Exact Solutions and Conservation Laws of a (2+1)-Dimensional Nonlinear KP-BBM Equation

In this paper, we study the two-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation. This equation is the Benjamin-Bona-Mahony equation formulated in the KP sense. We first obtain exact solutions of this equation using the Lie group analysis and the simplest equation method. The solutions obtained are solitary waves. In addition, the conservation laws for the KP-BBM equation are constructed by using the multiplier method.


Introduction
It is well known that the nonlinear evolution equations (NLEEs) are extensively used as models to describe physical phenomena in various fields of sciences, especially in fluid mechanics, solid state physics, plasma physics, plasma waves, and biology. One of the basic physical problems for such models is to obtain their exact solutions. In this respect, during the past few decades, various methods have been developed by researchers to find explicit solutions for the NLEEs. See, for example, Wang et al. [1] and references therein.
The purpose of this paper is to study one such NLEE, namely, the (2+1)-dimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation that is, given by Here, in (1), , , and are real valued constants. The solutions of (1) have been studied in various aspects. See, for example, the recent papers [2][3][4][5]. Wazwaz [2,3] used the sine-cosine method, the tanh method and the extended tanh method for finding solitonary solutions of this equation. Abdou [4] used the extended mapping method with symbolic computation to obtain some periodic solutions, solitary wave solution, and triangular wave solution.
In this paper, Lie group analysis [6][7][8][9] in conjunction with the simplest equation method [10] is employed to obtain some exact solutions of (1). In addition to this conservation laws will be derived for (1) using the multiplier method [11].

Symmetry Analysis
In this section, we first calculate the Lie point symmetries of (1) and later use them to construct exact solutions.

Lie Point Symmetries.
A Lie point symmetry of a partial differential equation (PDE) is an invertible transformation of the dependent and independent variables that leaves the equation unchanged. In general, determining all the symmetries of a partial differential equation is a formidable task. However, Sophus Lie observed that if we restrict ourselves to symmetries that depend continuously on a small parameter and that form a group (continuous one-parameter group of transformations), one can linearize the symmetry conditions and end up with an algorithm for calculating continuous symmetries.
The symmetry group of KP-BBM equation (1) will be generated by the vector field of the form where , = 1, 2, 3 and depend on , , , and . Applying the fourth prolongation pr (4) to (1), we obtain an overdetermined system of linear partial differential equations. Solving resultant system of linear overdetermined partial differential equations one obtains the following four Lie point symmetries:

Exact Solutions.
One of the main reasons for finding symmetries of a differential equation is to use them for finding exact solutions. In this subsection, we will utilize the symmetries calculated in the previous subsection to deduce exact solutions of (1). One way to obtain exact solutions of (1) is by reducing it to ordinary differential equations. This can be achieved with the use of Lie point symmetries admitted by (1). It is well known that the reduction of a partial differential equation with respect to -dimensional (solvable) subalgebra of its Lie symmetry algebra leads to reducing the number of independent variables by .
First of all, we utilize the symmetry = 1 + 2 + 3 and reduce the KP-BBM equation (1) to a PDE in two independent variables. It can be seen that the symmetry yields the following three invariants: Now treating as the new dependent variable and and as new independent variables, the KP-BBM equation (1) transforms to which is a nonlinear PDE in two independent variables. We now use the Lie point symmetries of (5) and transform it to an ordinary differential equation (ODE). Equation (5) has the following two translational symmetries: The combination Γ 1 +Γ 2 , where ̸ = 0 is a constant, of the two symmetries Γ 1 and Γ 2 yields the following two invariants: This gives rise to a group invariant solution = ( ) and consequently using these invariants, (5) is transformed into the fourth-order nonlinear ODE Integrating the above equation twice and taking the constants of integration to be zero, we obtain a second-order ODE Multiplying (9) by , integrating once and taking the constant of integration to be zero, we obtain the first-order ODE One can integrate the above equation by separating the variables. After integrating and reverting back to the original variables, we obtain the following group-invariant solutions of the KP-BBM equation (1): To find another group-invariant solution of the KP-BBM equation (1), we now make use of the symmetry Γ = 1 1 + 2 2 + 3 3 + 4 4 ( , = 1, . . . , 4 are constants). The symmetry Γ yields the following three invariants: Treating as the new dependent variable and and as new independent variables, the KP-BBM equation (1) transforms to a nonlinear fourth-order PDE in two independent variables, namely, Abstract and Applied Analysis 3 Equation (14) has a single Lie point symmetry, namely, and this symmetry yields the two invariants which gives a group-invariant solution = ( ) and consequently (14) is then transformed to which is a second-order linear ODE. Now, solving this equation and reverting back to the original variables, we obtain the following solutions of the KP-BBM equation (1): where 1 and 2 are constants of integration and

Simplest Equation Method
We now use the simplest equation method, which was introduced by Kudryashov [10] and modified by Vitanov [12], to solve the fourth-order ODE (8). The simplest equations that will be used are the Bernoulli and Riccati equations. Their solutions can be written in elementary functions [13]. Let us consider the solution of (8) in the form where ( ) satisfies the Bernoulli and Riccati equations, is a positive integer that can be determined by balancing procedure as in [12], and 0 , . . . , are parameters to be determined.

Solutions of (8) Using the Equation of Bernoulli as the Simplest Equation.
The balancing procedure [12] yields = 2 so the solutions of (20) are of the form Substituting (21) into (8) and making use of the Bernoulli equation [13] and then equating all coefficients of the functions to zero, we obtain an algebraic system of equations in terms of 0 , 1 , and 2 .
Solving the system of algebraic equations, we obtain Therefore, the solution of (1) is given by ) where = − + (1 − ) / , and is a constant of integration.

Solutions of (8) Using Riccati Equation as the Simplest Equation.
The balancing procedure yields = 2, so the solutions of (20) are of the form Substituting (24) into (8) and making use of the Riccati equation [13], we obtain algebraic equations in terms of 0 , 1 , and 2 by equating all coefficients of the functions to zero.
Solving the algebraic equations, one obtains and hence the solutions of (1) are where = − + (1 − ) / , and is a constant of integration.

Conservation Laws
In this section, we construct conservation laws for (1). The multiplier method will be used [11]. See also [14].
The following are known (see, e.g., [9] and the references therein).
The Euler-Lagrange operator, for each , is given by The holds identically. Here, we will consider multipliers of the second order, that is, Λ = Λ ( , , , V, , , V , , , V , ). The right hand side of (31) is a divergence expression. The determining equation for the multiplier Λ is Once the multipliers are obtained, the conserved vectors are calculated via a homotopy formula [11].

Construction of Conservation Laws for KP-BBM Equation.
We now construct conservation laws for the (2+1)dimensional nonlinear KP-BBM equation (1). The zerothorder multiplier for (1) is, Λ( , , , ), that is, given by where 1 is a constant, and 2 ( ) and 3 ( ) are arbitrary functions of . Corresponding to the above multiplier, we have the following conserved vectors of (33):

Concluding Remarks
In this paper, we obtained the solutions of the twodimensional nonlinear Kadomtsov-Petviashivilli-Benjamin-Bona-Mahony (KP-BBM) equation by employing the Lie group analysis and the simplest equation method. The solutions obtained are solitary waves and nontopological solutions. The conservation laws for the underlying equation were also derived by using the multiplier method.