1. Introduction

Nonlinear partial differential equations are very important in a variety of scientific fields, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, capillary-gravity waves and chemical physics. The nonlinear wave phenomena observed in the above mentioned scientific fields, are often modeled by the bell-shaped sech solutions and the kink-shaped tanh solutions. The availability of these exact solutions, for those nonlinear equations can greatly facilitate the verification of numerical solvers on the stability analysis of the solution. Nonlinear wave phenomena of dispersion, dissipation, diffusion, reaction and convection are very important in nonlinear wave equations. In recent years, new exact solutions may help to find new phenomena. Also, the explicit formulas may provide physical information and help us to understand the mechanism of related physical models.

In this paper, by means of the He’s semi-inverse method, we will obtain some Solitary solutions of the (2 + 1)-dimensional Boussinesq and breaking soliton equations.

2. Description of He’s Semi-inverse Method

We suppose that the given nonlinear partial differential equation for to be in the form

(1)

Step 1. Seek solitary wave solutions of Eq. (3) by taking and transform Eq. (3) to the ordinary differential equation

(2)

where prime denotes the derivative with respect to .

Step 2. If possible, integrate Eq. (3) term by term one or more times. This yields constant(s) of integration. For simplicity, the integration constant(s) can be set to zero.

Step 3. According to He’s semi-inverse method, we construct the following trial-functional

(3)

where is an unknown function of and its derivatives.

There exist alternative approaches to the construction of the trial-functionals, see Refs ^{[1, 2]}.

Step 4. By the Ritz method, we can obtain different forms of solitary wave solutions, in the form

(4)

where and are constants to be further determined.

Substituting Eq. (5) into Eq. (6) and making stationary with respect to and results in

(5)

(6)

Solving simultaneously Eqs. (5) and (6) we obtain and . Hence, the solitary wave solution (4) is well determined.

3. He’s Semi-inverse Method for (2 + 1)- Dimensional Boussinesq Equation

Consider the (2 + 1)-dimensional Boussinesq equation

(7)

Using the wave variable reduces it to an ODE

(8)

Integrating twice and setting the constants of integration to zero, we obtain

(9)

According to Ref. ^[1], By He’s semi-inverse method ^[2], we can arrive at the following variational formulation:

(10)

We assume the soliton solution in the following form

(11)

where , is an unknown constant to be further determined.

By Substituting Eq. (11) into Eq. (10) we obtain

(12)

For making stationary with respect to and results in

(13-1)

(13-2)

or simplifying

(14-1)

(14-2)

From Eqs. (14-1) and (14-2), we can easily obtain the following relations:

(15)

So the solitary wave solution can be approximated as

(16)

In this solution is an arbitrary complex parameter.( For , see Figure 1)

Figure 1. Peak solition solution of Eq. (7)

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4. He’s Semi-inverse Method for (2 + 1)-Dimensional Breaking Soliton Equation

We now consider the (2 + 1)-dimensional breaking soliton equations

(17)

Using the wave variable reduces it to an ODE

(18)

Integrating the second equation in the system and neglecting constants of integration we find

(19)

Substituting (32) into the first equation of the system and integrating we find

(20)

According to Ref. ^[1], By He’s semi-inverse method ^[2], we can arrive at the following variational formulation:

(21)

We assume the soliton solution in the following form

(22)

where , is an unknown constant to be further determined.

Figure 2. Peak solition solutions of Eqs. (17)

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By Substituting Eq. (22) into Eq. (21) we obtain

(23)

For making stationary with respect to and results in

(24-1)

(24-2)

or simplifying

(25-1)

(25-2)

From Eqs. (25-1) and (25-2), we can easily obtain the following relations:

(26)

So the solitary wave solution can be approximated as

(27)

And

(28)

In this solutions is an arbitrary complex parameter.( For , see Figure 2)

5. Conclusions

In this paper, by using the He’s semi-inverse method; we obtained some solitary solutions of (2+1)-dimensional Boussinesq and breaking soliton equations. He’s semi-inverse method is a very dominant instrument to find the solitary solutions for various nonlinear equations.

References

[1]	He, J. H., “Some asymptotic methods for strongly nonlinear equations”, Internat. J. Modern Phys. B, Vol. 20, 2006, 1141-1199.
	In article		CrossRef
[2]	He, J. H., “Variational principles for some nonlinear partial differential equations with variable coefficients”, Chaos, Solitons and Fractals, Vol. 19, 2004, 847-851.
	In article		CrossRef
[3]	Najafi, M., Arbabi, S., Najafi, M., “New application of sine-cosine method for the generalized (2+1)-dimensional nonlinear evolution equations”, International Journal of Advanced Mathematical Sciences, Vol. 1, No. 2, 2013, 45-49.
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