Q-Symmetry and Conditional Q-Symmetries for Boussinesq Equation

and Applied Analysis 3


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 Qsymmetry 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.

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 2

Abstract and Applied Analysis
The invariance condition leading to conditional Q-symmetries for (2) is given by Here (3) denotes the second prolongation of , namely, 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 1 = 1 ( , , ), 2 = 2 ( , , ), and = ( , , ). The invariance condition (5) leads to the following expression: This leads to In particular, from = 0 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 1 ( , , ), 2 ( , , ), and ( , , ), we obtain where 1 , 2 , and 3 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 ( , ) = 3 2 + 2 3 Substituting (15) into (2), we finally obtain the following nonlinear ordinary differential equation where ( ) = / , ( ) = 2 / 2 , and ( ) = 3 / 3 . Solving an ordinary differential equation (17), we have three cases of solutions for ( ).

Case 1. Consider
where 3 is an arbitrary constant.
where 1 and 3 are arbitrary constants.

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 + 1 independent variables ( 0 , 1 , . . . , ) and one field variable , that is, an equation of the form where 0 ≤ 1 ≤ 2 ≤ ⋅ ⋅ ⋅ ≤ ≤ , = 0, . . . , . 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 if̆= 0.
denotes the Lie derivative, and̆is found by prolonging the vertical generator ; that is, and is the total derivative operator. We give the definition for conditional invariance of (24) as follows.
under the condition is called the Q-symmetry generator and̆is 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 4 Abstract and Applied Analysis 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 1 , 2 , 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 1 ( ) is arbitrary functions of . Substituting from (42) into (1), we finally obtain nonlinear ordinary differential equation for 1 ( ) taking the form 36 3 1 ( ) + 2 2 1 ( ) + 18 1 ( ) 1 ( ) + 5 1 ( ) + 18 2 ( ) + 9 1 ( ) = 0, where = / , = 2 / 2 and = 3 / 3 ; ( = 1).
Solving a system of an ordinary differential equation (43), we have two cases of solutions for 1 ( ).