# Kink-Like Wave and Compacton-Like Wave Solutions for a Two-Component Fornberg-Whitham Equation

Authors: Shaoyong Li; Ming Song
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

Using bifurcation method and numerical simulation approach of dynamical
systems, we study a two-component Fornberg-Whitham equation. Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, two new phenomena are revealed; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. Exact implicit or parameter expressions of these solutions are given.


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## Body

## 1. Introduction

The Fornberg-Whitham equation(1)ut-uxxt+ux+uux=uuxxx+3uxuxx
was used to study the qualitative behaviors of wave breaking [1]. It admits a wave of the greatest height, as a peaked limiting form of the traveling wave solution [2], u(x,t)=A~exp⁡((1/2)|x-(4/3)t|), where A~ is an arbitrary constant. Recently, Zhou and Tian found that (1) possess kink-like wave solutions in [3]. They obtained some solitons, peakons, and periodic cusp wave solutions in [4]. Further, they obtained the smooth periodic wave solutions and loop-soliton solutions by using elliptic integral [5]. Feng and Wu [6] considered the classification of single traveling wave solutions to (1). Chen et al. [7] gave some smooth periodic wave, smooth solitary wave, periodic cusp wave, and loop-soliton solutions of (1) and made the numerical simulation.He et al. [8] studied the following modified Fornberg-Whitham equation:
(2)ut-uxxt+ux+u2ux=uuxxx+3uxuxx.
In some parametric conditions, some peakons and solitary waves were found and their exact parametric representations in explicit form were obtained.Jiang and Bi [9] considered the Fornberg-Whitham equation with linear dispersion term given by
(3)ut-uxxt+ux+uux=uuxxx+3uxuxx-γuxxx,
where γ is a real constant. When γ=0, (3) reduces to (1). They investigated the existence of the smooth and nonsmooth traveling wave solutions and gave some analytic expressions of smooth solitary wave, periodic cusp wave, and peakon solutions for (3).Fan et al. [10] presented a two-component Fornberg-Whitham equation given by
(4)ut=uxxt-ux-uux+uuxxx+3uxuxx+ρx,ρt=-(ρu)x,
where u=u(x,t) is the height of the water surface above a horizontal bottom and ρ=ρ(x,t) is related to the horizontal velocity field. When ρ=0, (4) reduces to (1). Parametric conditions to smooth soliton solution, kink solution, antikink solution, and uncountable infinite many smooth periodic wave solutions of (4) were given. Later, Wen [11] further studied (4). He presented all possible phase portraits determinately and gave all the exact explicit parametric conditions for various solutions.The concept of compacton: soliton with compact support or strict localization of solitary waves appeared in the work of Rosenau and Hyman [12], where a genuinely nonlinear dispersive equation K(n,n) defined by
(5)ut+a(un)x+(un)xxx=0
was subjected to experimental and analytical studies. They found certain solitary wave solutions which vanish identically outside a finite core region. These solutions are called compactons. Several studies have been conducted in [13–18]. The kink-like wave or generalized kink wave is discovered by Liu et al. [14], which is defined on semifinal bounded domain and possesses some properties of the kink wave.Many methods have been used to investigate traveling wave solutions to nonlinear equations, such as Jacobi elliptic function method [19, 20], F-expansion and extended F-expansion method [21, 22], and (G′/G)-expansion method [23, 24]. Here, our aim in this paper is to use the bifurcation method of dynamical systems [25–28] to investigate (4). We obtain the kink-like wave and compacton-like wave solutions with implicit or parameter expressions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, we point out that the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. To the best of our knowledge, these solutions and phenomena are new for (4). Our work may help people to know deeply the described physical process and possible applications of (4).The remainder of this paper is organized as follows. In Section2, we study the bifurcation phase portraits. In Section 3, we make the numerical simulation for bounded integral curves. In Section 4, we derive the exact implicit or parameter expressions of the kink-like wave and compacton-like wave solutions. A brief conclusion is given in Section 5.


## 2. Bifurcation Phase Portraits

We look for the traveling wave solutions of (4) in the form of
(6)u(x,t)=φ(ξ)+c~,ρ(x,t)=ψ(ξ),ξ=x-ct,
where c~ is the mean level and c is the wave speed.Substituting (6) into (4) and integrating once with respect to ξ, it follows that
(7)(c-c~-1)φ-12φ2+(φ′)2+(φ+c~-c)φ′′+ψ=θ,ψ=-gφ+c~-c,
where θ, g are two integral constants and g≠0 (if g=0, then ρ=0 from the second equation of (7) and (6). In this case (4) reduces to (1), which was studied in [3–7]).In order to study conveniently, we choosec~=c, and this only makes a translational movement of the singular line from φ=c-c~ to φ=0, so there is no essential difference for the results. Thus, substituting the second equation of (7) into the first equation of (7), we obtain
(8)φ2φ′′=12φ3+φ2+θφ+g-φ(φ′)2.Lettingy=φ′, we obtain the following planar system:
(9)dφdξ=y,dydξ=(1/2)φ3+φ2+θφ+g-φy2φ2,
under the transformation dξ=φ2dτ, and system (9) becomes
(10)dφdτ=yφ2,dydτ=12φ3+φ2+θφ+g-φy2.Obviously, system (9) and system (10) have the same first integral
(11)H(φ,y)=φ2y2-(14φ4+23φ3+θφ2+2gφ)=h,
where h is an integral constant. Consequently, these two systems have the same topological phase portraits except for the straight line φ=0. Thus, we can understand the phase portraits of system (9) from those of system (10).In order to state conveniently, for given constantsθ and g, let
(12)f0(φ)=12φ3+φ2+θφ+g,g1(θ)=227(-2(2+4-6θ)+3(3+4-6θ)θ),g2(θ)=-2729(4+16-27θ)(8+216-27θ-27θ),g3(θ)=-2729(-4+16-27θ)(-8+216-27θ+27θ),g4(θ)=-227(4-24-6θ+3(-3+4-6θ)θ),g5(θ)=227(-4+9θ).Assume thatφ1, φ2, and φ3 are three roots of equation f0(φ)=0, where
(13)φ1=-23+16(-1+3i)δ+(1+3i)(3θ-2)3δ,φ2=-23-16(1+3i)δ+(1-3i)(3θ-2)3δ,φ3=-23+4-6θ3δ+δ3,δ=(g(16+27g)-36gθ-4θ2+8θ3-8-27g+18θ+33g(16+27g)-36gθ-4θ2+8θ3)1/3.Meanwhile, we give conditions as follows.Case 1.
θ
<
(
2
/
3
), g<g1(θ) or θ≥(2/3), g<0.Case 2.
θ
<
(
2
/
3
)
(
θ
≠
(
1
/
2
)
) and g=g1(θ).Case 3.
θ
<
(
1
/
2
) and g1(θ)<g<g2(θ).Case 4.
θ
<
(
1
/
2
) and g=g2(θ).Case 5.
θ
<
(
1
/
2
) and g2(θ)<g<0.Case 6.
θ
≤
0 and 0<g<g3(θ).Case 7.
θ
≤
0 and g=g3(θ).Case 8.
θ
≤
0 and g3(θ)<g<g4(θ).Case 9.
θ
≤
0 and g=g4(θ).Case 10.
θ
<
(
2
/
3
), g>g4(θ) or θ≥(2/3), g>0.Case 11.
0
<
θ
<
(
1
/
2
), 0<g<g4(θ) or (1/2)≤θ<(2/3), g5(θ)<g<g4(θ).Case 12.
0
<
θ
<
(
2
/
3
) and g=g4(θ).Case 13.
θ
=
(
1
/
2
) and 0<g<g5(θ).Case 14.
(
1
/
2
)
≤
θ
<
(
2
/
3
) and g=g5(θ).Case 15.
(
1
/
2
)
<
θ
<
(
2
/
3
) and g=g1(θ).Case 16.
(
1
/
2
)
<
θ
<
(
2
/
3
) and g1(θ)<g<g5(θ).According to the qualitative theory of differential equations and the above conditions, we have the results as Table1.Table 1
The singular point and phase portrait for different cases.
Case
Root off0(φ)=0
Center
Saddle
Degenerate saddle
Phase portrait
Case  1
φ
3
>
0
—
(
φ
3
,
0
)
—
Figure1(a)
Case  2
φ
1
=
φ
2
<
0
<
φ
3
—
(
φ
3
,
0
)
(
φ
1
,
0
)
Figure1(b)
Case  3
φ
1
<
φ
2
<
0
<
φ
3
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(c)
Case  4
φ
1
<
φ
2
<
0
<
φ
3
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)  H(φ1,0)=0
—
Figure1(d)
Case  5
φ
1
<
φ
2
<
0
<
φ
3
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(e)
Case  6
φ
1
<
0
<
φ
2
<
φ
3
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(f)
Case  7
φ
1
<
0
<
φ
2
<
φ
3
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)  H(φ3,0)=0
—
Figure1(g)
Case  8
φ
1
<
0
<
φ
2
<
φ
3
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(h)
Case  9
φ
1
<
0
<
φ
2
=
φ
3
—
(
φ
1
,
0
)
(
φ
3
,
0
)
Figure1(i)
Case  10
φ
1
<
0
—
(
φ
1
,
0
)
—
Figure1(j)
Case  11
φ
1
<
φ
2
<
φ
3
<
0
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(k)
Case  12
φ
1
<
φ
2
=
φ
3
<
0
—
(
φ
1
,
0
)
(
φ
2
,
0
)
Figure1(l)
Case  13
φ
1
<
φ
2
<
φ
3
<
0
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(m)
Case  14
φ
1
<
φ
2
<
φ
3
<
0
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)  H(φ1,0)=H(φ3,0)
—
Figure1(n)
Case  15
φ
1
=
φ
2
<
φ
3
<
0
—
(
φ
3
,
0
)
(
φ
1
,
0
)
Figure1(o)
Case  16
φ
1
<
φ
2
<
φ
3
<
0
(
φ
2
,
0
)
(
φ
1
,
0
)  (φ3,0)
—
Figure1(p)


## 3. Numerical Simulation for Bounded Integral Curves

In this section, we make the numerical simulation for bounded integral curves. For convenience, throughout the following work we only discuss the solutionφ(ξ) with respect to the first component u=φ(ξ)+c and omit the second component ρ=ψ(ξ)=-g/φ of (4).From the derivation in Section2 we see that the bounded traveling waves of (4) correspond to the bounded integral curves of (8) and the bounded integral curves of (8) correspond to the orbits of system (9) in which φ=φ(ξ) is bounded. Therefore we can simulate the bounded integral curves of (8) by using the information of the phase portraits of system (9).It follows from [14–18] that the open orbits Ki  (i = 1–5) of system (9) correspond to the compacton-like waves of (4), the heteroclinic orbits Ji (i = 1,2) of system (9) correspond to the kink-like waves of (4), and the periodic orbits surrounding the center point (φ2,0) correspond to the periodic waves of (4). Here we only make the numerical simulation for Cases 4 and 5 as Examples 1 and 2. The other cases are similar to Examples 1 and 2, so we omit them.Example 1.
For Case4, taking θ=-1<1/2, then g=g2(θ)=-1.39361, φ1=-2.3461, φ2=-0.930567, and φ3=1.27666.
(
1
) From (11), the two heteroclinic orbits J1± (see Figure 1(d)) passing through the saddle point (φ3,0) have expressions, respectively,
(14)y=±(1/4)φ4+(2/3)φ3+θφ2+2gφ+h(φ3)φ2,
where h(φ3)=H(φ3,0) and 0<φ<φ3. We assume that φ(0) and φ′(0) are the initial values for the orbit of system (9). For any given φ1*  (0<φ1*<φ3), then from the first equation of system (9) we have φ′=y(φ1*) at φ=φ1*. For example, setting φ1*=1, we have y(φ1*)=±0.5160826789900144. Thus taking φ(0)=1 and φ′(0)=±0.5160826789900144 as initial values, respectively, we simulate the integral curves of (8) as (a) and (b) in Figure 2.
(
2
) From (11), the two heteroclinic orbits J2± (see Figure 1(d)) passing through the saddle point (φ1,0) have expressions, respectively,
(15)y=±(1/4)φ4+(2/3)φ3+θφ2+2gφ+h(φ1)φ2,
where h(φ1)=H(φ1,0)=H(0,0)=0 and φ1<φ<0. For any given φ2*(φ1<φ2*<0), then from the first equation of system (9) we have φ′=y(φ2*) at φ=φ2*. For example, setting φ2*=-1, we have y(φ2*)=±1.1707040231039854. Thus taking φ(0)=-1 and φ′(0)=±1.1707040231039854 as initial values, respectively, we simulate the integral curves of (8) as (c) and (d) in Figure 2.
(
3
) From (11), the orbit K1 (see Figure 1(d)) passing through (φ0,0) has expressions
(16)y=±(1/4)φ4+(2/3)φ3+θφ2+2gφ+h(φ0)φ2,
where h(φ0)=H(φ0,0), 0<φ≤φ0, and 0<φ0<φ3. Choosing φ0=1∈(0,φ3) and taking φ(0)=φ0 and φ′(0)=0 as initial values, respectively, we simulate the integral curve of (8) as (e) in Figure 2.The phase portraits of system (10) in different cases i (i = 1–16).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)
(m)
(n)
(o)
(p)The simulations of integral curves of (8) when θ=-1, g=-1.39361 (corresponding to Example 1), and g=-0.696804 (corresponding to Example 2). (a) φ(0)=1 and φ′(0)=-0.5160826789900144. (b) φ(0)=1 and φ′(0)=0.5160826789900144. (c) φ(0)=-1 and φ′(0)=-1.1707040231039854. (d) φ(0)=-1 and φ′(0)=1.1707040231039854. (e) φ(0)=1 and φ′(0)=0. (f) φ(0)=-1.624378546433569 and φ′(0)=0.
(a)
(b)
(c)
(d)
(e)
(f)Example 2.
For Case5, taking θ=-1<(1/2), then g=(1/2)g2(θ)=-0.696804∈(g2(θ),0), φ1=-2.56755, φ2=-0.50572, and φ3=1.07327. From (11), the orbit K3 (see Figure 1(e)) passing through (φ3r,0) has expressions
(17)y=±(1/4)φ4+(2/3)φ3+θφ2+2gφ+h(φ3)φ2,
where h(φ3)=H(φ3,0), φ3r≤φ<0, and φ3r=-1.624378546433569. Taking φ(0)=φ3r and φ′(0)=0 as initial values, respectively, we simulate the integral curve of (8) as (f) in Figure 2.Remark 3.
The kink-like waves in Figures2(a) and 2(b) are defined on (-∞,ξ~1) and (-ξ~1,+∞), respectively. The kink-like waves in Figures 2(c) and 2(d) are defined on (-ξ~2,+∞) and (-∞,ξ~2), respectively. The compacton-like wave in Figure 2(e) has peak form on (-ξ~3,ξ~3), where ξ~1, ξ~2, and ξ~3 satisfy
(18)ξ~1=∫0φ1*‍sds(1/4)s4+(2/3)s3+θs2+2gs+h(φ3),for0<φ1*<φ3,ξ~2=∫φ2*0‍-sds(1/4)s4+(2/3)s3+θs2+2gs+h(φ1),forφ1<φ2*<0,ξ~3=∫0φ0‍sds(1/4)s4+(2/3)s3+θs2+2gs+h(φ0),for0<φ0<φ3,
where h(φ0)=H(φ0,0), h(φ1)=H(φ1,0), and h(φ3)=H(φ3,0). Take the data of Example 1, that is, θ=-1, g=-1.39361, φ1*=1, φ2*=-1, φ3=1.27666, φ1=-2.3461, and φ0=1, then from (18) we obtain ξ~1=0.54746, ξ~2=0.480057, and ξ~3=0.915546 which are identical with the simulations (see Figures 2(a)–2(e)).Remark 4.
The compacton-like wave in Figure2(f) is defined on (-ξ~4,ξ~4), where ξ~4 satisfies
(19)ξ~4=∫φ3r0‍-sds(1/4)s4+(2/3)s3+θs2+2gs+h(φ3),
where h(φ3)=H(φ3,0). Take the data of Example 2; that is, θ=-1, g=-0.696804, φ3=1.07327, and φ3r=-1.624378546433569, and then from (19) we obtain ξ~4=1.73392 which is identical with the simulation (see Figure 2(f)).Remark 5.
For Cases4 and 5, there are a family of periodic orbits surrounding the center point (φ2,0), but the boundaries of the periodic orbits are different. For Case 4, the boundaries of the periodic orbits are the two heteroclinic orbits J2± (see Figure 3(a)), while for Case 5, the boundary of the periodic orbits is the open orbit K4 (see Figure 3(b)). Taking the data of Example 1 and a set of initial values (φ(0),φ′(0)), that is, θ=-1, g=-1.39361 and (φ(0),φ′(0))=(-1,0.7),(-1,1.1),(-1,1.17),(-1,1.170704), we simulate the periodic orbits of (8) as Figure 4. Similarly, taking the data of Example 2 and a set of initial values (φ(0),φ′(0)), that is, θ=-1, g=-0.696804, and (φ(0),φ′(0))=(-0.7,0),(-0.9,0),(-0.98,0),(-0.9854465,0), we simulate the periodic orbits of (8) as Figure 5. The simulations in Figure 4 imply that the periodic waves tend to two kink-like waves when the periodic orbits tend to the heteroclinic orbits J2±. The simulations in Figure 5 imply that the periodic waves tend to the periodic compacton-like wave when the periodic orbits tend to the open orbit K4.The periodic orbits surrounding the center point(φ2,0) and their boundaries, where (a) for Case 4 and (b) for Case 5.
(a)
(b)The simulations of periodic integral curves of (8) for Case 4, where the initial values (φ(0),φ′(0)) = (-1,0.7),(-1,1.1),(-1,1.17),(-1,1.170704).
(a)
(b)
(c)
(d)The simulations of periodic integral curves of (8) for Case 5, where the initial values (φ(0),φ′(0)) = (-0.7,0),(-0.9,0),(-0.98,0),(-0.9854465,0).
(a)
(b)
(c)
(d)


## 4. The Expressions of Kink-Like and Compacton-Like Waves

In this section we derive the exact expressions of the kink-like and compacton-like waves in different casesi (i= 1–16). Assuming that (φ(0),φ′(0)) is the initial point of an orbit of system (9). Let
(20)F(φ)=14φ4+23φ3+θφ2+2gφ+h0,
where h0=H(φ(0),φ′(0)), then form H(φ,y)=h0, the following equation:
(21)φ2y2=F(φ)
determines the orbit passing through (φ(0),φ′(0)).


### 4.1. Solutions of Kink-Like Wave

(
1
)
In Cases i (i = 1–4) corresponding to phase portraits in Figures 1(a)–1(d), (20) becomes
(22)F(φ)=14(φ3-φ)2(φ2+m1φ+n1),
where m1=(8/3)+2φ3, n1=4θ+(1/3)φ3(16+9φ3). Thus the orbits J1± passing through saddle point (φ3,0) have expressions
(23)y=±(φ3-φ)φ2+m1φ+n12φ,
where 0<φ<φ3. Substituting (23) into dφ/dξ=y and integrating along J1+ and J1- for initial value φ(0)=φ1*, where 0<φ1*<φ3, we obtain two kink-like wave solutions of implicit expression as follows:
(24)f1(φ)=f1(φ1*)e(1/2)ξ,-ξ1<ξ<∞,f1(φ)=f1(φ1*)e-(1/2)ξ,-∞<ξ<ξ1,
where
(25)ξ1=2ln⁡f1(φ1*)f1(0),f1(s)=(2a1s2+m1s+n1+b1(φ3-s)+2a1)μ1(φ3-s)μ1(2s2+m1s+n1+2s+m1),μ1=φ3a1,a1=φ32+m1φ3+n1,b1=-m1-2φ3.The derivations of other kink-like wave solutions are similar to the above case, so we omit the details and only list the results.(
2
) In Cases i (i=5,6) corresponding to phase portraits in Figures 1(e) and 1(f), we obtain two kink-like wave solutions of implicit expression as follows:
(26)f2(φ)=f2(φ1*)e(1/2)ξ,-ξ2<ξ<∞,f2(φ)=f2(φ1*)e-(1/2)ξ,-∞<ξ<ξ2,
where
(27)ξ2=2ln⁡f2(φ1*)f2(0),f2(s)=(2a2(s-φ3l)(s-φ3r)+b2(φ3-s)+2a2)μ2(φ3-s)μ2(2(s-φ3l)(s-φ3r)+2s-φ3l-φ3r),μ2=φ3a2,a2=(φ3-φ3l)(φ3-φ3r),b2=φ3l+φ3r-2φ3,φ3l=13(-4-3φ3-28-18θ-3φ3(4+3φ3)),φ3r=13(-4-3φ3+28-18θ-3φ3(4+3φ3)).(
3
) In Case 7 corresponding to phase portrait in Figure 1(g), we obtain two kink-like wave solutions of implicit expression as follows:
(28)f3(φ)=f3(φ1*)e(1/2)ξ,-ξ3<ξ<∞,f3(φ)=f3(φ1*)e-(1/2)ξ,-∞<ξ<ξ3,
where
(29)ξ3=2ln⁡f3(φ1*)f3(0),f3(s)=(s-φ3∘-s)(φ3(s-φ3∘)+(φ3-φ3∘)s)μ3(s-φ3∘+s)(φ3(s-φ3∘)-(φ3-φ3∘)s)μ3,μ3=φ3φ3-φ3∘,φ3∘=-83-2φ3.(
4
) In Case 11 corresponding to phase portraits in Figure 1(k), we obtain two kink-like wave solutions of implicit expression as follows:
(30)f4(φ)=f4(φ1*)e(1/2)ξ,-∞<ξ<ξ4,f4(φ)=f4(φ1*)e-(1/2)ξ,-ξ4<ξ<∞,
where
(31)ξ4=2ln⁡f4(0)f4(φ1*),f4(s)=(2a2(s-φ3l)(s-φ3r)-b2(s-φ3)+2a2)μ2(s-φ3)μ2(2(s-φ3l)(s-φ3r)+2s-φ3l-φ3r).(
5
) In Case 12 corresponding to phase portraits in Figure 1(l), we obtain two kink-like wave solutions of implicit expression as follows:
(32)f5(φ)=f5(φ1*)+12ξ,-∞<ξ<ξ5,f5(φ)=f5(φ1*)-12ξ,-ξ5<ξ<∞,
where
(33)ξ5=2(f5(0)-f5(φ1*)),f5(s)=2φ3φ3-φ32s-φ32s-φ3-ln⁡(2(s-φ3)(s-φ32)+2s-φ3-φ32),φ3=13(-2+4-6θ),φ32=-83-3φ3.(
6
) In Cases i (i=13,15,16) corresponding to phase portraits in Figures 1(m), 1(o), and 1(p), we obtain two kink-like wave solutions of implicit expression as follows:
(34)f6(φ)=f6(φ1*)e(1/2)ξ,-∞<ξ<ξ6,f6(φ)=f6(φ1*)e-(1/2)ξ,-ξ6<ξ<∞,
where
(35)ξ6=2ln⁡f6(0)f6(φ1*),f6(s)=(2a1s2+m1s+n1-b1(s-φ3)+2a1)μ1(s-φ3)μ1(2s2+m1s+n1+2s+m1).(
7
) In Case 14 corresponding to phase portrait in Figure 1(n), we obtain two kink-like wave solutions of implicit expression as follows:
(36)f7(φ)=f7(φ1*)+12ξ,-∞<ξ<ξ7,f7(φ)=f7(φ1*)-12ξ,-ξ7<ξ<∞,
where
(37)ξ7=2ln⁡f7(0)f7(φ1*),f7(s)=1φ3-φ1ln⁡(s-φ1)φ1(s-φ3)φ3.(
8
) In Case 4 corresponding to phase portrait in Figure 1(d), we obtain two kink-like wave solutions of implicit expression as follows:
(38)f8(φ)=f8(φ2*)e(1/2)ξ,-∞<ξ<ξ8,f8(φ)=f8(φ2*)e-(1/2)ξ,-ξ8<ξ<∞,
where φ1<φ2*<0 and
(39)ξ8=2ln⁡f8(0)f8(φ2*),f8(s)=(φ1∘-s+-s)(φ1(s-φ1∘)-(φ1-φ1∘)s)μ4(φ1∘-s--s)(φ1(s-φ1∘)+(φ1-φ1∘)s)μ4,μ4=φ1φ1-φ1∘,φ1∘=-83-2φ1.(
9
) In Cases i (i = 5–12) corresponding to phase portraits in Figures 1(e)–1(l), we obtain two kink-like wave solutions of implicit expression as follows:
(40)f9(φ)=f9(φ2*)e(1/2)ξ,-∞<ξ<ξ9,f9(φ)=f9(φ2*)e-(1/2)ξ,-ξ9<ξ<∞,
where
(41)ξ9=2ln⁡f9(0)f9(φ2*),f9(s)=(2a3s2+m2s+n2+b3(s-φ1)+2a3)μ5(s-φ1)μ5(2s2+m2s+n2+2s+m2),μ5=φ1a3,a3=φ12+m2φ1+n2,b3=2φ1+m2,m2=83+2φ1,n2=4θ+13φ1(16+9φ1).


### 4.2.  Solutions of Compacton-Like Wave

(
1
) In Cases i (i = 1–4) corresponding to phase portraits in Figures 1(a)–1(d) and φ(0)=φ0, (20) becomes
(42)F(φ)=14(φ0-φ)(φ01-φ)(φ-φ02)(φ-φ¯02),
where φ0 and φ01 are real roots and φ02 and φ¯02 are conjugate complex roots of F(φ)=0 and 0<φ0<φ3<φ01. Thus the orbit K1 has expressions:
(43)y=±(φ0-φ)(φ01-φ)(φ-φ02)(φ-φ¯02)2φ,
where 0<φ≤φ0. By applying transformation dξ=2φdv to dφ/dξ=y, we have
(44)dφdv=2φy.Substituting (43) into (44) and integrating along K1, we get
(45)φ=pφ01-qφ0-(pφ01+qφ0)cn(w,k1)p-q-(p+q)cn(w,k1),|w|≤w1,
where
(46)k1=(p+q)2-(φ0-φ01)24pq,p=(A-φ01)2+B2,q=(A-φ0)2+B2,A=φ02-φ¯022,B=-(φ02-φ¯02)24,w=pqv,w1=cn-1(pφ01-qφ0pφ01+qφ0,k1).Substituting (45) into dξ=2φdv and integrating once, we get
(47)ξ=2pq((α12α12-1)pφ01+qφ0p+qw+(p-q)(φ0-φ01)2(p+q)1111111×(Π(sin-1(sn(w,k1)),α12α12-1,k1)-α1β)),
where
(48)α1=p+qq-p,β=12α12-1k12+(k1′α1)2×ln⁡k12+(k1′α1)2dn(w,k1)+α12-1sn(w,k1)k12+(k1′α1)2dn(w,k1)-α12-1sn(w,k1),k1′=1-k12.Thus we obtain a compacton-like wave solution of parametric expression as follows:(49)φ=pφ01-qφ0-(pφ01+qφ0)cn(w,k1)p-q-(p+q)cn(w,k1),ξ=2pq(pφ01+qφ0p+qw+(p-q)(φ0-φ01)2(p+q)111111×(α12α12-1Π(sin-1(sn(w,k1)),α12α12-1,k1)111111111-α1βα12α12-1)α12α12-1),
where w is a parameter variable and -w1≤w≤w1, 0<φ≤φ0.The derivations of other compacton-like solutions are similar to the above case, so we omit the details and only list the results.(
2
) In Cases i (i = 7–10) corresponding to phase portraits in Figures 1(g)–1(j), we obtain a compacton-like wave solution of parametric expression as follows:
(50)φ=pφ0-qφ01+(pφ0+qφ01)cn(w,k1)p-q+(p+q)cn(w,k1),ξ=2pq(α12α12-1pφ0+qφ01p+qw-(p-q)(φ0-φ01)2(p+q)11112gg111×(Π(sin-1(sn(w,k1)),α12α12-1,k1)11111g111111111+α1βα12α12-1)),
where w is a parameter variable, -w2≤w≤w2, and w2=cn-1(((pφ0-qφ01)/(pφ0+qφ01))·k1), φ0≤φ<0,(
3
) In Cases i (i=5,6) corresponding to phase portraits in Figures 1(e) and 1(f), we obtain a compacton-like wave solution of implicit expression as follows:
(51)f2(φ)=f2(φ3r)e-(1/2)|ξ|,|ξ|≤ξ10,
where
(52)ξ10=2ln⁡f2(φ3r)f2(0).(
4
) In Case 5 corresponding to phase portraits in Figure 1(e), we obtain a compacton-like wave solution of implicit expression as follows:
(53)4-φ0l∘φ0r∘-φ0∘(sn-1(φ0l∘(φ-φ0∘)φ0∘(φ-φ0l∘),k2)-φ0l∘-φ0∘φ0l∘1111111111111×Π(sin-1(φ0l∘(φ-φ0∘)φ0∘(φ-φ0l∘)),α22,k2))=|ξ|,|ξ|≤ξ11,
where φ0∘ is a root of equation F(φ)=0 and
(54)ξ11=4-φ0l∘φ0r∘-φ0∘(sn-1(1,k2)-φ0l∘-φ0∘φ0l∘Π(π2,α22,k2)),α2=φ0∘φ0l∘,k2=(φ0r∘-φ0l∘)φ0∘(φ0r∘-φ0∘)φ0l∘,φ0l∘=16(-8-3φ0∘--144θ-(8+3φ0∘)(-8+9φ0∘)),φ0r∘=16(-8-3φ0∘+-144θ-(8+3φ0∘)(-8+9φ0∘)).(
5
) In Case 6 corresponding to phase portraits in Figure 1(f), we obtain a compacton-like wave solution of implicit expression as follows:
(55)4φ0r∘φ0∘-φ0l∘(sn-1(φ0r∘(φ0∘-φ)φ0∘(φ0r∘-φ),k3)+φ0∘-φ0r∘φ0r∘1111111111f11×Π(sin-1(φ0r∘(φ0∘-φ)φ0∘(φ0r∘-φ)),α32,k3))=|ξ|,|ξ|≤ξ12,
where
(56)ξ12=4φ0r∘φ0∘-φ0l∘(sn-1(1,k3)+φ0∘-φ0r∘φ0r∘Π(π2,α32,k3)),α3=φ0∘φ0r∘,k3=(φ0r∘-φ0l∘)φ0∘(φ0∘-φ0l∘)φ0r∘.(
6
) In Case 16 corresponding to phase portraits in Figure 1(p), we obtain a compacton-like wave solution of implicit expression as follows:
(57)f10(φ)=f10(φ1r)e(1/2)|ξ|,|ξ|≤ξ13,
where
(58)ξ13=2ln⁡f10(0)f10(φ1r),f10(s)=(2a4(s-φ1l)(s-φ1r)+b4(s-φ1)+2a4)μ6(s-φ1)μ6(2(s-φ1l)(s-φ1r)+2s-φ1l-φ1r),μ6=φ1a4,a4=(φ1-φ1l)(φ1-φ1r),b4=2φ1-φ1l+φ1r,φ1l=13(-4-3φ1-28-18θ-3φ1(4+3φ1)),φ1r=13(-4-3φ1+28-18θ-3φ1(4+3φ1)).


## 4.1. Solutions of Kink-Like Wave

(
1
)
In Cases i (i = 1–4) corresponding to phase portraits in Figures 1(a)–1(d), (20) becomes
(22)F(φ)=14(φ3-φ)2(φ2+m1φ+n1),
where m1=(8/3)+2φ3, n1=4θ+(1/3)φ3(16+9φ3). Thus the orbits J1± passing through saddle point (φ3,0) have expressions
(23)y=±(φ3-φ)φ2+m1φ+n12φ,
where 0<φ<φ3. Substituting (23) into dφ/dξ=y and integrating along J1+ and J1- for initial value φ(0)=φ1*, where 0<φ1*<φ3, we obtain two kink-like wave solutions of implicit expression as follows:
(24)f1(φ)=f1(φ1*)e(1/2)ξ,-ξ1<ξ<∞,f1(φ)=f1(φ1*)e-(1/2)ξ,-∞<ξ<ξ1,
where
(25)ξ1=2ln⁡f1(φ1*)f1(0),f1(s)=(2a1s2+m1s+n1+b1(φ3-s)+2a1)μ1(φ3-s)μ1(2s2+m1s+n1+2s+m1),μ1=φ3a1,a1=φ32+m1φ3+n1,b1=-m1-2φ3.The derivations of other kink-like wave solutions are similar to the above case, so we omit the details and only list the results.(
2
) In Cases i (i=5,6) corresponding to phase portraits in Figures 1(e) and 1(f), we obtain two kink-like wave solutions of implicit expression as follows:
(26)f2(φ)=f2(φ1*)e(1/2)ξ,-ξ2<ξ<∞,f2(φ)=f2(φ1*)e-(1/2)ξ,-∞<ξ<ξ2,
where
(27)ξ2=2ln⁡f2(φ1*)f2(0),f2(s)=(2a2(s-φ3l)(s-φ3r)+b2(φ3-s)+2a2)μ2(φ3-s)μ2(2(s-φ3l)(s-φ3r)+2s-φ3l-φ3r),μ2=φ3a2,a2=(φ3-φ3l)(φ3-φ3r),b2=φ3l+φ3r-2φ3,φ3l=13(-4-3φ3-28-18θ-3φ3(4+3φ3)),φ3r=13(-4-3φ3+28-18θ-3φ3(4+3φ3)).(
3
) In Case 7 corresponding to phase portrait in Figure 1(g), we obtain two kink-like wave solutions of implicit expression as follows:
(28)f3(φ)=f3(φ1*)e(1/2)ξ,-ξ3<ξ<∞,f3(φ)=f3(φ1*)e-(1/2)ξ,-∞<ξ<ξ3,
where
(29)ξ3=2ln⁡f3(φ1*)f3(0),f3(s)=(s-φ3∘-s)(φ3(s-φ3∘)+(φ3-φ3∘)s)μ3(s-φ3∘+s)(φ3(s-φ3∘)-(φ3-φ3∘)s)μ3,μ3=φ3φ3-φ3∘,φ3∘=-83-2φ3.(
4
) In Case 11 corresponding to phase portraits in Figure 1(k), we obtain two kink-like wave solutions of implicit expression as follows:
(30)f4(φ)=f4(φ1*)e(1/2)ξ,-∞<ξ<ξ4,f4(φ)=f4(φ1*)e-(1/2)ξ,-ξ4<ξ<∞,
where
(31)ξ4=2ln⁡f4(0)f4(φ1*),f4(s)=(2a2(s-φ3l)(s-φ3r)-b2(s-φ3)+2a2)μ2(s-φ3)μ2(2(s-φ3l)(s-φ3r)+2s-φ3l-φ3r).(
5
) In Case 12 corresponding to phase portraits in Figure 1(l), we obtain two kink-like wave solutions of implicit expression as follows:
(32)f5(φ)=f5(φ1*)+12ξ,-∞<ξ<ξ5,f5(φ)=f5(φ1*)-12ξ,-ξ5<ξ<∞,
where
(33)ξ5=2(f5(0)-f5(φ1*)),f5(s)=2φ3φ3-φ32s-φ32s-φ3-ln⁡(2(s-φ3)(s-φ32)+2s-φ3-φ32),φ3=13(-2+4-6θ),φ32=-83-3φ3.(
6
) In Cases i (i=13,15,16) corresponding to phase portraits in Figures 1(m), 1(o), and 1(p), we obtain two kink-like wave solutions of implicit expression as follows:
(34)f6(φ)=f6(φ1*)e(1/2)ξ,-∞<ξ<ξ6,f6(φ)=f6(φ1*)e-(1/2)ξ,-ξ6<ξ<∞,
where
(35)ξ6=2ln⁡f6(0)f6(φ1*),f6(s)=(2a1s2+m1s+n1-b1(s-φ3)+2a1)μ1(s-φ3)μ1(2s2+m1s+n1+2s+m1).(
7
) In Case 14 corresponding to phase portrait in Figure 1(n), we obtain two kink-like wave solutions of implicit expression as follows:
(36)f7(φ)=f7(φ1*)+12ξ,-∞<ξ<ξ7,f7(φ)=f7(φ1*)-12ξ,-ξ7<ξ<∞,
where
(37)ξ7=2ln⁡f7(0)f7(φ1*),f7(s)=1φ3-φ1ln⁡(s-φ1)φ1(s-φ3)φ3.(
8
) In Case 4 corresponding to phase portrait in Figure 1(d), we obtain two kink-like wave solutions of implicit expression as follows:
(38)f8(φ)=f8(φ2*)e(1/2)ξ,-∞<ξ<ξ8,f8(φ)=f8(φ2*)e-(1/2)ξ,-ξ8<ξ<∞,
where φ1<φ2*<0 and
(39)ξ8=2ln⁡f8(0)f8(φ2*),f8(s)=(φ1∘-s+-s)(φ1(s-φ1∘)-(φ1-φ1∘)s)μ4(φ1∘-s--s)(φ1(s-φ1∘)+(φ1-φ1∘)s)μ4,μ4=φ1φ1-φ1∘,φ1∘=-83-2φ1.(
9
) In Cases i (i = 5–12) corresponding to phase portraits in Figures 1(e)–1(l), we obtain two kink-like wave solutions of implicit expression as follows:
(40)f9(φ)=f9(φ2*)e(1/2)ξ,-∞<ξ<ξ9,f9(φ)=f9(φ2*)e-(1/2)ξ,-ξ9<ξ<∞,
where
(41)ξ9=2ln⁡f9(0)f9(φ2*),f9(s)=(2a3s2+m2s+n2+b3(s-φ1)+2a3)μ5(s-φ1)μ5(2s2+m2s+n2+2s+m2),μ5=φ1a3,a3=φ12+m2φ1+n2,b3=2φ1+m2,m2=83+2φ1,n2=4θ+13φ1(16+9φ1).


## 4.2.  Solutions of Compacton-Like Wave

(
1
) In Cases i (i = 1–4) corresponding to phase portraits in Figures 1(a)–1(d) and φ(0)=φ0, (20) becomes
(42)F(φ)=14(φ0-φ)(φ01-φ)(φ-φ02)(φ-φ¯02),
where φ0 and φ01 are real roots and φ02 and φ¯02 are conjugate complex roots of F(φ)=0 and 0<φ0<φ3<φ01. Thus the orbit K1 has expressions:
(43)y=±(φ0-φ)(φ01-φ)(φ-φ02)(φ-φ¯02)2φ,
where 0<φ≤φ0. By applying transformation dξ=2φdv to dφ/dξ=y, we have
(44)dφdv=2φy.Substituting (43) into (44) and integrating along K1, we get
(45)φ=pφ01-qφ0-(pφ01+qφ0)cn(w,k1)p-q-(p+q)cn(w,k1),|w|≤w1,
where
(46)k1=(p+q)2-(φ0-φ01)24pq,p=(A-φ01)2+B2,q=(A-φ0)2+B2,A=φ02-φ¯022,B=-(φ02-φ¯02)24,w=pqv,w1=cn-1(pφ01-qφ0pφ01+qφ0,k1).Substituting (45) into dξ=2φdv and integrating once, we get
(47)ξ=2pq((α12α12-1)pφ01+qφ0p+qw+(p-q)(φ0-φ01)2(p+q)1111111×(Π(sin-1(sn(w,k1)),α12α12-1,k1)-α1β)),
where
(48)α1=p+qq-p,β=12α12-1k12+(k1′α1)2×ln⁡k12+(k1′α1)2dn(w,k1)+α12-1sn(w,k1)k12+(k1′α1)2dn(w,k1)-α12-1sn(w,k1),k1′=1-k12.Thus we obtain a compacton-like wave solution of parametric expression as follows:(49)φ=pφ01-qφ0-(pφ01+qφ0)cn(w,k1)p-q-(p+q)cn(w,k1),ξ=2pq(pφ01+qφ0p+qw+(p-q)(φ0-φ01)2(p+q)111111×(α12α12-1Π(sin-1(sn(w,k1)),α12α12-1,k1)111111111-α1βα12α12-1)α12α12-1),
where w is a parameter variable and -w1≤w≤w1, 0<φ≤φ0.The derivations of other compacton-like solutions are similar to the above case, so we omit the details and only list the results.(
2
) In Cases i (i = 7–10) corresponding to phase portraits in Figures 1(g)–1(j), we obtain a compacton-like wave solution of parametric expression as follows:
(50)φ=pφ0-qφ01+(pφ0+qφ01)cn(w,k1)p-q+(p+q)cn(w,k1),ξ=2pq(α12α12-1pφ0+qφ01p+qw-(p-q)(φ0-φ01)2(p+q)11112gg111×(Π(sin-1(sn(w,k1)),α12α12-1,k1)11111g111111111+α1βα12α12-1)),
where w is a parameter variable, -w2≤w≤w2, and w2=cn-1(((pφ0-qφ01)/(pφ0+qφ01))·k1), φ0≤φ<0,(
3
) In Cases i (i=5,6) corresponding to phase portraits in Figures 1(e) and 1(f), we obtain a compacton-like wave solution of implicit expression as follows:
(51)f2(φ)=f2(φ3r)e-(1/2)|ξ|,|ξ|≤ξ10,
where
(52)ξ10=2ln⁡f2(φ3r)f2(0).(
4
) In Case 5 corresponding to phase portraits in Figure 1(e), we obtain a compacton-like wave solution of implicit expression as follows:
(53)4-φ0l∘φ0r∘-φ0∘(sn-1(φ0l∘(φ-φ0∘)φ0∘(φ-φ0l∘),k2)-φ0l∘-φ0∘φ0l∘1111111111111×Π(sin-1(φ0l∘(φ-φ0∘)φ0∘(φ-φ0l∘)),α22,k2))=|ξ|,|ξ|≤ξ11,
where φ0∘ is a root of equation F(φ)=0 and
(54)ξ11=4-φ0l∘φ0r∘-φ0∘(sn-1(1,k2)-φ0l∘-φ0∘φ0l∘Π(π2,α22,k2)),α2=φ0∘φ0l∘,k2=(φ0r∘-φ0l∘)φ0∘(φ0r∘-φ0∘)φ0l∘,φ0l∘=16(-8-3φ0∘--144θ-(8+3φ0∘)(-8+9φ0∘)),φ0r∘=16(-8-3φ0∘+-144θ-(8+3φ0∘)(-8+9φ0∘)).(
5
) In Case 6 corresponding to phase portraits in Figure 1(f), we obtain a compacton-like wave solution of implicit expression as follows:
(55)4φ0r∘φ0∘-φ0l∘(sn-1(φ0r∘(φ0∘-φ)φ0∘(φ0r∘-φ),k3)+φ0∘-φ0r∘φ0r∘1111111111f11×Π(sin-1(φ0r∘(φ0∘-φ)φ0∘(φ0r∘-φ)),α32,k3))=|ξ|,|ξ|≤ξ12,
where
(56)ξ12=4φ0r∘φ0∘-φ0l∘(sn-1(1,k3)+φ0∘-φ0r∘φ0r∘Π(π2,α32,k3)),α3=φ0∘φ0r∘,k3=(φ0r∘-φ0l∘)φ0∘(φ0∘-φ0l∘)φ0r∘.(
6
) In Case 16 corresponding to phase portraits in Figure 1(p), we obtain a compacton-like wave solution of implicit expression as follows:
(57)f10(φ)=f10(φ1r)e(1/2)|ξ|,|ξ|≤ξ13,
where
(58)ξ13=2ln⁡f10(0)f10(φ1r),f10(s)=(2a4(s-φ1l)(s-φ1r)+b4(s-φ1)+2a4)μ6(s-φ1)μ6(2(s-φ1l)(s-φ1r)+2s-φ1l-φ1r),μ6=φ1a4,a4=(φ1-φ1l)(φ1-φ1r),b4=2φ1-φ1l+φ1r,φ1l=13(-4-3φ1-28-18θ-3φ1(4+3φ1)),φ1r=13(-4-3φ1+28-18θ-3φ1(4+3φ1)).


## 5.  Conclusion

In this paper, we have found two types of bounded traveling wave solutions for a two-component Fornberg-Whitham equation (see (4)), that is, the compacton-like wave and kink-like wave solutions. Their planar graphs of these solutions are simulated by using software Mathematica (see Figure 2). Meanwhile we reveal two kinds of new phenomena; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively (see Figures 4 and 5). In different Cases i (i = 1–16), the exact implicit or parameter expressions of kink-like wave and compacton-like wave solutions are given.



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*Source: 621918-2014-03-16.xml*