# Q-Symmetry and Conditional Q-Symmetries for Boussinesq Equation

Authors: Hassan A. Zedan; Seham Sh. Tantawy
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We study in this paper theQ-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.


---

## Body

## 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(1)
u
t
t
+
u
x
2
+
u
u
x
x
+
u
x
x
x
=
0
,
where u
(
x
,
t
) 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 (
G
′
/
G
) expansion method [5], optimal and symmetry reductions [6], and projective Riccati equations method [7].The aim of this paper is to calculate and list theQ-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. ConditionalQ-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
(2)
u
t
t
+
u
x
2
+
u
u
x
x
+
u
x
x
x
=
0
,
(3)
η
(
x
,
t
,
u
)
-
ξ
1
(
x
,
t
,
u
)
u
x
-
ξ
2
(
x
,
t
,
u
)
u
t
=
0
,
where (3) is the invariant surfaces corresponding to the Lie symmetry group generator
(4)
Z
=
ξ
1
(
x
,
t
,
u
)
∂
∂
x
+
ξ
2
(
x
,
t
,
u
)
∂
∂
t
+
η
(
x
,
t
,
u
)
∂
∂
u
.
The invariance condition leading to conditional Q-symmetries for (2) is given by
(5)
Z
(
3
)
F
|
{
F
(
j
)
=
0
,
Q
(
k
)
=
0
}
=
0
,
where
(6)
F
1
=
u
t
t
+
u
x
2
+
u
u
x
x
+
u
x
x
x
,
Q
=
η
(
x
,
t
,
u
)
-
ξ
1
(
x
,
t
,
u
)
u
x
-
ξ
2
(
x
,
t
,
u
)
u
t
.
Here Z
(
3
) denotes the second prolongation of Z, namely,
(7)
Z
(
3
)
=
Z
+
γ
1
∂
∂
u
x
+
γ
2
∂
∂
u
t
+
γ
11
∂
∂
u
x
x
+
γ
22
∂
∂
u
t
t
+
γ
111
∂
∂
u
x
x
x
,
where
(8)
γ
1
=
D
x
(
η
1
)
-
u
x
D
x
(
ξ
1
)
-
u
t
D
x
(
ξ
2
)
,
γ
2
=
D
t
(
η
1
)
-
u
x
D
t
(
ξ
1
)
-
u
t
D
t
(
ξ
2
)
,
γ
11
=
D
x
(
γ
1
)
-
u
x
x
D
x
(
ξ
1
)
-
u
x
t
D
x
(
ξ
2
)
,
γ
22
=
D
t
(
γ
2
)
-
u
x
t
D
t
(
ξ
1
)
-
u
t
t
D
t
(
ξ
2
)
,
γ
111
=
D
x
(
γ
11
)
-
u
x
x
x
D
x
(
ξ
1
)
-
u
x
x
t
D
x
(
ξ
2
)
.A generatorZ which satisfies condition (5) is called a conditional Q-symmetry generator, where by the invariant surface (3). The F
(
j
) and Q
(
k
) denote the jth and kth prolongations, respectively. D
x and D
t denote the total derivative with respect to x and with respect to t, respectively.We now derive the general determining equations for the conditionalQ-symmetry generators for (2). We set ξ
1
=
ξ
1
(
x
,
t
,
u
),  ξ
2
=
ξ
2
(
x
,
t
,
u
),  and η
=
η
(
x
,
t
,
u
). The invariance condition (5) leads to the following expression:
(9)
γ
22
+
u
γ
11
+
η
u
x
x
+
2
u
x
γ
1
+
γ
111
=
0
.
This leads to
(10)
u
x
x
η
+
u
t
t
η
u
+
η
u
u
x
x
x
-
2
u
x
t
ξ
t
-
u
x
u
t
t
ξ
u
-
2
u
t
u
x
t
ξ
u
-
3
u
x
x
2
ξ
u
-
4
u
x
u
x
x
x
ξ
u
-
3
u
x
x
x
ξ
x
+
2
u
x
u
x
x
(
u
x
η
u
+
η
x
-
u
x
2
ξ
u
-
u
x
ξ
x
)
+
η
t
t
+
2
u
t
η
t
u
+
u
t
2
η
u
u
+
3
u
x
u
x
x
η
u
u
+
3
u
x
x
η
x
u
-
u
x
ξ
t
t
-
2
u
t
u
x
ξ
t
u
-
u
t
2
u
x
ξ
u
u
-
6
u
x
2
u
x
x
ξ
u
u
-
9
u
x
u
x
x
ξ
x
u
-
3
u
x
x
ξ
x
x
+
u
(
u
x
x
η
u
-
3
u
x
u
x
x
ξ
u
-
2
u
x
x
ξ
x
+
u
x
2
η
u
u
iiiiiiiiiii
+
2
u
x
η
x
u
+
η
x
x
-
u
x
3
ξ
u
u
-
2
u
x
2
ξ
x
u
-
u
x
ξ
x
x
)
+
u
x
3
η
u
u
u
+
3
u
x
2
η
x
u
u
+
3
u
x
η
x
x
u
+
η
x
x
x
-
u
x
4
ξ
u
u
u
-
3
u
x
3
ξ
x
u
u
-
3
u
x
2
ξ
x
x
u
-
u
x
ξ
x
x
x
=
0
.
In particular, from Q
=
0 follows
(11)
ξ
2
(
x
,
t
,
u
)
u
t
=
η
(
x
,
t
,
u
)
-
ξ
1
(
x
,
t
,
u
)
u
x
.The determining equations for the conditionalQ-symmetry generator Z 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 ξ
1
(
x
,
t
,
u
), ξ
2
(
x
,
t
,
u
), and η
(
x
,
t
,
u
), we obtain
(12)
η
(
x
,
t
,
u
)
=
-
2
k
3
u
3
,
ξ
1
(
x
,
t
,
u
)
=
k
1
+
2
k
3
x
3
,
ξ
2
(
x
,
t
,
u
)
=
k
2
+
k
3
t
,
where k
1,  k
2, and k
3 are arbitrary constants.The conditionalQ-symmetry is given by
(13)
Z
=
(
k
1
+
2
k
3
x
3
)
∂
∂
x
+
(
k
2
+
k
3
t
)
∂
∂
t
-
2
k
3
u
3
∂
∂
u
.
The general solution of the associated invariant surface condition,
(14)
(
k
1
+
2
k
3
x
3
)
∂
u
∂
x
+
(
k
2
+
k
3
t
)
∂
u
∂
t
=
-
2
k
3
u
3
,
is
(15)
u
(
x
,
t
)
=
φ
(
z
)
(
3
k
1
+
2
k
3
x
)
,
where φ
(
z
) is arbitrary function of z and
(16)
z
(
x
,
t
)
=
3
k
2
+
2
k
3
x
(
k
2
+
k
3
t
)
2
/
3
.
Substituting (15) into (2), we finally obtain the following nonlinear ordinary differential equation for φ
(
z
) taking the form
(17)
-
216
k
3
φ
(
z
)
+
54
φ
2
(
z
)
+
216
k
3
z
φ
′
(
z
)
+
5
z
4
φ
′
(
z
)
-
72
z
φ
(
z
)
φ
′
(
z
)
+
18
z
2
φ
′
2
(
z
)
-
108
k
3
z
2
φ
′′
(
z
)
+
2
z
5
φ
′′
(
z
)
+
18
z
2
φ
(
z
)
φ
′′
(
z
)
+
36
k
3
z
3
φ
′′′
(
z
)
=
0
,
where  φ
′
(
z
)
=
d
φ
i
/
d
z,  φ
′′
(
z
)
=
d
2
φ
i
/
d
z
2, and φ
′′′
(
z
)
=
d
3
φ
i
/
d
z
3.Solving an ordinary differential equation (17), we have three cases of solutions for φ
(
z
).Case 1.
Consider(18)
φ
(
z
)
=
4
k
3
,
where k
3 is an arbitrary constant.Case 2.
Consider(19)
φ
(
z
)
=
-
1
4
z
3
,
k
3
=
-
3
.Case 3.
Consider(20)
φ
(
z
)
=
4
k
3
-
1
4
z
3
,
k
3
=
-
1
4
.By using (18)–(20) into (15), we have solutions for Boussinesq equation (1) in the following forms.Family.
Consider(21)
u
(
x
,
t
)
=
4
k
3
(
3
k
1
+
2
k
3
x
)
,
where k
1 and k
3 are arbitrary constants.Family.
Consider(22)
u
(
x
,
t
)
=
-
(
1
/
4
)
z
3
(
3
k
1
-
6
x
)
,
where z
=
(
3
k
1
-
6
x
)
/
(
k
2
-
3
t
)
2
/
3 and k
1 is an arbitrary constant.Family.
Consider(23)
u
(
x
,
t
)
=
-
1
+
(
1
/
4
)
z
3
(
3
k
1
-
x
/
2
)
,
where z
=
(
3
k
1
-
(
1
/
2
)
x
)
/
(
k
2
-
(
1
/
4
)
t
)
2
/
3 and k
1 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 r with m
+
1 independent variables (
x
0
,
x
1
,
…
,
x
m
) and one field variable u, that is, an equation of the form
(24)
F
(
x
0
,
x
1
,
…
,
x
m
,
u
,
∂
u
∂
x
0
,
…
,
∂
r
u
∂
x
j
1
⋯
∂
x
j
r
)
=
0
,
where 0
≤
j
1
≤
j
2
≤
⋯
≤
j
r
≤
m, j
=
0
,
…
,
m. A Lie transformation group that leaves (24) invariant is generated by a Lie symmetry generator Z, defined by
(25)
Z
=
∑
j
=
0
m
ξ
j
(
x
0
,
x
1
,
…
,
x
m
,
u
,
v
)
∂
∂
x
j
+
η
(
x
0
,
x
1
,
…
,
x
m
,
u
)
∂
∂
u
.
Z
w is the associated vertical form of (25), defined by
(26)
Z
w
=
(
η
-
∑
j
=
0
m
ξ
j
u
j
)
∂
∂
u
,
where Z
w
|
θ
=
Z
|
θ. Here θ is a differential 1-form, called the contact form, which is defined by
(27)
θ
=
d
u
-
∑
j
=
0
m
u
j
d
x
j
.
Equation (24) is called invariant under the prolonged Lie symmetry generators Z
w if
(28)
L
Z
˘
w
F
=
0
.
L denotes the Lie derivative, and Z
˘
w is found by prolonging the vertical generator Z
w; that is,
(29)
Z
˘
w
=
∑
j
=
0
m
D
j
(
U
1
)
∂
∂
u
j
+
⋯
+
∑
j
1
,
…
,
j
r
=
0
m
D
j
1
,
…
,
j
r
(
U
1
)
∂
∂
u
j
1
,
…
,
j
r
,
where
(30)
U
=
(
η
-
∑
j
=
0
m
ξ
j
u
j
)
,
and D
j 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
(31)
L
Z
˘
w
F
=
0
,
under the condition
(32)
Z
w
|
θ
=
0
.Z
w is called the Q-symmetry generator and Z
˘
w 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
(33)
F
≡
u
t
t
+
u
x
2
+
u
u
x
x
+
u
x
x
x
=
0
,
under the condition
(34)
Z
w
|
θ
=
η
-
ξ
1
u
x
-
ξ
2
u
t
=
0
,
has to be studied. Let us consider the Q-symmetry generator in the form
(35)
Z
=
ξ
1
(
x
,
t
,
u
)
∂
∂
x
+
ξ
2
(
x
,
t
,
u
)
∂
∂
t
+
η
(
x
,
t
,
u
)
∂
∂
u
.
By applying the Lie derivative (31) and condition (32), we get
(36)
D
t
t
(
U
)
+
u
D
x
x
(
U
)
+
η
u
x
x
+
2
u
x
D
x
(
U
)
+
D
x
x
x
(
U
)
=
0
,
where
(37)
D
x
=
∂
∂
x
+
u
x
∂
∂
u
+
u
x
x
∂
∂
u
x
+
u
x
t
∂
∂
u
t
+
⋯
D
t
=
∂
∂
t
+
u
t
∂
∂
u
+
u
t
t
∂
∂
u
t
+
u
x
t
∂
∂
u
x
+
⋯
.The determining equations for theQ-symmetry generator Z 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 ξ
1, ξ
2, and η, we obtain
(38)
η
(
x
,
t
,
u
)
=
-
2
k
3
u
3
,
ξ
1
(
x
,
t
,
u
)
=
k
1
+
2
k
3
x
3
,
ξ
2
(
x
,
t
,
u
)
=
k
2
+
k
3
t
.All of the similarity variables associated with the Lie symmetries (38) can be derived by solving the following characteristic equation:
(39)
d
x
ξ
1
=
d
t
ξ
2
=
d
u
η
.
Consequently
(40)
d
x
(
k
1
+
2
k
3
x
/
3
)
=
d
t
(
k
2
+
k
3
t
)
=
d
u
-
2
k
3
u
/
3
.
We obtain the following similarity variable:
(41)
z
(
x
,
t
)
=
3
k
1
+
2
k
3
x
(
k
2
+
k
3
t
)
2
/
3
,
and the similarity solutions take the form
(42)
u
(
x
,
t
)
=
F
1
(
z
)
(
k
2
+
k
3
t
)
2
/
3
,
where F
1
(
z
) is arbitrary functions of z. Substituting from (42) into (1), we finally obtain nonlinear ordinary differential equation for F
1
(
z
) taking the form
(43)
36
k
3
F
1
′′′
(
z
)
+
2
z
2
F
1
′′
(
z
)
+
18
F
1
(
z
)
F
1
′′
(
z
)
+
5
F
1
(
z
)
+
18
F
′
2
(
z
)
+
9
z
F
1
′
(
z
)
=
0
,
where  F
i
′
=
d
φ
i
/
d
z,  F
i
′′
=
d
2
φ
i
/
d
z
2 and F
i
′′′
=
d
3
φ
i
/
d
z
3;  (
i
=
1
).Solving a system of an ordinary differential equation (43), we have two cases of solutions for F
1
(
z
).Case 1.
Consider(44)
F
1
(
z
)
=
4
k
3
z
,
where k
3 is an arbitrary constant.Case 2.
Consider(45)
F
1
(
z
)
=
-
1
4
z
2
.Substitut from (44)-(45) into (42) to obtain the solutions for the Boussinesq equation (1) in the following forms.Family.
Consider(46)
u
(
x
,
t
)
=
4
k
3
z
(
k
2
+
k
3
t
)
2
/
3
,
where k
2 and k
3 are an arbitrary constants.Family.
Consider(47)
u
(
x
,
t
)
=
-
z
2
4
(
k
2
+
k
3
t
)
2
/
3
,
where z
=
(
3
k
1
+
2
k
3
x
)
/
(
k
2
+
k
3
t
)
2
/
3 and k
2 and k
3 are arbitrary constants.



---

*Source: 175982-2014-01-16.xml*