#### Abstract

We study in this paper the *Q*-symmetry and conditional *Q*-symmetries of Boussinesq equation. The solutions which we obtain, in this case, are in the form of convergent power series with easily computable components.

#### 1. Introduction

The Boussinesq equation, which belongs to the KdV family of equations and describes motions of long waves in shallow water under gravity propagating in both directions, is given by where is a sufficiently often differentiable function.

A great deal of research work has been invested in recent years for the study of the Boussinesq equation. Many effective methods for obtaining exact solutions of Boussinesq equation have been proposed, such as variational iteration method [1], Travelling wave solutions [2], potential method [3], scattering method [4], the expansion method [5], optimal and symmetry reductions [6], and projective Riccati equations method [7].

The aim of this paper is to calculate and list the *Q*-symmetry and conditional *Q*-symmetries of Boussinesq equation. We can say today that many mathematicians, mechanicians and physicists, such as Euler, D’Alembert, Poincare, Volterra, Whittaker, Bateman, implicitly used conditional symmetries for the construction of exact solutions of the linear wave equation.

Nontrivial conditional symmetries of a PDE (partial differential equation) allow us to obtain in explicit form such solutions which cannot be found by using the symmetries of the whole set of solutions of the given PDE [8]. Moreover, conditional symmetries make the class of PDEs reduce to a system of ODEs (ordinary differential equations). As a rule, the reduced equations one obtains from conditional symmetries and from *Q*-symmetry are significantly simpler than those found by reduction using symmetries of the full set of solutions. This allows us to construct exact solutions of the reduced equations.

#### 2. Conditional *Q*-Symmetries

The classical symmetry properties can be extended if one studies (1) together with the invariant surface of the symmetry generator as an overdetermined system of partial differential equations [9]. That is, one studies the Lie symmetry properties of the system
where (3) is the invariant surfaces corresponding to the Lie symmetry group generator
The invariance condition leading to conditional *Q*-symmetries for (2) is given by
where
Here denotes the second prolongation of , namely,
where

A generator which satisfies condition (5) is called a conditional *Q*-symmetry generator, where by the invariant surface (3). The and denote the th and th prolongations, respectively. and denote the total derivative with respect to and with respect to , respectively.

We now derive the general determining equations for the conditional *Q*-symmetry generators for (2). We set , , and . The invariance condition (5) leads to the following expression:
This leads to
In particular, from follows

The determining equations for the conditional *Q*-symmetry generator are now obtained by equating to zero the coefficients of the independent coordinates. By solving this system of linear partial differential equations for the infinitesimal , , and , we obtain
where , , and are arbitrary constants.

The conditional *Q*-symmetry is given by
The general solution of the associated invariant surface condition,
is
where is arbitrary function of and
Substituting (15) into (2), we finally obtain the following nonlinear ordinary differential equation for taking the form
where , , and .

Solving an ordinary differential equation (17), we have three cases of solutions for .

*Case 1. *Consider
where is an arbitrary constant.

*Case 2. *Consider

*Case 3. *Consider

By using (18)–(20) into (15), we have solutions for Boussinesq equation (1) in the following forms.

*Family. *Consider
where and are arbitrary constants.

*Family. *Consider
where and is an arbitrary constant.

*Family. *Consider
where and is an arbitrary constant.

#### 3. *Q*-Symmetry Generators

Before we consider conditional symmetries of (1), let us briefly describe the classical Lie approach and introduce our notation [10]. We are concerned with a partial differential equation of order with independent variables and one field variable , that is, an equation of the form where , . A Lie transformation group that leaves (24) invariant is generated by a Lie symmetry generator , defined by is the associated vertical form of (25), defined by where . Here is a differential 1-form, called the contact form, which is defined by Equation (24) is called invariant under the prolonged Lie symmetry generators if denotes the Lie derivative, and is found by prolonging the vertical generator ; that is, where and is the total derivative operator. We give the definition for conditional invariance of (24) as follows.

*Definition 1. *Equation (24) is called *Q*-conditionally invariant if
under the condition

is called the *Q*-symmetry generator and is called the prolonged vertical *Q*-symmetry generator. Let us now study (1) by the use of the above definition. From the definition it follows that the Lie derivative (31), for equations
under the condition
has to be studied. Let us consider the *Q*-symmetry generator in the form
By applying the Lie derivative (31) and condition (32), we get
where

The determining equations for the *Q*-symmetry generator are now obtained by equating to zero the coefficients of the independent coordinates. By solving this system of linear partial differential equations for the infinitesimal , , and , we obtain

All of the similarity variables associated with the Lie symmetries (38) can be derived by solving the following characteristic equation: Consequently We obtain the following similarity variable: and the similarity solutions take the form where is arbitrary functions of . Substituting from (42) into (1), we finally obtain nonlinear ordinary differential equation for taking the form where , and ; .

Solving a system of an ordinary differential equation (43), we have two cases of solutions for .

*Case 1. *Consider
where is an arbitrary constant.

*Case 2. *Consider

Substitut from (44)-(45) into (42) to obtain the solutions for the Boussinesq equation (1) in the following forms.

*Family. *Consider
where and are an arbitrary constants.

*Family. *Consider
where and and are arbitrary constants.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 109/130/1433. The authors, therefore, acknowledge with thanks DSR technical and financial support.