝑧 𝜓 = 0 (3) where h(x,y,z) is the wave amplitude and 𝜓(𝑥, 𝑦, 𝑧) in the system (2) and (3) is eigen functions. The present work is organized as follows; section 2, is devoted to the mathematical formulation of the problem .We start with an initial solution h and recursively obtain via the system eigen functions 𝜓, 𝜓
1
, an improved solution h[1]. Applying DT, N-times, produces N-soliton solutions. In Section 3, we explicitly detail, the explicit solitary wave solutions for different seeds form and plot them. In Section
4, two soliton solutions are derived and plotted, applying the DT method. The paper ends with a conclusion, in section (5).
Corresponding author. Tel.: +201096140008, +201151444075.
E-mail address: ah_halim@hotmail.com (A.A.Halim), eng_rahmaelsadat@yahoo.com (R.Sadat)

Darboux transformation is a recursive algorithm, deriving a series of explicit solutions from a trivial one. Applying it to the Lax pair (2) and (3) results in two eigenfunction 𝜓, 𝜓
1
. These are used together, with an initial seed solution h in the following equations; 𝜓[1] = ( 𝑑 𝑑𝑥
− 𝜓
′
1 ) 𝜓 (4) 𝜓
1 ℎ[1] = ℎ + 𝜓
′
1 (5) 𝜓
1 where 𝜓[1] satisfies (2) and (3) and h [1] is a new solution for equation (1). Replacing for 𝜓[1] in (2) and (3) yields;
−𝜓 𝑦
[1] + 𝜓 𝑥𝑥𝑥
[1] + 3ℎ 𝑥
[1] 𝜓 𝑥
[1] +
3
2 ℎ 𝑥𝑥
[1] 𝜓[1] = 0 (6)
2ℎ 𝑧
[1] 𝜓 𝑥𝑧
[1] − ℎ 𝑥𝑧
[1] 𝜓 𝑧
[1] + 2ℎ 2 𝑧
[1] 𝜓[1] = 0 (7)
Applying DT, once more we have; 𝑑
𝜓[2] = [ 𝑑𝑥
− 𝜓
′
2
[1] 𝜓
2
[1]
] 𝜓[1] (8)
ℎ[2] = ℎ[1] + 𝑑 𝑑𝑥 ln 𝜓
2
[1] (9) where 𝜓
2
[1] is defined as; 𝜓
2
[1] = [ 𝑑 𝑑𝑥
− 𝜓
′
1 𝜓
1
] 𝜓
2
(10) where 𝜓
2
is additional solution of (2) and (3) using a different constant of integration. From (8) into equations (6) and (7) we obtain;
−𝜓 𝑦
[2] + 𝜓 𝑥𝑥𝑥
[2] + 3ℎ 𝑥
[2] 𝜓 𝑥
[2] +
3
2 ℎ 𝑥𝑥
[2] 𝜓[2] = 0 (11)
2ℎ 𝑧
[2]𝜓 𝑥𝑧
[2] − ℎ 𝑥𝑧
[2]𝜓 𝑧
[2] + 2ℎ 2 𝑧
[2] 𝜓[2] = 0 (12)
Where ℎ[2] is a new solution of equation (1). Applying DT N-time gives the following forms for 𝜓[𝑁], ℎ[𝑁] ;
𝜓[𝑁] =
𝑊(𝜓
1
,𝜓
2
,…,𝜓
𝑁
,𝜓)
𝑊(𝜓
1
,𝜓
2
,…,𝜓
𝑁
)
(13)
ℎ[𝑁] = ℎ + 𝑑 𝑑𝑥 ln 𝑊(𝜓
1
, 𝜓
2
, … , 𝜓
𝑁
) (14) where W is the Wronskian of the eigen functions; 𝜓
1
, 𝜓
2
, … , 𝜓
𝑁
, 𝜓 .
This section gives a single soliton (solitary wave) solution for both the nonlinear equation (1) and its
Lax pair (2) and (3). To simplify the solution of this system, we use a simple seed field (h). Some seed fields are chosen and the explicit solutions are given below.
3.1 First initial (seed) solution
Consider an initial wave form;
ℎ = 𝑥 + 𝑦 + 𝑧 (15)

Substituting h into equations (2) and (3) gives;
−𝜓 𝑦
+ 𝜓 𝑥𝑥𝑥
+ 3𝜓 𝑥
= 0 (16) 𝜓 𝑥𝑧
+ 𝜓 = 0
Let in equation (16) the solitary wave solution be;
(17) 𝜓(𝑥, 𝑦, 𝑧) = 𝜙(𝜁) ,
Where, 𝜁 = 𝑥 + 𝑐
1 𝑦 + 𝑐
1 𝑧
−3
, to reduce equation (16) as follow;
(18) 𝜙 𝜁𝜁𝜁
+ (3 − 𝑐
1
)𝜙 𝜁
= 0 (19)
Integrating with respect to 𝜁 we obtain; 𝜙 𝜁𝜁
+ (3 − 𝑐
1
) 𝜙 = 𝑐 (20)
Where, c is an arbitrary constant of integration. As the boundary conditions for solitary wave are; 𝜙, 𝜙 𝜁
, 𝜙 𝜁𝜁
→ 0 as 𝜁 → ±∞ , thus c =0.
;
𝜙(𝜁) = 𝐴(𝑧)𝑒 𝜆
1 𝜁 + 𝐵(𝑧)𝑒 −𝜆
1 𝜁
(21)
Where, 𝜆
1
= √𝑐
1
− 3, 𝑐
1
> 3 (22) and 𝐴(𝑧), 𝐵(𝑧) are function to be determined later. Returning to the original variables x, y and 𝜓 , we have;
𝜓(𝑥, 𝑦, 𝑧) = 𝑘
1 𝑒 𝜆
1 𝑥+𝑐
1 𝜆
1
1 𝑦− 𝜆1 𝑧
+ 𝑘
2 𝑒
−𝜆
1 𝑥+𝑐
1 𝜆
1 𝑦−
1 𝜆1 𝑧
(23)
Where, 𝑘
1
, 𝑘
2
are two arbitrary constants.
let 𝑘
1
=
1
2
, k
2
= −
1
2
we obtain;
Others values are 𝑘
1
=
1
2
, 𝑘
2
=
1
𝜓(𝑥, 𝑦, 𝑧) = 𝑠𝑖𝑛ℎ (𝜆
1
(𝑥 + 𝑐
1 𝑦) − 𝑧) (24) 𝜆
1
1
2 gives;
𝜓
1
(𝑥, 𝑦, 𝑧) = 𝑐𝑜𝑠ℎ (𝜆
1
(𝑥 + 𝑐
1 𝑦) −
1 𝜆
1
Hence from formula (4), 𝜓[1] will be 𝑧) (25)
𝜓[1] = 𝜆
1
𝑠𝑒𝑐ℎ (𝜆
1
(𝑥 + 𝑐
1 𝑦) −
1 𝜆
1 𝑧) (26)
Which is a solution of the new pair (6) and (7) with new field ℎ[1] from (5) given by
ℎ[1] = 𝑥 + 𝑦 + 𝑧 + 𝜆
1
𝑡𝑎𝑛ℎ (𝜆
1
(𝑥 + 𝑐
1 𝑦) −
1 𝜆
1 𝑧) (27) h[1], the
solution of equation (1) is plotted below in Fig.1 .
Corresponding author. Tel.: +201096140008, +201151444075.
E-mail address: ah_halim@hotmail.com (A.A.Halim), eng_rahmaelsadat@yahoo.com (R.Sadat)

Figure 1: (a) Solitary wave solution, h[1] at c1=12, 𝝀
𝟏
=3,z=2 in (27)
6 h
6 h
4
4
2
2
1.0
0.5
0.5
1.0
y
1.0
0.5
0.5
1.0
x
2 2
(b) Plane graph of h[1] in (27) at y=0 (c) Plane graph of h[1]in (27)at x=0
3.2 Second seed solution
Consider an initial wave form
ℎ = xy + (
1 z
) (28)
Substituting h from (28) into system (2, 3) reduces it to the form
−𝜓 𝑦
+ 𝜓 𝑥𝑥𝑥
+ 3𝑦𝜓 𝑥
= 0 (29)
−𝑧 2 𝜓 𝑥𝑧
+ 𝜓 = 0 (30)
Solving the system (29) and (30) yields;

𝜓
1
= cosh (𝜆
1 𝑥 + (𝜆
1
2
+
3
2 𝑦) 𝜆
1 𝑦 −
1 𝜆
1 𝑧
) (31)
From (31) in (7) we obtain the solution; h[1] = xy + (
1 z
) + 𝜆
1
𝑡𝑎𝑛ℎ [𝜆
1 𝑥 + 𝜆
1
𝟑 𝑦 +
3
2 𝜆
1 𝑦 2 −
1
λ
1 z
] (32)
, This solution is plotted below in Fig. 2 for 𝜆
1
= 3, 𝑧 = 2
Figure 2: Solitary wave solution h[1] for 𝜆
1
= 3, 𝑧 = 2
To derive a two-soliton solution; h[2], we apply the DT formula for two soliton solution,
𝜓[2] =
𝑊(𝜓
1
,𝜓
𝑊(𝜓
1
2
,𝜓
2
,𝜓)
(33)
) ℎ[2] = ℎ + 𝑑 𝑑𝑥 ln 𝑊(𝜓
1
, 𝜓
2
) (34) where W is the wronskian of three eigenfunctions; 𝜓, 𝜓
1
, 𝜓
2
, while h is the seed solution.
4.1 First seed solution
The first seed solution has the form; ℎ = 𝑥 + 𝑦 + 𝑧 , while 𝜓, 𝜓
1
are obtained from equations (24) and
(25). To obtain 𝜓
2
, we set 𝑘
1
= 𝑘
2
=
1
2
in (23), and choose another 𝜆
2
parameter as 𝜆
2
=
√𝑐
2
− 3, 𝑐
2
> 3 . This result in;
𝜓
2
(𝑥, 𝑦, 𝑧) = 𝑐𝑜𝑠ℎ (𝜆
2
(𝑥 + 𝑐
2 𝑦) −
1 𝜆
2 𝑧) (35)
Replacing for 𝜓
2 in (33),(34)we obtain;
𝜓[2] = 𝜆
2 𝜆
1
𝑡𝑎𝑛ℎ(𝜆
2
(𝜆
2
1
−𝜆
2
2
) 𝑠𝑒𝑐ℎ[𝜆
1
(𝑥+𝑐
1 𝑦)− 𝑧 𝜆1
]
(𝑥+𝑐
2 𝑦)− 𝑧 𝜆2
)−𝜆
1
𝑡𝑎𝑛ℎ(𝜆
1
(𝑥+𝑐
1 𝑦)− 𝑧 𝜆1
)
(36)
Corresponding author. Tel.: +201096140008, +201151444075.
E-mail address: ah_halim@hotmail.com (A.A.Halim), eng_rahmaelsadat@yahoo.com (R.Sadat)

ℎ[2] = 𝑥 + 𝑐
1 𝑦 + 𝑧 + 𝜆
2
𝑡𝑎𝑛ℎ(𝜆
2
(𝑥+𝑐
2 𝜆
2
2 𝑦)− 𝑧 𝜆2
−𝜆
2
1
)−𝜆
1
𝑡𝑎𝑛ℎ(𝜆
1
(𝑥+𝑐
1 𝑦)− 𝑧 𝜆1
)
(37)
The two-soliton solution, h[2] is plotted in Fig.3 for different c’s.
Figure 3: (a) Solitary wave solution, h[2] for 𝑐
1
= 12, 𝑐
2
= 39, 𝜆
1
= 3, 𝜆
2
= 6, 𝑧 = 2
50 h
50 h
1.0
0.5
0.5
1.0
x
1.0
0.5
0.5
1.0
y
50
50
100 100
(b) Plane graph of h[2] in (37) at y=0 (c) Plane graph of h[2] in (37) at x=0
4.2 Second seed solution h
Starting with another initial wave form;

ℎ = 𝑥𝑦 +
1 𝑧
(38)
And following steps similar to those in § 4.1, we obtain; h[2] = xy + (
1 z
) + 𝜆
1
𝑡𝑎𝑛ℎ (𝜆
1 𝑥 + 𝜆
1
𝟑 𝑦 +
3
2 𝜆
1 𝑦 2 −
1
λ
1 z
)
+ 𝜆
2
2
−𝜆
2
1
+𝜆
2
2 tanh
2
(𝜆
2 𝑥+𝜆 𝜆
2
2
𝟑 𝑦+
3
2 𝜆
𝑡𝑎𝑛ℎ(𝜆
2
2 𝑦
2
− 𝑥+𝜆
2
𝟑
1
λ2z 𝑦+
)−𝜆
1
3
2 𝜆
2 𝑦 𝜆
2
2
𝑡𝑎𝑛ℎ(𝜆
1
1
−
λ2z
)−𝜆
1 𝑥+𝜆
1
𝟑 𝑦+
𝑡𝑎𝑛ℎ(𝜆
1
3
2 𝜆
1 𝑥+𝜆 𝑦
1
2
𝟑
−
1
λ1z
3 𝑦+
2 𝜆
)𝑡𝑎𝑛ℎ(𝜆
2
1 𝑦 2
1
−
λ1z
) 𝑥+𝜆
2
𝟑 𝑦+
3
2 𝜆
2 𝑦
2
−
1
λ2z
)
(39)
, This solution h[2] is plotted in Fig. 4 for 𝜆
1
= 0.5, 𝜆
2
= 0.25, 𝑧 = 1
Figure 4: Two-soliton solution, h[2] for 𝜆
1
= 0.5, 𝜆
2
= 0.25, 𝑧 = 1
We did here solve Hirota Satsuma equation (1) using Darboux transformation. We tested different seed solutions and applying one and two solitons DT, we derive new solutions.
[1] M.J. Ablowitz and P.A.Clarkson, Solitons, Nonlinear Evolution Equations and Inverse Scattering,
Cambridge University Press (1991)
[2] R.Hirota, Exact solutions of the KdV equation for multiple collisions of solitons. Physics Rev.
Letters. 27, 1192-1199
[3] V.A.Matveev and M.A.Salle, Darboux transformation and solitons. Berlin Heidelberg: Springer-
Verlag. (1991)
Corresponding author. Tel.: +201096140008, +201151444075.
E-mail address: ah_halim@hotmail.com (A.A.Halim), eng_rahmaelsadat@yahoo.com (R.Sadat)

[4] O.H.El-Kalaawy and R.S. Ibrahim, Backlund Transformation and soliton solutions for the higher order and inhomogeneous nonlinear Schrodinger equations. International Journal of Nonlinear
Science, 7 (2009) 3-11.
[5] F.Cariello and M. Tabor, Painleve expansions for non integrable evolution equations.Physica D.
39 (1989) 77-94.
[6] M. Wang, Y. Zhou and Z. Li, Applications of a homogenous balance method to exact solutions of nonlinear equation in mathematical physics. Physics Letters A., 216 (1996) 67-75.
[7] Z.T.Fu, S.K. Liu, S.D. Liu and Q. Zhao, New Jacobi elliptic function expansion and new periodic solutions of nonlinear wave equations. Physics Letters A., 290 (2001) 72-76.
[8] E.Fan, Extended tanh function method and its applications to nonlinear equations. Physics Letters
A.,277 (2000) 212-218
[9] E.Yosufoglu and A. Bekir, Symbolic computation and new families of exact traveling solutions for the Kawahara and modified Kawahara equations. Computer and Mathematics with
Application, 55 (2008) 1113-1121.
[10] X. Zheng, , Y. Chen and H. Zhang, Generalized extended tanh-function method and its application to(1+1) dimensional dispersive wave equation. Physics Letters A., 311(2003) 145-
157.
[11] Z.Lu and H. Zhang, On a further tanh method. Physics Letters A., 307 (2003) 269-273.
[12] Y. Shen, and N. Cao, The double Expansions approach and novel nonlinear wave solutions of soliton equation in(2+1) dimension. Applied Mathematics and Computation, 198 (2008) 683-690.
[13] S.Zhang and T.Xia, An improved generalized F-expansion method and its application to the(2+1) dimensional KdV equations. Communications in nonlinear Science and Numerical Simulation,
13 (2008) 1294-1301.
[14] M.Wang and X. Li, Application of F- Expansion to periodic wave solutions of a new
Hamiltonian amplitude equation. Chaos soliton Fractal, 24 (2005)1257-1268.
[15] M.Wang, and X. Li, Extended F-expansion method and period wave solutions for the generalized
Zakharov equations. Physics Letters A., 343 ( 2005) 48-54.
[16] Y.Wang , C. Dai, L. Wu and J. Zhang,Exact and numerical solitary wave solutions of generalized
Zakharov equation by the Adomian decomposition method. Chaos SolitonFract., 32 ( 2007)
1208-1214.
[17] A.T.Abassy, A.M.El-Tawil and K.H.Saleh, The solution of Burgers and good Boussinesq equations using Admpad technique. Chaos Soliton Fract., 32 (2007)1008-1026.
[18] J.He and M.A. Abdou, New periodic solution for nonlinear evolution equations using Exp function method.Choas Soliton and Fractals, 34 (2007)1421-1429.
[19] J.He and X. Wu, Exp -function method for nonlinear wave equations. Choas Soliton and Fractals,
30 (2007) 700-708.
[20] Y.Z. Peng, Exact solutions for some nonlinear partial differential equations. Physics Letters A.,
314 ( 2003) 401-408.
[21] A. Elgarayhi , New periodic wave solutions for the shallow water equations and the generalized
Klein-Gordon equation. Communications in Nonlinear Science and Numerical Simulation.
13(2008) 877-888.
[22] S.A.Elwakil, A. Elgarayhi and A. Elhanbaly, Exact periodic wave solutions for some nonlinear partial differential equations. Chaos, Soliton Fractal, 29 (2006) 1037-1044.
[23] A Elgarayhi. New solitons and periodic wave solutions for the dispersive long wave equations.Physica A, 36 (2006) 416-428.
[24] A. Elgarayhi, and A. Elhanbaly,.New exact traveling wave solutions for the two-dimensional
Kadomtsev-Petviashvili and Boussinesq Equations. Physics Letter A, 343 )2005( 85-89.
[25] C.H.Gu, H.S.Hu and Z.X. Zhou, Darboux Transformation in Soliton Theory and its Geometric
Applications, Shanghai Scientic and Technical Publishers, Shanghai,( 2005).
[26] E.G. Fan, Integrable evolution systems based on Gerdjikov–Ivanov equations, bi-Hamiltonian structure, finite-dimensional integrable systems and N-fold Darboux transformation, J.
Mathematical Physics. 41 (2000) 7769–7778.

[27] J. Wang, Darboux transformation and soliton solutions for the Boiti-Pempinelli-Tu (BPT) hierarchy, J. Physics. A ,38 (2005) 8367–8377.
[28] Yu-Shan Xue , Bo Tian, Wen-Bao Ai , Yan Jiang , Darboux transformation and Hamiltonian structure for the Jaulent–Miodek hierarchy, J.
Applied Mathematics and Computation, 218 (2012)
11738–11750
[29] A.A. Halim, S. B. Leble, Chaos solitons and fractals, 19( 2004) 99-108.
[30] A.A. Halim, Chaos solitons and fractals, 36 (2008) 646-653.
[31] P.G. Estevez and J Prada. Singular manifold method for an equation in 2 + 1 dimensions. J.
Nonlinear Mathematical Physics. 12 (2005) 266–279.
Corresponding author. Tel.: +201096140008, +201151444075.
E-mail address: ah_halim@hotmail.com (A.A.Halim), eng_rahmaelsadat@yahoo.com (R.Sadat)