Exact Solutions of Generalized Boussinesq-burgers Equations and (2+1)-dimensional Davey-stewartson Equations

under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We study two coupled systems of nonlinear partial differential equations, namely, generalized Boussinesq-Burgers equations and 21-dimensional Davey-Stewartson equations. The Lie symmetry method is utilized to obtain exact solutions of the generalized Boussinesq-Burgers equations. The travelling wave hypothesis approach is used to find exact solutions of the 21-dimensional Davey-Stewartson equations.


Introduction
Most nonlinear physical phenomena that appear in many areas of scientific fields such as plasma physics, solid state physics, fluid dynamics, optical fibers, mathematical biology, and chemical kinetics can be modelled by nonlinear partial differential equations NLPDEs .The investigation of exact travelling wave solutions of these NLPDEs is important for the understanding of most nonlinear physical phenomena and possible applications.To address this issue, various methods for finding travelling wave solutions to NLPDEs have been proposed.Some of the most important methods include homogeneous balance method 1 , the ansatz method 2, 3 , variable separation approach 4 , inverse scattering transform method 5 , Bäcklund transformation 6 , Darboux transformation 7 , Hirota's bilinear method 8 , the G /G -expansion method 9 , the reduction mKdV equation method 10 , the tri-function method 11, 12 , the projective Riccati equation method 13 , the sine-cosine method 14, 15 , the Jacobi elliptic function expansion method 16, 17 , the F-expansion method 18 , the exp-function expansion method 19 , and Lie symmetry method 20-24 .

The Generalized Boussinesq-Burgers Equations
We first consider the generalized Boussinesq-Burgers equations 25 given by u t auu x bv x 0, 2.1 where a, b, c, and d are real nonzero constants.These equations arise in the study of fluid flow and describe the propagation of shallow water waves, where x and t represent the normalized space and time, respectively.Here u x, t represents the horizontal velocity and at the leading order it is the depth averaged horizontal field, while v x, t denotes the height of the water surface above the horizontal level at the bottom 25 .
The Boussinesq-Burgers equations given by 2.1 -2.2 will be solved by the Lie symmetry approach.The symmetry group of the generalized Boussinesq-Burgers equations 2.1 -2.2 will be generated by the vector field given by Applying the third prolongation pr 3 X 21 to 2.1 -2.2 and solving the resultant overdetermined system of linear partial differential equations, one obtains the following three Lie point symmetries:

2.4
Journal of Applied Mathematics 3 We now consider the symmetry X 1 νX 2 , where ν is an arbitrary constant.This symmetry gives rise to the group-invariant solution as where z x − νt is an invariant of the symmetry X 1 νX 2 .Substitution of 2.5 into 2.1 -2.2 results in the system of ordinary differential equations as Integration of 2.6 with respect to z yields where the constant of integration is chosen to be zero, since we are looking for a soliton solution.Solving for G z , we obtain Substituting this value of G z into 2.7 gives the third-order nonlinear ordinary differential equation as which can be integrated twice to obtain Here again the constants of integration are taken to be zero for the same reason as given above.Integrating 2.11 and reverting back to our original variables, we obtain where α, β, γ, and δ are constants given by α a 2 − 5ac 4c 2 1,

2.14
A profile of the solution 2.12 -2.13 is given in Figure 1.

The (2+1)-Dimensional Davey-Stewartson Equations
The 2 1 -dimensional Davey-Stewartson equations were first introduced by Davey and Stewartson in 1974 26 .This system of equations is completely integrable and is often used to describe the long-time evolution of a twodimensional wave packet 27-29 .We first transform the 2 1 -dimensional Davey-Stewartson equations 3.1 -3.2 to a system of nonlinear ordinary differential equations in order to derive its exact solutions.
We make the following transformation: where p, q, r, k, c, and d are real constants.Using this transformation, the 2 1 -dimensional Davey-Stewartson equations 3.1 -3.2 transform to Integration of 3.5 twice and taking the constants of integration to be zero, one obtains Now substituting 3.6 into 3.4 , we get which can be written in the following form: 3.9 Solving 3.8 , with the aid of Mathematica, we obtain the following solution: where sn P 1 | ω is a Jacobian elliptic function of the sine-amplitude 30 , and is the modulus of the elliptic function with 0 < ω < 1.Here c 1 and c 2 are constants of integration.Reverting back to our original variables, we can now write the solution of our It should be noted that the solution 3.12 is valid for 0 < ω < 1 and as ω approaches zero, the solution becomes the normal sine function, sin z, and as ω approaches 1, the solution tends to the tanh function, tanh z.
The profile of the solution 3.12 is given in Figure 2.

Conclusion
In this paper, we studied two systems of nonlinear partial differential equations.Firstly, we obtained exact solutions of the generalized Boussinesq-Burgers equations given by 2.1 -2.2 using the Lie symmetry method.The solutions obtained were travelling wave solutions.Secondly, we found exact solutions of the 2 1 -dimensional Davey-Stewartson equations 3.1 -3.2 using the travelling wave hypothesis.The Davey-Stewartson system was first transformed to a system of nonlinear ordinary differential equations, which was then solved to obtain the exact solutions.

Figure 2 :
Figure 2: Profile of the solution 3.12 .