Solitary and periodic wave solutions of the loaded modi(cid:28)ed Benjamin-Bona-Mahony equation via the functional variable method

. In this article, we establish new travelling wave solutions for the loaded Benjamin-Bona-Mahony and the loaded modi(cid:28)ed Benjamin-Bona-Mahony equation by the functional variable method. The performance of this method is reliable and e(cid:27)ective and gives the exact solitary wave solutions and periodic wave solutions. All solutions of these equations have been examined and three dimensional graphics of the obtained solutions have been drawn by using the Matlab program. We get some traveling wave solutions, which are expressed by the hyperbolic functions and trigonometric functions. This method is e(cid:27)ective to (cid:28)nd exact solutions of many other similar equations.

where u(x, t) is an unknown function, x ∈ R, t ≥ 0, α is any constant.
The BBM equation has been investigated as a regularized version of the KdV equation for shallow water waves [3]. In certain theoretical investigations the equation is studied as a model for long waves and from the standpoint of existence and stability, the equation oers considerable technical advantages over the KdV equation [4]. In addition to shallow water waves, the equation is applicable to the study of drift waves in plasma or the Rossby waves in rotating uids. Under certain conditions, it also provides a model of onedimensional transmitted waves.
The modied Benjamin-Bona-Mahony equation is a special type of the BBM equation. By changing nonlinear term of the form αu n u x (n = 2), the new modied form is obtained as follows: where u(x, t) is an unknown function, x ∈ R, t ≥ 0, α is constant, γ(t) is the given real continuous function.
We construct exact travelling wave solutions of the loaded BBM equation  Nakhushev [24,25], and others. It is known that the loaded dierential equations contain some of the traces of an unknown function. In [26,27,28,29], the term of loaded equation was used for the rst time, the most general denitions of the loaded dierential equation were given and also a detailed classications of the dierential loaded equations as well as their numerous applications were presented. A complete description of solutions of the nonlinear loaded equations and their applications can be found in papers [30,31,32,33,34,35].

Description of the functional variable method
Consider nonlinear evolution equations with independent variables x, y and t is of the form where F is a polynomial in u = u(x, y, t) and its partial derivatives. Zerarka and others in [36,37] have summarized the functional variable method in the following.
Step 1. We use the wave transformation where p and q are constants, k is the speed of the traveling wave.
Next, we can introduce the following transformation for a travelling wave solution of eq. (5) u(x, y, t) = u(ξ), and the chain Using eq. (7) and (8), the nonlinear partial dierential eq. (5) can be transformed into an ordinary dierential equation of the form where P is a polynomial in u(ξ) and its total derivatives, u = du dξ .
Step 2. Then we make a transformation in which the unknown function u is considered as a functional variable in the form then, the solution can be found by the relation here ξ 0 is a constant of integration which is set equal to zero for convenience. Some successive dierentiations of u in terms of F are given as Step 3. The ordinary dierential eq. (9) can be reduced in terms of u, F and its derivatives upon using the expressions of eq. (12) into eq. (5) gives The key idea of this particular form eq. (13) is of special interest because it admits analytical solutions for a large class of nonlinear wave type equations.
After integration, eq. (13) provides the expression of F and this, together with eq. (10), give appropriate solutions to the original problem.

Solutions of the loaded Benjamin-Bona-Mahony equation via the functional variable method
Using the wave variable that will convert eq. (3) to an ordinary dierential equation integrating once eq. (15) with respect to ξ, we have Following eq. (12), it is easy to deduce from eq. (16) an expression for the Integrating eq. (17) and setting the constant of integration to zero yields From eq. (10) and eq. (18) we deduce that After integrating eq. (19), with zero constant of integration, we have following It is obvious that the function u(0, t) can be easily found based on expression eq. (20).
From (20) 2) when k−p−pγ(t)u(0,t) kp 2 < 0, we have the periodic wave solutions Now, by choosing free parameters we will write the simple form of solitary and periodic wave solutions of the loaded BBM equation which can be used for the graphical illustrations.
It is obvious that the function u(0, t) can be easily found based on expression eq. (31).