1. Introduction

The dynamics of physical systems is in general described by nonlinear partial differential equations (NPDE). If the increase of non-linearity on the other hand gives supplementary information in the understanding of the system, it complicates the analytic resolution of these equations. These NPDEs are seen in Mechanics of continuous media, in fluid mechanics, in nonlinear optic, in thermodynamics, kinetic chemistry… If the obtaining of these equations is one’s in a while easy, the resolution is not always easy and at times constituted a veritable challenge. It is in this light that for the past years, researchers have been working hard and also proposing solutions and methods of resolution. In case of dynamics of solitary waves, many efficient methods have been stated. We can mention among others the tanh-sech method[1 -3], the homogeneous balance method[4-7], the extended tanh method[8,9], thetanh-cothmethod[10], the exp-function method[11-15], the jacobi’s elliptic function method[16,17], the F-expansion method [18-20]. Beyond a multitude of methods many results were published in order to ameliorate the above mention methods or to extend them to other forms of equations[21-28]. In this work, we will use the principle that consists in decomposing the equation that we want to construct the solutions under the shape [22-26]

(1)

where , , , and are functions of the coefficients to determine.

The paper is organized as follows: In section 2 , we look for the ranges of equations, in section 3, we solve the obtained ranges of equations and in section 4, we conclude the work.

2. Implementations of Equations of Coefficients

Sasa-Satsuma’s equation is a well developed one of higher order obtained in certain order of nonlinearity. This equation is obtained principally in surface hydrodynamic wave paquets when perturbation development extends to the 4 order. We obtain similar equations in optics for waves of great velocity. The equation of Sasa-Satsuma presents itself in two forms [27, 28], the form that interest us is the modified one presented as follow

(2)

where represents the first derivative of the envelope with respect to time, represents the first derivative of the envelope with respect to variable and stands for a conjugate complex of . We propose to construct a solution made up of a combination of analytic forms of solitary waves. Not knowing the exact form capable of producing good results, we opt for the construction of the solution of the form

(3)

where , , , and can be real or complex. Given that the choice of the exact solution form of equation (2) is not always easy, we suppose that all the coefficients are complexes such that

, , , and where , and are real constants and . The choice of the solution in this form permits to have maximum possibilities as regard to the choice of the form of equation (3) which verifies best equation (2). Considering this, the introduction of equation (3) in equation (2) gives

(4)

where and are functions of the coefficients to determine. Equation (4) is called the range equations of coefficients with factors where and . This equation has a real part and the imaginary part. On identifying the two parts of equation (4) equal to zero, we obtain the following equations classified in order of priority [24,25]. Hence the real part of equation (4) gives as:

Term in ,

(5)

Term in ,

(6)

Term in ,

(7)

Term in ,

(8)

Term in ,

(9)

The imaginary part of equation (4) leads to the following equations such as:

Term in ,

(10)

Term in ,

(11)

Term in ,

(12)

Term in ,

(13)

Term in ,

(14)

As mentioned above, the choice of the form of solution when we use the BDK method [34,35] is not always easy. In the setting of this work, we wanted in a first time that the coefficients and are complex (); this in the goal to multiply the possibilities of obtaining the shape of the most suitable solution. This being, the sets of equations (10), (11), (12), (13) and (14) possess 8 unknowns and whose resolution is not easy because of their nonlinearity. Of all considered hypotheses, we got two that allowed us to get acceptable solutions.

The groups of equations (10), (11), (12), (13) and (14) lead to identities while the groups of equations (5), …, (9) become

Term in ,

(15)

Term in ,

(16)

Term in ,

(17)

Term in ,

(18)

Term in ,

(19)

3. Resolution of the Range Equations

The equation (15) imposes us three possibilities, the case where and the case where and , and the case where and But in the continuation we are going to be interested in the last two quoted cases.

First case

When and , the equations (16), (17),..., (19) become respectively

(20)

(21)

(22)

(23)

Taking into account of equation (21) in equation (20) gives us the constraint and thereafter, the combination of equations (20), (22) and (23) yields

(24)

(25)

(26)

So when we consider equations (24), (25) and (26) in equation (3) we obtain the solution

(27)

While observing the solution equations (11), we note that we should choose the constants of the initial solution (3) as they verify the following criteria, , , and

Second case: and

The gotten below equations essentially come from the real part of equation (4) because its imaginary part produced merely identities. Thus, it follows:

Term in ,

(28)

Term in ,

(29)

Term in

(30)

Term in ,

(31)

From equation (28) we consider the case where and . Hence, the equations (29), (30) and (31) respectively write

(32)

(33)

(34)

Solving the equations (32), (33) and (34) we obtain

(35)

(36)

(37)

with . Substituting equations (35), (36) and (37) in equation (3) we obtain the solution

(38)

According to the solution (38), one realizes that one should have merely chosen to construct the solution of the equation (2) under the shape of equation (3) as the following conditions are verified: and .

4. Conclusions

The Sasa-Satsuma equation under its modified shape as considered in this work is not always easy to integrate. Not knowing initially the shape of solution which is susceptible to verify this equation, we proposed to construct a solution that is the combination of the hyperbolic functions of type solitary wave. The use of the BDK method allowed us to obtain successfully the equations of range of coefficients that allowed assigning some values to these coefficients. These values of determined coefficients also permit to give some particular solutions. We note that beyond the calculations that require a lot of concentrations, the obtained solutions confirm the fact that Sasa-Satsuma equation is integrable. It is sufficient to really choose the shape of solution to construct. Our satisfaction is especially due to the fact that the obtaining of these solutions was possible thanks to the choice of the method that we had used. This method is very adapted to the greatly nonlinear partial differential equation which present the scattering terms.

ACKNOWLEDGEMENTS

We acknowledge support from the ministry of Higher Education of Cameroon through its program of support to research, which enabled us to carry out this work. We also thank Mr Kenfack Christopher Nguimatio and Tsapgou Jean Jeremie for some corrections after the reading of the work.