Symmetry and Group Theory and Its Application to Few-Body PhysicsView this Special Issue
Symmetries, Traveling Wave Solutions, and Conservation Laws of a -Dimensional Boussinesq Equation
We analyze the -dimensional Boussinesq equation, which has applications in fluid mechanics. We find exact solutions of the -dimensional Boussinesq equation by utilizing the Lie symmetry method along with the simplest equation method. The solutions obtained are traveling wave solutions. Moreover, we construct the conservation laws of the -dimensional Boussinesq equation using the new conservation theorem, which is due to Ibragimov.
It is well known that the -dimensional Boussinesq equation , describes the propagation of long waves on the surface of water with a small amplitude and plays a vital part in fluid mechanics . It is completely integrable and admits multiple soliton solutions.
The -dimensional Boussinesq equation is given by
In , the author obtained one-periodic wave solution, two-periodic wave solution, and soliton solution for (3) by means of Hirota’s bilinear method and the Riemann theta function. Wazwaz  employed a combination of Hirota’s method and Hereman’s method to formally study (3) and derived two soliton solutions of (3). Some other work concerning symmetries and exact solutions of some Boussinesq equations can be seen in [9–12].
In the last few decades several methods have appeared in the literature, which can be used to find exact solutions of nonlinear evolution equations (NLEEs). Some of these methods are the inverse scattering transform method , the Darboux transformation method , the sine-cosine method , Hirota’s bilinear method , Jacobi elliptic function expansion method , Lie group analysis [18–20], and the exp-function expansion method .
In this paper we use Lie group method along with the simplest equation method [22, 23] to construct some exact solutions of (3). Furthermore, we employ the new conservation theorem due to Ibragimov  to derive conservation laws for (3).
Lie group method, which was developed by Sophus Lie (1842–1899) in the nineteenth century, is a systematic method that can be used to find solutions of nonlinear partial differential equations (PDEs). It is based upon the study of the invariance under one-parameter Lie group of point transformations [18, 19].
Conservation laws play a very important role in the solution process and the reduction of PDEs [25–27]. They have been used in investigating the existence, uniqueness, and stability of solutions of certain nonlinear PDEs [28–30] and also in the development of numerical methods [31, 32].
2. Traveling Wave Solutions of (3)
We obtain exact solutions of (3) using Lie group method along with the simplest equation method.
2.1. Non-Topological Soliton Solutions Using Lie Point Symmetries
The vector field where , and depend on , , , , and , is a generator of Lie point symmetries of the -dimensional Boussinesq equation (3) if and only if Here is the fourth prolongation of the vector field . The invariance condition (5) yields the determining equations, which are a system of linear partial differential equations. Solving this system we obtain the following eight Lie point symmetries: To obtain the nontopological soliton solution of (3), we use the combination of the four translation symmetries, namely, , where is a constant. Solving the associated Lagrange system for , we obtain the four invariants Now considering as the new dependent variable and , , and as new independent variables, (3) transforms to a nonlinear PDE in three independent variables, namely, The Lie point symmetries of (8) are The use of the combination , ( is a constant) of the three translation symmetries, gives us the three invariants Treating as the new dependent variable and and as new independent variables, (8) transforms to which is a nonlinear PDE in two independent variables. Equation (11) has three Lie point symmetries, namely, and the symmetry ( is a constant) provides the two invariants which gives rise to a group invariant solution . Using these invariants, the PDE (11) transforms to which is a fourth-order nonlinear ODE. This ODE can be integrated easily. Integrating it four times while choosing the constants of integration to be zero (because we are looking for soliton solutions) and then reverting back to our original variables , we obtain the following group-invariant (nontopological soliton) solutions of the Boussinesq equation (3): where is a constant of integration and
2.2. Exact Solutions of (3) Using Simplest Equation Method
We now use the simplest equation method to obtain more solutions of the nonlinear ODE (14), which will then give us more exact solutions for our Boussinesq equation (3). Bernoulli and Riccati equations will be used as the simplest equations [22, 23].
2.2.1. Solutions of (3) Using the Bernoulli Equation as the Simplest Equation
In this case the balancing procedure yields so the solutions of (14) have the form Inserting (17) into (14) and using the Bernoulli equation  and then equating the coefficients of powers of to zero gives us the following algebraic system of six equations: These equations can be solved with the aid of Mathematica and one possible solution for , , and is Consequently, returning back to the original variables, a solution of (3) is  where and is an arbitrary constant of integration.
2.2.2. Solutions of (3) Using the Riccati Equation as the Simplest Equation
Here the balancing procedure gives so the solutions of (14) are of the form Substituting (21) into (14) and using the Riccati equation , as before, we obtain the following algebraic system of equations in terms of , , and : Solving the above equations yields and, consequently, the solutions of (3) are where and is an arbitrary constant of integration.
3. Conservation Laws for (3)
We utilize the new conservation theorem due to Ibragimov  to obtain conservation laws for the -dimensional Boussinesq equation (3) written as For details of notations, definitions, and theorems the reader is referred to .
In Section 2.1 we derived the following eight Lie point symmetries of equation (25): Corresponding to each of these eight Lie point symmetries we shall construct eight conserved vectors. By definition  the adjoint equation of (25) is given by which gives Here is a new dependent variable. Clearly, (25) is not self-adjoint. The Lagrangian for the system of (25) and (28) is given by (i) Consider first the translation symmetry . In this case the operator  is the same as and the Lie characteristic function . Thus the components  , of the conserved vector are given by (ii) The second translation symmetry gives . Hence the symmetry generator gives rise to the following components of the conserved vector: (iii) For the third symmetry , we have and the corresponding components of the conserved vector are (iv) The fourth symmetry gives and the corresponding components of the conserved vector are (v) For the symmetry , we have and the corresponding components of the conserved vector, as before, are given by (vi) Likewise, the symmetry gives and the corresponding components of the conserved vector are given by (vii) As before, the symmetry yields and the corresponding components of the conserved vector are given by (viii) Finally, for the symmetry the value of is not the same as and in fact is given by The Lie characteristic function and, consequently, the conserved vector has components given by
Remark. Each conserved vector obtained above contains the arbitrary solution of the adjoint equation (28) and hence gives an infinite number of conservation laws.
Exact solutions of the -dimensional Boussinesq equation (3) were obtained with the aid of Lie point symmetries of (3) as well as the simplest equation method. The solutions obtained were solitary waves and nontopological soliton. Furthermore, the conservation laws for the -dimensional Boussinesq equation were also constructed by utilizing the new conservation theorem due to Ibragimov .
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Chaudry Masood Khalique would like to thank the North-West University, Mafikeng Campus, for its continued support.
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