PACS: 02.30.Jr; 02.30.Ik

1. Introduction

Nonlinear partial differential equations (NPDE) have been analyzed by different type of mathematical approach, among which include the Darboux transformation, the inverse scattering method, the Hirota method, the Backlund transformation, the tanh method, the sine-cosine method, the exp-function method, the variational iteration method, the homogenous balance method and among others ^{1}^{-}^{41}.

In this article, the *D*(*m*, *n*) system ^{1} are considered:

For *m* = 2 and *n* = 1, the system (1) is called “The normal Drinfel’d-Sokolov system”

where *a*, *b*, *c* are unchanged. The system (2) is considered as an example of a system of nonlinear equations possessing Lax pairs of a special form ^{13}. Wang obtained its Hamiltonian, recursion, symplectric and cosymplectric operators and roots of its symmetries and scaling symmetry of the system (2) ^{14}.

For *n* = 1 the system (1) changes to “The generalized Drinfel’d-Sokolov system” as:

Wazwaz obtained some exact traveling wave solutions of the *D*(*m*, *n*) system with compact and noncompact structures by applying the tanh method and the sine-cosine method ^{15}.

We firstly apply the ansatz method ^{16}^{,}^{17}^{,}^{18}^{,}^{19}^{,}^{20} to obtain the exact special solutions to the *D*(*m*, *n*) system. Then, we use the He’s variational approach ^{21} to obtain unknown traveling wave solution to the subsidiaries of the *D*(*m*, *n*) system. We give a comparison between the obtained solutions and those exist in the literature.

2. Ansatz method

We start by considering the solution of the equation

where *a*_{0} ≠ 0 and *b*_{0} ≠ 0 are constants. When *b*_{0} > 0, Eq. (4) admits two solutions as:

where *A* is an arbitrary unchanged. If *b*_{0} < 0,
recognizing that cosh^{2}
*z* + sinh^{2}
*z* = 1 we know that Eq.
(4) has two solutions of the form as:

Secondly, we make a consideration of the solutions of the equation of the form

where *c*_{0} ≠ 0 and *d*_{0} ≠ 0 are constants. If *c*_{0} < 0, Eq. (7) confesses two solutions

If *c*_{0} > 0 in Eq. (7), then the equation offers two solutions of the form

**2.1. Nonlinear Dispersion D(m, n) System**

We assume that the traveling wave solution has the form *u*(*x*, *t*) = *u*(*ξ*) with wave variable *ξ* = *k*(*x* - *λt*), (*k*, *λ* ≠ 0). Then, we get the following ordinary differential equation:

We get

by (11), where *c*_{1} is arbitrary constant. Substituting (12) into (11) we obtain

By integrating Eq. (13), we get

where *c*_{2} is integration constant.

**Case 1.** When *c* = -*bm* =
λ/*c*_{1}, the nonlinear ODE (14) becomes

Therefore, we get the rational solution of Eq. (15),

and

where *c*_{3} and *c*_{4} are arbitrary constants.

**Case 2.** If we take *m* = *n* - 1 and
*c* = λ/*c*_{1} in Eq. (14), the following traveling
wave solutions are obtained

and

where *c*_{5} and *c*_{6} arbitrary constants. In view of (18) and (19), we clearly see that these solutions exist provided that (*bm* + *c*)/(*aλ*(*m* + 1)) > 0

and

where *c*_{7} and *c*_{8} arbitrary constants. In view of (20) and (21), we clearly see that these solutions exist provided that (*bm* + *c*)/(*aλ*(*m* + 1)) < 0.

**Case 3.**
*c* + *bm* ≠ 0, *cc*_{1} -
*λ* ≠ 0 and specially *c*_{2} = 0.

Let

Substituting (22) into (14) leads to the following equation

Letting *ν* = *w*^{2/m}, we have

which changes Eq. (23) to

If we take *n* = 2*m* + 1 in Eq. (25), we get the algebraic traveling wave solution of the form:

and

where

and

**Case 4.** If *m* = *n* - 1, we know that Eq. (25) becomes

where *n* ≠ 1 and *a*, *k*, *n*, λ ≠ 0.

If we take *anλ*(*c* + *bn* - *b*) > 0, then we acquire from Eqs. (5) and (28)

and

**Theorem 1.** The *D*(*m*, *n*) system
has solutions in Eq. (28)
described as follows:

1. When *anλ*(*c* + *bn* - *b*) > 0,

is a solitary wave solution with compact support.

2. When *anλ*(*c* + *bn* - *b*) > 0,

is a compacton solution for Eq. (1) and

3. Equation (25) can be written as following

If *n* = 1in Eq. (33) that yields as

When *a*(*cc*_{1} - λ) < 0,

which is a singular soliton solution for the *D*(*m*, *n*) equation for

4. When *a*(*cc*_{1} - λ) > 0 and *m* < 0,

is a traveling wave solution for the *D*(*m*, *n*) equation for

**Remark 1.** If (*bn* - *b* -
*c*)/(*λan*) < 0, it follows from (6) and
(28) that

and

**Theorem 2.** The *D*(*m*, *n*)
equation with when *m* = *n* equation has the
following solutions:

1. When (*c* + *bn* - *b*) < 0, *anλ* > 0, (*λ* - *cc*_{1})(*n* + 1) < 0 and *n* ≠ 1

is a solitary solution of Eq. (1).

2. When (*c* + *bn* - *b*) < 0, *anλ* > 0, (*λ* - *cc*_{1})(*n* + 1) > 0 and *n* ≠ 1

is a solitary solution of Eq. (1). If we take *n* < 1, then Eq. (40) is a bounded solution.

3. *m* = *n* < 1solutions (39) and (40) turns to solitary wave solutions

and

**Case 5.**
*a*(*cc*_{1} - λ) > 0 and
*m* > 0,

Thus, by using (8) and (34), the periodic solutions of Eq. (1) are obtained as:

and

**Case 6.**
*a*(*cc*_{1} - λ) > 0 and
*m* > 0,

Therefore, by considering (9) and (34), solitary pattern and bell-shaped solitary wave solutions of (1) are obtained as:

and

**Case 7.**
*a*(*cc*_{1} - λ) > 0 and
*m* ≠ 0,

Using the solutions of (43) and (44), gives the following compacton solutions as:

for

and *u* = 0, otherwise.

for

and *u* = 0, otherwise.

**Case 8.**
*a*(*cc*_{1} - λ) > 0;
*m* < 0 and *m* ≠ 1.

Using cosh(*x*) = cos(*ix*) and sinh(*x*) = -sin(*ix*) we have the following solitary pattern solutions of Eq. (1):

and

**Remark 2.** When *m* = *n* < 1, the obtained
solution (42) agrees with the outcomes (2.13a), (2.13b) in ^{36} and (25) in ^{37}. The solution (42) changes to the compacton
solution (2.12a) and the periodic solution (2.12b) in ^{36}^{,}^{37}.

If *m* > 0, the obtained solution (46) is consented with the outcomes (3.18a) and (3.18b) described in ^{36} and (26) in ^{37}. The solution (46) changes to the solitary pattern solution (3.17a) and solitary wave solution (3.17b) in ^{36}.

**Remark 3.** If *a*(*cc*_{1} - λ) < 0, the
obtained solution (35) agrees with the outcomes (3.9a) and (3.9b) in ^{36} and (32) in ^{37}. The solution (35) changes to
the singular solitary wave solution.

**Remark 4.** If we take (*c* + *bn* -
*b*) < 0, *anλ* > 0, (*λ*
- *cc*_{1})(n + 1) < 0 then the obtained solution (39)
similar to the solitary pattern solutions (3.7a) and (3.7b) in ^{36} and (46) in ^{37}.

3. Variational principle

In this Section, He’s variational principle will be applied to the system (1). This technique was first proposed by He^{21} and it is popularly known as He’s semi-inverse variational principle. Some years back, it was applied mainly to extract soliton solutions of nonlinear PDEs and systems by many authors ^{16}^{,}^{17}^{,}^{18}^{,}^{19}^{,}^{20}^{,}^{21}^{,}^{22}^{,}^{23}^{,}^{24}. Biswas and co-workers ^{17}^{,}^{18}^{,}^{19}^{,}^{20} obtained optical solitons and soliton solutions with higher order dispersion by applying the He’s variational principle. Xu and Zhang’s^{25} used a variational principle to construct catalytic reactions in short monoliths by He’s semi-inverse approach. He’s variational method was used to the effective nonlinear oscillators with high nonlinearity by Liu ^{26}. Zheng *et al*. ^{27} established a class of generalized variational principles for the initial-boundary value problem of micromorphic magneto electrodynamics by He’s semi-inverse technique. In order to seek traveling wave solutions of the system (1). We consider

Let *ν*^{n} = *V*

the 1-soliton solution ansatz, given by

is substituted into (51). Here, in (53), the parameters *p* and *q* represent the amplitude and inverse width of the soliton, respectively.

where *p*_{1} = *cc*_{1} - *λ*, *p*_{2} = *ak*^{2} and

From the above equation it is obtained as

where *F* is Gauss’ hypergeometric function defined as

and Re[(2 + *m*)*q*] > 0, Re[(1 + *n*)*q*] > 0, Re[*q*] > 0.

Making *J* stationary with respect to *p* and *q* results in

Solving Eqs. (57) and (58) for *m* = *n* = 2 simultaneously, we get

Therefore, by substituting *p* and *q* in (52) we have the following a new solitary wave solution for the system (1) as:

So, the solitary wave solution (60) will exist for *ap*(*λ* - *cc*_{1}) > 0.

4. Results and Discussions

In this article, we investigated the nonlinear dispersion *D*(*m*, *n*) system and obtained some traveling wave solutions by applying the ansatz technique and the He’s variational principle. Several forms of solutions including topological, non-topological, compacton, solitary pattern, singular soliton, algebraic and periodic wave solutions were acquired. The approaches can be used to a lot of other nonlinear differential equations and coupled systems. Some new obtained exact solutions were previously unknown by other methods. We proved the existence of these solutions for a generalized form of the *D*(*m*, *n*) system under specific conditions. In general, the outcome expose that the ansatz approach and the He’s variational principle are important mathematical techniques for solving nonlinear partial differential equations in terms of correctness and ability to avoid errors.