AbstractBy variational iteration method, we obtain new solitary solutions for non-linear dispersive equations. Particularly, shock-peakon solutions in K(2,2) equation and shock-compacton solutions in K(3,3) equation are found by this simple method. These two types of solutions are new solitary wave solutions which have the shapes of shock solutions and compacton solutions (or peakon solutions)

Tian, Lixin

Yin, Jiuli

Elsevier - Publisher Connector 

Journal of Computational and Applied Mathematics 207 (2007) 46–52
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Shock-peakon and shock-compacton solutions for K(p, q) equation
by variational iteration method
Lixin Tian∗, Jiuli Yin
Nonlinear Scientiﬁc Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212003, PR China
Received 10 April 2006; received in revised form 6 May 2006
Abstract
By variational iteration method, we obtain new solitary solutions for non-linear dispersive equations. Particularly, shock-peakon
solutions in K(2, 2) equation and shock-compacton solutions in K(3, 3) equation are found by this simple method. These two types
of solutions are new solitary wave solutions which have the shapes of shock solutions and compacton solutions (or peakon solutions).
© 2006 Elsevier B.V. All rights reserved.
MSC: 35Q51; 35Q53
Keywords: K(p, q) equation; Variational iteration method; Shock-peakon; Shock-compacton
1. Introduction
In order to study the role of non-linear dispersion in the formation of patterns in liquid drops, Rosenau and Hyman
[9] studied the genuinely non-linear dispersive equation K(p, q) given by
ut + a(up)x + (uq)3x = 0. (1)
Eq. (1) has been widely studied by many authors [11–18,20,21]. Recently, two special solitary wave solutions were dis-
covered [1,9]. Tian andYin [11–13] obtained compacton solutions for generalized Camassa–Holm equation,K(m, n, 1)
equation and fully non-linear sine-Gordon. Peakon solutions were found in the Camassa–Holm equation [1] which are
called peakons because they have a discontinuous ﬁrst-order derivative at the wave peak. Camassa–Holm equation was
studied by different methods [10,2,8].
Peakon and compacton have the characters of soliton, and both are non-local. They are continuous waves and can be
expressed by  function. When waves appear discontinuously, there exists a shock wave (see [19]). It is a kind of wave
which exists in many mathematical physics equations. In this paper, we will study the K(p, q) equation and obtain its
new shock solitary waves, having the characters of peakon and compacton. This is a new type of solitary wave. At the
same time, we ﬁnd that it is non-local as well. Hence, this solution is of very high signiﬁcance.
The variational iteration method (see [3–7]) has been shown to solve effectively, easily, and accurately a large class
of non-linear problems with approximations converging rapidly to accurate solutions. Particularly, He and Wu [7]
obtained solitary solutions and compacton-like solutions for K(3, 1) equation by variational iteration method.
∗ Corresponding author. Tel./fax: +86 5118791467.
E-mail address: tianlx@ujs.edu.cn (L. Tian).
0377-0427/$ - see front matter © 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.cam.2006.07.026
L. Tian, J. Yin / Journal of Computational and Applied Mathematics 207 (2007) 46–52 47
In the next section, we obtain new solitary solutions and many exact solutions for non-linear dispersive equations
by the variational iteration method. Particularly, shock-peakon solutions in K(2, 2) equation and shock-compacton
solutions in K(3, 3) equation are found by this simple method.
2. Exact solutions by variational iteration method
According to the variational iteration method [3–7], we construct a correction functional for Eq. (1), which reads
un+1(x, t) = un(x, t) +
∫ t
0
{(un)t + a(u˜pn )x + (u˜qn)3x} d = 0, (2)
where  is a general Lagrange multiplier, and u˜n denotes restricted variation [6], i.e., u˜=n 0.
With the help of the above correction functional stationary, we obtain the following stationary conditions
′() = 0, (3)
1 + ()|t= = 0. (4)
The Lagrangian multiplier, therefore, can be identiﬁed as
 = −1. (5)
Substituting Eq. (5) into the correction functional (2) results in the following iteration formula
un+1(x, t) = un(x, t) −
∫ t
0
{(un)t + a(upn )x + (uqn )3x} d. (6)
As illustrating examples, we only consider K(2, 2) and K(3, 3) in the following sections.
2.1. Shock-peakon solution in K(2, 2) equation
Now we consider K(2, 2) equation:
ut + a(u2)x + (u2)3x = 0 (7)
where  plays an important role in determining the shape of solutions. In view of the iteration formula, Eq. (6), we have
un+1(x, t) = un(x, t) −
∫ t
0
{(un)t + a(u2n)x + (u2n )3x} d. (8)
To search for its compacton solution, we can assume an initial condition in the form
u(x, 0) = − 4c
3a
cos2
√
a
4
(x + x0), (9)
where x0 and c are constants. With the initial approximation given by (9) and using the above iteration formula (8), we
can obtain the rest of the other components
u0 = − 4c3a cos
2
√
a
4
(x + x0),
u1(x, t) = − 4c3a cos
2
√
a
4
(x + x0) + c
2
3
√
a
t sin
√
a
2
(x + x0),
u2(x, t) = − 4c3a cos
2
√
a
4
(x + x0) + c
2
3
√
a
t sin
√
a
2
(x + x0) − c
3
12
t2 cos
√
a
2
(x + x0)
48 L. Tian, J. Yin / Journal of Computational and Applied Mathematics 207 (2007) 46–52
1
0.5
0
−40
−4
−2
−20 −10 10 20
−20
0
20
40
0
2
4
0.2
0.4
0.6
0.8
1
1.2
(a) (b)
Fig. 1. Compacton solution.
2
1.75
1.25
1
−5
0
5
10
15 −10
−5
−4 −2 2 4
0
5
1.5
1.8
1.6
1.4
1.2
(a) (b)
Fig. 2. Flowing peakon solution.
and so on, the rest of components of the iteration formula of (8) can be deduced byMathematical package. The solutions
u(x, t) are readily found in a closed form
u() =
{
− 4c
3a
cos2
√
a
4
, || 
2
,
0 otherwise,
(10)
where = x + ct + x0. The obtained compacton solution, Eq. (10), has the same expression with that in Ref. [1]. The
solution is shown in Fig. 1 with a = 1, c = −1, x0 = 0.
From (10), we can ﬁnd compacton solutions arise as a > 0. Therefore, we pay more attention to what happens to the
solution when a < 0. Assuming another initial condition as u(x, 0) = Ae±(
√−a/2)(x+x0) + c0, we can then obtain the
solutions u(x, t) in a closed form as
u(x, t) = Ae−(
√−a/2)|(x−(3/2)ac0t+x0)| + c0, (11)
where A, x0 and c0 are arbitrary constants, which are ﬂowing peakon solutions as shown in Fig. 2 with A = 1,
a = −1, c0 = 1, x0 = 0.
Note that A in (11) is an arbitrary constant, hence we can obtain a new solitary solution called shock-peakon solution
which can be written in the form
u() = A sign()e−(
√−a/2)|| + c0, (12)
L. Tian, J. Yin / Journal of Computational and Applied Mathematics 207 (2007) 46–52 49
2
1.5
1
0.5
0
−5
0
5
−2
−1
0
1
2
2
1.5
1
0.5
−10 −5 5 10
(a) (b)
Fig. 3. Shock-peakon solution.
where  = x − (3/2)ac0t + x0 and sign() = /||. The shock-peakon solution is illustrated in Fig. 3 with A = 1,
a = −1, c0 = 1, x0 = 0. This is a new type of solitary waves and is a discontinuous wave. Hence it is a shock wave. At
the same time, it is a peakon as well. In fact, from the graphs and computation we can ﬁnd that it has a discontinuous
ﬁrst-order derivative at  = 0. But, this solitary wave is non-local. It can be expressed by  function. Note that
the fact
e−(
√−a/2)|| = e−(
√−a/2)sign(), d
d
sign() = ().
Hence
d
d
u() = A()e(−
√−a/2)sign() −
√−a
2
A[sign()() + 1]e−(
√−a/2)sign()
,
where () is  function.
Assuming different initial conditions, we may obtain different exact solutions in a closed form as follows:
u(x, 0) − 4c
3a
sin2
√
a
4
(x + x0) − 4c3a cosh
2
√−a
4
(x + x0) 4c3a sinh
2
√−a
4
(x + x0)
u(x, t) − 4c
3a
sin2
√
a
4
(x + ct + x0) − 4c3a cosh
2
√−a
4
(x + ct + x0) 4c3a sinh
2
√−a
4
(x + ct + x0)
2.2. Shock-compacton solutions in K(3, 3) equation
Now we consider K(3, 3) equation:
ut + a(u3)x + (u3)3x = 0. (13)
Its correction iteration formula reads
un+1(x, t) = un(x, t) −
∫ t
0
{(un)t + a(u3n)x + (u3n)3x} d = 0. (14)
Its initial condition is assumed to have the form
u(x, 0) =
√
− 3c
2a
cos
√
a
3
(x + x0), (15)
50 L. Tian, J. Yin / Journal of Computational and Applied Mathematics 207 (2007) 46–52
1
0.5
0
−100
−50
0
50
100
150 −10
−5
0
5
10
1.2
1
0.8
0.6
0.4
0.2
−40 −20 20 40 60 80 100
(a) (b)
Fig. 4. Compacton solution.
where x0 and c are constants. The rest of the other components are deduced as follows
u0 =
√
− 3c
2a
cos
√
a
3
(x + x0),
u1(x, t) =
√
− 3c
2a
cos
√
a
3
(x + x0) −
√
−c
3
6
t sin
√
a
3
(x + x0),
u2(x, t) =
√
− 3c
2a
cos
√
a
3
(x + x0) −
√
−c
3
6
t sin
√
a
3
(x + x0) −
√
− ac
5
216
t2 cos
√
a
3
(x + x0)
and so on. The solutions u(x, t) in a closed form are obtained by the Mathematica package
u() =
⎧⎨
⎩
√
− 3c
2a
cos
√
a
3
, || 
2
,
0 otherwise,
(16)
where  = x + ct + x0. The compacton solution is shown in Fig. 4 with a = 1, c = −1, x0 = 0.
By the same method, we can obtain the solution u(x, t) in a closed form as
u() =
⎧⎨
⎩−
√
− 3c
2a
cos
√
a
3
, || 
2
,
0 otherwise,
(17)
So we can obtain a new solitary solution called the shock-compacton solution as follows
u() =
⎧⎨
⎩
√
− 3c
2a
sign() cos
√
a
3
, || 
2
,
0 otherwise
(18)
which is shown in Fig. 5 with a = 1, c = −1, x0 = 0.
From the graphs and computation we can ﬁnd that it has a discontinuous ﬁrst-order derivative at = 0,±/2. Note
that the fact
u() =
√
− 3c
2a
sign()A() cos
√
a
3
,
d
d
A() = 
(
 + 
2
)
− 
(
 − 
2
)
and
d
d
sign() = (),
L. Tian, J. Yin / Journal of Computational and Applied Mathematics 207 (2007) 46–52 51
1
0
−1
−10
−5
0
5
10
−4
−2
0
2
4
−20 −10
−0.5
0.5
1
−1
10 20
(a) (b)
Fig. 5. Shock-compacton solution.
we have
d
d
u() =
√
− 3c
2a
()A() cos
√
a
3
 +
√
− 3c
2a
sign()
[

(
 + 
2
)
− 
(
 − 
2
)]
cos
√
a
3

−
√
− c
6
sign()A() sin
√
a
3
,
where
A() =
{
1, || 
2
,
0 otherwise,
() is  function. Hence shock-compacton solution is non-local and new type of solitary wave. It has the characters
of shock and compacton.
3. Conclusion
In this paper, we obtain two new types of solitary wave solution: shock-peakon and shock-compacton for K(p, q)
equation. They are non-local shock wave solutions, having the characters of peakon and compacton.
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Shock-peakon and shock-compacton solutions for K(p,q) equation by variational iteration method 

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Shock-peakon and shock-compacton solutions for K(p,q) equation by variational iteration method

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