Abstract: This paper introduces a family of evolutionary 1+1 PDEs that describe the balance between convection and stretching in the dynamics of 1D nonlinear waves in fluids. It is reversible in time and parity invariant. In the paper, special solutions are discussed for = 0 and for ∕ = 0, the general solutions are given.When = 3 and = −1 , the paper obtains the exact solutions of the generalized −family equation

Changxing Wu

Chunxiang Feng

CiteSeerX

ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.8(2009) No.2,pp.213-217
Traveling Wave Solutions of the Generalized b-family Equation with
Dispersion
Changxing Wu ∗,Chunxiang Feng
Nonlinear Scientific Research Center, Jiangsu University
Zhenjiang, Jiangsu, 212013, P.R. China
(Received 10 April 2009, accepted 27 June 2009)
Abstract: This paper introduces a family of evolutionary 1+1 PDEs that describe the balance
between convection and stretching in the dynamics of 1D nonlinear waves in fluids. It is re-
versible in time and parity invariant. In the paper, special solutions are discussed for 𝑏 = 0 and
for 𝑏 ∕= 0, the general solutions are given.When 𝑏 = 3 and 𝑏 = −1 , the paper obtains the exact
solutions of the generalized 𝑏−family equation.
Keywords: b-family equation, traveling wave solution
1 Introduction
In [1]-[3], Degasperis and Procesi firstly studied the following family of third order dispersive PDE conser-
vation laws
𝑢𝑡 + 𝑐0𝑢𝑥 + 𝛾𝑢𝑥𝑥𝑥 − 𝛼2𝑢𝑡𝑥𝑥 =
(
𝑐1𝑢
2 + 𝑐2𝑢
2
𝑥 + 𝑐3𝑢𝑢𝑥𝑥
)
𝑥
(1)
where 𝛼, 𝑐0, 𝑐1, 𝑐2 and 𝑐3 are real constants and indices denote partial derivatives. When 𝑐1 = −𝛼2 ,
𝑐2 =
𝜀(𝛽−1)
2 , 𝑐3 = 𝜀 and replacing 𝑐0 with 𝑘, and 𝛼
2 with 𝜀 in the equation above, we obtain the following
equation {
(𝑢− 𝜀𝑢𝑥𝑥)𝑡 + 𝑘𝑢𝑥 + 𝛼𝑢𝑢𝑥 + 𝛾𝑢𝑥𝑥𝑥 = 𝜀(𝛽𝑢𝑥𝑢𝑥𝑥 + 𝑢𝑢𝑥𝑥𝑥), 𝑥 ∈ 𝑅, 𝑡 > 0,
𝑢(𝑥, 0) = 𝑢0(𝑥),
(2)
where 𝑢(𝑥, 𝑡) stands for the fluid velocity in the 𝑥 direction (or equivalently the height of the free surface
of water above a float bottom), 𝑘 is a constant related to the critical shallow water wave speed, and 𝛼,
𝛽, 𝜀 are dispersion parameters. It is necessary to point out that Eq.(2) is equivalent to Eq.(1) since when
𝜀 = 𝛼2 = 𝑐3, 𝑘 = 𝑐0, 𝛼 = −2𝑐1 and 𝛽 = 1 + 2𝑐2𝜀 , Eq.(2) turns out to be Eq.(1). To better understand the
common properties of the equation (1), we resort to study Eq.(2), which is convenient for us to research. We
call it the general shallow water wave equation.
In Eq.(2), if 𝛼 − 𝛽 = 1, 𝛾 = −𝑘𝜀,then this equation includes the KdV equation , the Camassa-Holm
equation, the Degasperis-Procesi equation and the 𝑏−family of equations. For convenience, we suppose that
𝛼− 𝛽 = 1, 𝛾 = −𝑘𝜀 from Eq.(2). Thus Eq.(2) changes into the following equation (𝑏 = 𝛽)
𝑚𝑡 + 𝑢𝑚𝑥︸︷︷︸
𝑐𝑜𝑛𝑣𝑒𝑐𝑡𝑖𝑜𝑛
+ 𝑏𝑢𝑥𝑚︸ ︷︷ ︸
𝑠𝑡𝑟𝑒𝑐ℎ𝑖𝑛𝑔
= −𝑘𝑚𝑥︸ ︷︷ ︸
𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑖𝑜𝑛
, (3)
in independent variables time and one spatial coordinate 𝑥.
Zhou and Tian [4] employ the bifurcation method to dynamical systems to investigate the exact travel-
ing wave solutions for the Fornberg-Whitham equation. The explicit expressions for peakons and periodic
∗Corresponding author. E-mail address: wcx.fly@163.com .
Copyright c⃝World Academic Press, World Academic Union
IJNS.2009.10.15/269
214 International Journal of Nonlinear Science,Vol.8(2009),No.2,pp. 213-217
cusp wave solutions are also obtained. By using bifurcation method, Zhou and Tian [5] successfully find
the Fornberg-Whitham equation has a type of traveling wave solutions called kink-like wave solutions and
antikink-like wave solutions. Tian, et al [6] converge to the solution to the corresponding BBM equation as
the parameter converges to zero. Zhou and Tian [7] obtain a conserbation law which enable us to present
a blow-up result by using multiplier technique. A remarkable feature of the physical model is that it has
peakon solution which has peak form[8]. Tian and Yin [9] introduce the concept of nonlinear intensity,
study a fully nonlinear sine-Gordon equation SG(m, n, p) and obtain a new type of peakon solutions and
kink solutions by a direct method. Tian and Yin [10] introduce the fully nonlinear generalized Camassa-
Holm equation C(m,n,p) and by using four direct ansatzs, we obtain abundant solutions. Tian and Song[11]
consider generalized Camassa-Holm equations and the generalized weakly dissipative Camassa-Holm equa-
tions and derive some new exact peaked solitary wave solutions.
The paper is organized as follows. In Section 2, we obtain the conservation law and some general prop-
erties. In Section 3, peakons, ramps and cliffs are discussed for 𝑏 = 0 and for 𝑏 ∕= 0 , general solution is
given. With special cases 𝑏 = −1 and 𝑏 = 3 , we obtain the exact solutions of eq.(3).
2 Basic notations and general properties
We seek solutions for the fluid velocity 𝑢 that are defined either on the real line and vanishing at spatial
infinity, or on a periodic one dimensional domain. Here 𝑢 = 𝑔 ∗𝑚 denotes the convolution (or filtering),
𝑢(𝑥) =
∫ ∞
−∞
𝑔(𝑥− 𝑦)𝑚(𝑦)𝑑𝑦, (4)
which relates velocity 𝑢 to momentum density 𝑚 by integration against kernel 𝑔(𝑥) over the real line. We
shall choose 𝑔(𝑥) to be an even function, so that 𝑢 and 𝑚 have the same parity.
The family of Eq.(3) is characterized by the kernel 𝑔 and the real dimensionless constant 𝑏, which is
the ratio of stretching to convective transport. As we see, 𝑏 is also the number of covariant dimensions
associated with the momentum density 𝑚. The function 𝑔(𝑥) will determine the traveling wave shape and
length scale for Eq.(3), while the constant 𝑏 will provide a balance or bifurcation parameter for the nonlinear
solution behavior. Special values of 𝑏 will include the first few positive and negative integers.
Its invariance under space and time translations ensures that Eq.(3) admits traveling wave solutions for
any 𝑏. We write the traveling wave solutions as 𝑢 = 𝑢(𝑧) and 𝑚 = 𝑚(𝑧), where 𝑧 = 𝑥− 𝑐𝑡 , and let prime
′ denote 𝑑/𝑑𝑧.
Eq.(3) implies a similar reversible, parity invariant equation for the absolute value ∣𝑚∣,
(1− 𝑘/𝑐)∣𝑚∣𝑡 + 𝑢∣𝑚∣𝑥 + 𝑏𝑢𝑥∣𝑚∣ = 0 and 𝑢 = 𝑔 ∗𝑚. (5)
If 𝑚1/𝑏 is well-defined, Eq.(3) may be written as the conservation law
∂𝑡𝑇𝑚
1/𝑏 + ∂𝑥(𝑚
1/𝑏𝑢), 𝑇 = 1− 𝑘/𝑐. (6)
Eq.(3) for 𝑚 is reversible, or invariant under 𝑡 → −𝑡 , 𝑢 → −𝑢. The latter implies 𝑚 → −𝑚. Hence,
the transformation 𝑢(𝑥, 𝑡) → −𝑢(𝑥,−𝑡) takes solutions into solutions, and in particular, it reverses the
direction and amplitude of the traveling wave 𝑢(𝑥, 𝑡) = 𝑐𝑔(𝑥− 𝑐𝑡).
We choose 𝑔(𝑥) to be an even function so that 𝑚 and 𝑢 = 𝑔 ∗ 𝑚 both have odd parity under mirror
reflections. Hence, Eq.(3) is invariant under the parity reflections 𝑢(𝑥, 𝑡) → −𝑢(−𝑥, 𝑡).
Eq.(3) implies a similar reversible, parity invariant equation for the absolute value ∣𝑚∣,
∣𝑚∣𝑡 + (𝑢+ 𝑘)∣𝑚∣𝑥 + 𝑏𝑢𝑥∣𝑚∣ = 0, 𝑢 = 𝑔 ∗𝑚. (7)
Theregfore, the positive and negative components 𝑚± = 12(𝑚 ± ∣𝑚∣) satisfy equation separately. Also, if
𝑚 is initially zero, it remains so. This is conservation of the signature of 𝑚 .
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C.Wu,C.Feng: Traveling Wave Solutions of the Generalized b-family Equation with Dispersion 215
3 Traveling wave solutions
For 𝑏 = 0 , Eq.(3) is Galilean invariant and its traveling wave solutions satisfy
(𝑢(𝑧)− 𝑐+ 𝑘)𝑚′(𝑧) = 0, 𝑧 = 𝑥− 𝑐𝑡, (8)
where prime ′ denotes 𝑑/𝑑𝑧 . Eq.(8) admits generalized functions 𝑚′(𝑧) ≃ 𝛿(𝑧) matched by 𝑢− 𝑐+ 𝑘 = 0
at 𝑧 = 0 . The velocity 𝑢 is given by the integral of the Green’s function that relates 𝑚 and 𝑢 = 𝑔 ∗𝑚
𝑢− 𝑐+ 𝑘 = (𝑐− 𝑘)[
∫
𝑔(𝑦)𝑑𝑦]𝑧0. (9)
When 𝑔(𝑥) = 𝑒−∣𝑥∣/𝛼 (the Green’s function for the 1D Helmholtz operator), we have 𝑚 = 𝑢 − 𝛼2𝑢𝑥𝑥.
Consequently, the equation 𝑚′ = 𝑢′ − 𝛼2𝑢′′′ = ±2𝛿(𝑧) with 𝑢− 𝑐+ 𝑘 = 0 at 𝑧 = 0 is satisfied by
𝑢− 𝑐+ 𝑘 = ±(𝑐− 𝑘)[
∫
𝑒−∣𝑦∣/𝛼𝑑𝑦]𝑧0 = ±(𝑐− 𝑘)𝑠𝑔𝑛(𝑧)(1− 𝑒−∣𝑧∣/𝛼). (10)
This represents a rightward moving traveling wave that connects the left state 𝑢− 𝑐+ 𝑘 = ±(𝑐− 𝑘) to the
same two right states.
From what has been discussed above, we have the following theorem:
Theorem 1 Assume 𝑏 = 0, 𝑢 = 𝑔 ∗𝑚, 𝑚 = 𝑢− 𝛼2𝑢𝑥𝑥, Eq.(3) has the solution 𝑢 = ±(𝑐− 𝑘)𝑠𝑔𝑛(𝑧)(1−
𝑒−∣𝑥∣/𝛼) + 𝑐− 𝑘.
We define 𝑝(𝑥) = 1
2𝛼2
𝑒−∣
𝑥
𝛼2
∣ ,𝑥 ∈ 𝑅 , then (1− 𝛼2∂2𝑥)−1𝑓 = 𝑝 ∗ 𝑓 for all 𝑓 ∈ 𝐿2(𝑅) , where ∗ denotes
convolution. Using this identity, we can rewrite Eq.(1) as the following nonlocal form
𝑢𝑡 + 𝑢𝑢𝑥 − 𝑘𝑢𝑥 + ∂𝑥𝑝 ∗ [3𝛼
2
2
𝑢2𝑥 + 2𝑘𝑢] = 0. (11)
The antisymmetric connections 𝑢 = ±(𝑐−𝑘)𝑠𝑔𝑛(𝑧)(1−𝑒−∣𝑧∣/𝛼) (with 𝑢−𝑐+𝑘 = ∓(𝑐−𝑘) connecting
to 𝑢 − 𝑐 + 𝑘 = ±(𝑐 − 𝑘)), with no jump in derivative at 𝑧 = 0, are the regularized shocks (cliffs). While
the symmetric connections 𝑢 = ±(𝑐− 𝑎)𝑒−∣𝑧∣/𝛼, with a jump in derivative at 𝑧 = 0, are the peakons. They
propagate rightward but may face either leftward or rightward, because Eq.(3) in the absence of viscosity
has no entropy condition that would distinguish between leftward and rightward facing solutions. Besides
Eq.(3) also has ramp-like similarity solutions 𝑢 ≃ 𝑥/𝑡 when 𝑔(𝑥) = 𝑒−∣𝑥∣/𝛼 for any 𝑏. They may emerge in
the initial value problem for the peakon case of Eq.(3) and interact with the peakons and cliffs.
For 𝑏 = 0 , the traveling wave Eq.(8) apparently has only the first integral for 𝑚 = 𝑢− 𝛼2𝑢𝑥𝑥
(𝑢− 𝑐+ 𝑘)(𝑢− 𝛼2𝑢′′)− 𝑢
2
2
+
𝛼2
2
𝑢′2 = 𝐾. (12)
Thus, perhaps surprisingly, we have been unable to find a second integral for the traveling wave equation
for peakons when 𝑏 = 0.
Reversibility means that Eq.(3) is invariant under the transformation 𝑢(𝑥, 𝑡) → −𝑢(𝑥,−𝑡) . Conse-
quently, the rightward traveling waves have leftward moving counterparts under the symmetry 𝑐 − 𝑘 →
−𝑐+ 𝑘 . The case of constant velocity 𝑢 = ±(𝑐− 𝑘) is also a solution.
For 𝑏 ∕= 0 , the conservation law (6) for traveling waves becomes
((𝑢− 𝑐+ 𝑘)𝑚1/𝑏)′ = 0, (13)
which yields after one integration
(𝑢− 𝑐+ 𝑘)𝑏𝑚 = 𝐾, (14)
where 𝐾 is the first integral.
For 𝑔(𝑥) = 𝑒−∣𝑥∣/𝛼 , so 𝑚 = 𝑢− 𝛼2𝑢𝑥𝑥 , Eq.(14) becomes
(𝑢− 𝑐+ 𝑘)𝑏(𝑢− 𝛼2𝑢′′) = 𝐾. (15)
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216 International Journal of Nonlinear Science,Vol.8(2009),No.2,pp. 213-217
For 𝑢− 𝑐+ 𝑘 ∕= 0 we rewrite Eq.(15) as
𝛼2𝑢
′′
= 𝑢−𝐾(𝑢− 𝑐+ 𝑘)−𝑏 (16)
and integrate again to give the second integral in two separate cases,
𝛼2𝑢
′2 =
{
𝑢2 − 2𝐾1−𝑏(𝑢− 𝑐+ 𝑘)1−𝑏 + 2𝐻, 𝑏 ∕= 1
𝑢2 − 2𝐾𝑙𝑛(𝑢− 𝑐+ 𝑘) + 2𝐻, 𝑏 = 1. (17)
We shall rearrange this into quadratures:
±𝑑𝑧
𝛼
=
𝑑𝑢
[𝑢2 − 2𝐾𝑙𝑛(𝑢− 𝑐+ 𝑘) + 2𝐻] 12
for 𝑏 = 1. (18)
and
±𝑑𝑧
𝛼
=
𝑑𝑢
[𝑢2 − 2𝐾1−𝑏(𝑢− 𝑐+ 𝑘)1−𝑏 + 2𝐻]
1
2
for 𝑏 ∕= 1, (19)
For 𝑏 = 1 and 𝐾 ∕= 0 , the integral in Eq.(18) is transcendental.
For 𝐾 = 0, the two quadratures Eq.(19) and Eq.(18) are equal, independent of 𝑏 , and elementary,
thereby yielding the traveling wave solutions
𝑒−∣𝑧∣/𝛼 =
𝑢+
√
𝑢2 + 2𝐻
𝑐− 𝑘 +√(𝑐− 𝑘)2 + 2𝐻 , (20)
with 𝑢− 𝑐+ 𝑘 = 0 at 𝑧 = 0 .
For 𝐻 = 0 , Eq.(20) recovers the peakon traveling wave.
For 𝐻 > 0, Eq.(20) gives a rightward moving traveling wave that is a continuous deformation of the
peakon.
For 𝐻 > 0 and 𝑐 = 𝑘, Eq.(20) gives stationary solutions of the form
𝑢+
√
𝑢2 + 2𝐻√
2𝐻
= 𝑒−∣𝑧∣/𝛼, (21)
then
𝑢 =
√
2𝐻
2
𝑒−∣𝑧∣/𝛼 −
√
2𝐻
2
𝑒∣𝑧∣/𝛼. (22)
For 𝑏 = 3, Eq.(19) is
±𝑑𝑧
𝛼
=
𝑑𝑢
[𝑢2 +𝐾(𝑢− 𝑐+ 𝑘)−2 + 2𝐻] 12
, (23)
which for the hyperbolic limit 𝐻 = 0 and 𝑐 = 𝑘 is
±𝑑𝑧
𝛼
=
𝑑𝑢
[𝑢2 +𝐾𝑢−2]
1
2
. (24)
Eq.(24) gives solutions of the form
𝑢 = (
1
2
𝑒−2∣𝑧∣/𝛼 − 𝐾
2
𝑒−2∣𝑧∣/𝛼)
1
2
. (25)
Theorem 2 Assume 𝑏 = 3, 𝑢 = 𝑔 ∗ 𝑚, 𝑚 = 𝑢 − 𝛼2𝑢𝑥𝑥, and 𝐻 = 0, 𝑐 = 𝑘 , Eq.(3) has the solution
𝑢 = (12𝑒
−2∣𝑧∣/𝛼 − 𝐾2 𝑒−2∣𝑧∣/𝛼)
1
2 .
For 𝑏 = −1, Eq.(19) becomes
±𝑑𝑧
𝛼
=
𝑑𝑢
[𝑢2 −𝐾(𝑢− 𝑐+ 𝑘)2 + 2𝐻] 12
, (26)
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C.Wu,C.Feng: Traveling Wave Solutions of the Generalized b-family Equation with Dispersion 217
which integrates to
𝑒−∣𝑧∣/𝛼 =
𝑢+
√
𝑢2 −𝐾(𝑢− 𝑐+ 𝑘)2 + 2𝐻 +𝐾(𝑐− 𝑘)
𝑐− 𝑘 +√(𝑐− 𝑘)2 + 2𝐻 +𝐾(𝑐− 𝑘) (27)
with 𝑢 = 𝑐− 𝑘 at 𝑧 = 0, then
𝑢 =
1
2𝐾
(−2𝐾𝑒−∣𝑧∣/𝛼 − 2𝑘𝐾𝑒−∣𝑧∣/𝛼 + 2
√
𝑘2 + 2𝐻𝑒−∣𝑧∣/𝛼 ± 2(2𝑘2𝑒−2∣𝑧∣/𝛼 − 2𝑘𝑒−2∣𝑧∣/𝛼
√
𝑘2 + 2𝐻
−𝑘2𝐾2𝑒−2∣𝑧∣/𝛼 + 2𝐻𝑒−2∣𝑧∣/𝛼 − 𝑘2𝐾3 − 𝑘2𝐾2 + 2𝐾𝐻 + 2𝑘2𝐾3𝑒−∣𝑧∣/𝛼 − 2𝑘𝐾2𝑒−∣𝑧∣/𝛼√
𝑘2 + 2𝐻 + 2𝑘𝐾2𝑒−2∣𝑧∣/𝛼
√
𝑘2 + 2𝐻 + 2𝑘2𝐾2𝑒−∣𝑧∣/𝛼 − 𝑘2𝐾3𝑒−2∣𝑧∣/𝛼 − 2𝐾𝐻𝑒−∣𝑧∣/𝛼) 12 ). (28)
Theorem 3 Assume 𝑏 = −1, 𝑢 = 𝑔 ∗𝑚, 𝑚 = 𝑢− 𝛼2𝑢𝑥𝑥, Eq.(3) has the solution
𝑢 =
1
2𝐾
(−2𝐾𝑒−∣𝑧∣/𝛼 − 2𝑘𝐾𝑒−∣𝑧∣/𝛼 + 2
√
𝑘2 + 2𝐻𝑒−∣𝑧∣/𝛼 ± 2(2𝑘2𝑒−2∣𝑧∣/𝛼 − 2𝑘𝑒−2∣𝑧∣/𝛼
√
𝑘2 + 2𝐻
−𝑘2𝐾2𝑒−2∣𝑧∣/𝛼 + 2𝐻𝑒−2∣𝑧∣/𝛼 − 𝑘2𝐾3 − 𝑘2𝐾2 + 2𝐾𝐻 + 2𝑘2𝐾3𝑒−∣𝑧∣/𝛼 − 2𝑘𝐾2𝑒−∣𝑧∣/𝛼√
𝑘2 + 2𝐻 + 2𝑘𝐾2𝑒−2∣𝑧∣/𝛼
√
𝑘2 + 2𝐻 + 2𝑘2𝐾2𝑒−∣𝑧∣/𝛼 − 𝑘2𝐾3𝑒−2∣𝑧∣/𝛼 − 2𝐾𝐻𝑒−∣𝑧∣/𝛼) 12 ).
Acknowledgements
Research was supported by the National Nature Science Foundation of China (No: 10771088) and the
Nature Science Foundation of Jiangsu (No:2007098) and the Outstanding Personnel Program in Six Fields
of Jiangsu (No: 6-A-029).
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IJNS homepage:http://www.nonlinearscience.org.uk/


Traveling Wave Solutions of the Generalized b-family Equation with Dispersion

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Traveling Wave Solutions of the Generalized b-family Equation with Dispersion

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