# Frobenius’ Idea Together with Integral Bifurcation Method for Investigating Exact Solutions to a Water Wave Model of the Generalized mKdV Equation

Authors: Weiguo Rui
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

By using Frobenius’ idea together with integral bifurcation method, we study a third order nonlinear equation of generalization form of the modified KdV equation, which is an important water wave model. Some exact traveling wave solutions such as smooth solitary wave solutions, nonsmooth peakon solutions, kink and antikink wave solutions, periodic wave solutions of Jacobian elliptic function type, and rational function solution are obtained. And we show their profiles and discuss their dynamic properties aim at some typical solutions. Though the types of these solutions obtained in this work are not new and they are familiar types, they did not appear in any existing literatures because the equationut+ux+νuxxt+βuxxx + αuux+1/3να(uuxxx+2uxuxx)+3μα2u2ux+νμα2(u2uxxx+ux3+4uuxuxx) + ν2μα2(ux2uxxx+2uxuxx2) = 0 is very complex. Particularly, compared with the cited references, all results obtained in this paper are new.


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## Body

## 1. Introduction

It has come to light that many problems in nonlinear science associated with mechanical, structural, aeronautical, oceanic, electrical, and control systems can be summarized as solving nonlinear partial differential equations (PDEs) which arise from important models with mathematical and physical significances. Finding their exact solutions has extensive applications in many scientific fields such as hydrodynamics, condensed matter physics, solid-state physics, nonlinear optics, neurodynamics, crystal dislocation, model of meteorology, water wave model of oceanography, and fibre-optic communication. The research methods for solving nonlinear PDEs deal with the inverse scattering transformation [1, 2], the Darboux transformation [3–5], the Bäcklund transformation [5–8], the bilinear method and multilinear method [9, 10], the classical and nonclassical Lie group approaches [11, 12], the Clarkson-Kruskal direct method [13–15], the mixing exponential method [16], the geometrical method [17, 18], the truncated Painleve´ expansion [19, 20], the function expansion method (including tanh expansion method [21, 22], sine-cosine expansion method [23, 24], exp-function method [25], and multiple exp-function method [26]), the bifurcation theory of planar dynamical system [27, 28], the F-expansion type method [29, 30], G′/G method [31, 32], and the integral bifurcation method [33–36]. Among these available methods for solving nonlinear PDEs, some of them employed Frobenius’ idea directly or indirectly. Frobenius’ idea is aso called Frobenius’ integrable decompositions [37]; it can reduce a partial differential equation (PDE) to an ordinary differential equation (ODE) under investigation for solution. Indeed, the F-expansion type methods indirectly employed Frobenius’ idea; crucial points of this method are to choose integrable ODE to start investigation for solution. In fact, the tanh function method and G′/G method are special cases of such an idea or general Frobenius’ idea. Direct Frobenius’ idea was also used to establish the transformed rational function method [38] and to solve the KPP equation [39].In this paper, we will employ Frobenius’ idea together with integral bifurcation method to investigate exact traveling wave solutions of the following integrable generalization of the modified KdV equation:(1)ut+ux+νuxxt+βuxxx+αuux+13ναuuxxx+2uxuxx+3μα2u2ux+νμα2u2uxxx+ux3+4uuxuxx+ν2μα2ux2uxxx+2uxuxx2=0,
where α,β,μ, and ν are constants and 0<α<1. The model (1) comes from the physical and asymptotic considerations via the methodology introduced by Fokas [40] in 1995; it can be regarded as a water wave model to describe the motion of water wave. It is worth to point out that the special case of (1),
(2)ut+ux+νuxxt+βuxxx+αuux+13ναuuxxx+2uxuxx+3μα2u2ux=0,
is also an important physical model. The above two equations were studied by many authors. Equation (2) was introduced by Fuchssteiner and Fokas in their previous works [41, 42] in 1981. The Lax pairs of (2) were given by Fokas in [40]. New Lax pairs and Darboux transformation of (2) were introduced by Yang and Rui in [43] recently. In [44], by using the bifurcation theory of dynamical system, the existence conditions of different kinds of traveling wave solutions of (2) were presented by Bi. In [45], by using the same method, Li and Zhang studied (1), the existence of solitary wave, kink and antikink wave solutions, uncountably infinite many smooth, and nonsmooth periodic wave solutions were discussed. However, exact travelling wave solutions of (1) were not obtained in [45] though the authors obtained some results of numerical simulation for smooth and nonsmooth periodic wave solutions in this work. Moreover, the investigations on exact solutions of (1) are few in the existing literatures because (1) is more complex than (2). Therefore, in this paper, employing Frobenius’ idea together with integral bifurcation method, we will investigate different kinds of exact traveling wave solutions of (1).The rest of this paper is organized as follows. In Section2, by using Frobenius’ idea, we will derive ordinary differential equation (ODE) which is equivalent to (1). In Section 3, by using the integral bifurcation method combined with factoring technique, we will investigate different kinds of exact traveling wave solutions of (1) and discuss their dynamic properties when the integral constants satisfy different conditions. In Section 4, we will discuss different kinds of exact traveling wave solutions of (1) under the special case of the parameter ν=0.


## 2. Direct Application of Frobenius’ Idea on Reducing the PDE (1) to an Integrable ODE

Frobenius’ idea is about changing a partial differential equation (PDE) into an ordinary differential equation (ODE) and then using integrable decomposition method to investigate its exact solutions. Thus, in this section, we first employ the direct Frobenius’ idea to change (1) into an integrable ordinary differential equation; see the following discussions.Making a traveling wave transformationu(x,t)=u(ξ) with ξ=x-ct, (1) can be reduced to the following ordinary differential equation (ODE):
(3)1-cdudξ+β-cνd3udξ3+αududξ+13ναud3udξ3+2dudξd2udξ2+3μα2u2dudξ+νμα2u2d3udξ3+dudξ3+4ududξd2udξ2+μν2α2dudξ2d3udξ3+2dudξd2udξ22=0,
where c is wave velocity which moves along the direction of x-axis and c≠0. Equation (3) can be rewritten as
(4)1-cdudξ+β-cνd3udξ3+12αdu2dξ+13ναddξud2udξ2+12dudξ2+μα2du3dξ+νμα2ddξu2d2udξ2+ududξ2+μν2α2ddξdudξ2d2udξ2=0.
Integrating (4) once, we obtain
(5)1-cu+β-cνd2udξ2+12αu2+13ναud2udξ2+12dudξ2+μα2u3+νμα2u2d2udξ2+ududξ2+μν2α2dudξ2d2udξ2=g,
where g is an integral constant. Employing direct Frobenius’ idea, we need not change (5) into a 2-dimensional planar system as the method in [33–36]. we can directly integrate (5) again; see the following calculus.Multiplyingdu/dξ to the both sides of (5) yields
(6)1-cududξ+β-cνdudξd2udξ2+12αu2dudξ+13ναududξd2udξ2+12dudξ3+μα2u3dudξ+νμα2u2dudξd2udξ2+ududξ3+μν2α2dudξ3d2udξ2=gdudξ.
Integrating (6) once, we obtain
(7)-gu+12(1-c)u2+12(β-cν)dudξ2+16αu3+16ναududξ2+14μα2u4+14νμ2α2dudξ4+12νμα2u2dudξ2=h,
where h is another arbitrary integral constant. When ν≠0, (7) can be rewritten as
(8)-4hν-4gνu+21-cνu2+2α3νu3+μα2νu4+2β-cννdudξ2+2α3ududξ2+2μα2u2dudξ2+μ2α2dudξ4=0.


## 3. Different Kinds of Exact Traveling Wave Solutions of (1)

In this section, by using the integral bifurcation method combined with factoring technique as in [36], we will investigate different kinds of exact traveling wave solutions of (1) and discuss their dynamic properties via (7) and (8).


### 3.1. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants g≠0 and h≠0

Whenν(ν-μ)>0 and h=(324νμ2(c-1)2 + μ648cν2-648νβ+324β2+36cν-36ν-324c2ν2 + ν + 36νβ-cν)/1296ν(μ-ν)α2μ2≠0, g=(18μ(c-1)+1)/108αμ2≠0, (8) can be decomposed in the following form:
(9)r0+r1u+r2u2+μ2dudξ2×s0+s1u+s2u2+α2dudξ2=0,
where the coefficients r0,r1,r2,s0,s1, and s2 are defined by the following expressions:
(10)r0=νcν-β+μ1-c-1/18-cν-βνν-μνα2νν-μ,r1=μ-ν+νν-μ3ανν-μ,r2=1-μννν-μ,s0=-νcν-β+μ1-c-1/18+cν-βνν-μνμ2νν-μ,s1=αν-μ+νν-μ3μ2νν-μ,s2=α2μ-α2μννν-μ.
Thus (9) can be reduced to the following two ordinary differential equations:
(11)r0+r1u+r2u2+μ2dudξ2=0,
or
(12)s0+s1u+s2u2+α2dudξ2=0,
where the coefficients r0,r1,r2,s0,s1, and s2 are given by (10).Solving (11) under the conditions r2<0 and q=4r0r2-r12>0, we obtain two hyperbolic function solutions of (1) as follows:
(13)u=±12r2qsinh⁡-r2μ(x-ct)∓r1.Solving (11) under the conditions r2>0 and q=4r0r2-r12<0, we obtain two periodic wave solutions of (1) as follows:
(14)u=∓12r2-qsinr2μx-ct±r1.Similarly, solving (12) under the conditions s2<0 and q~=4s0s2-s12>0, we obtain two hyperbolic function solutions of (1) as follows:
(15)u=±12s2q~sinh⁡-s2αx-ct∓s1.Solving (12) under the conditions s2>0 and q~=4s0s2-s12<0, we obtain two periodic wave solutions of (1) as follows:
(16)u=∓12s2-q~sins2αx-ct±s1.


### 3.2.  Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants g=0 and h≠0

Whenν2-μν>0 and g=0,h=(18μν-18μβ-ν)2/1296να2μ3(ν-μ)≠0, c=1-1/18μ, (8) can be decomposed in the following form:
(17)μν-μβ-ν/18νμ2ν-μ+ν2-μν+αμ-ν+ν2-μν3μ2ν2-μνu+α2μ1-ν2-μννu2+α2dudξ2×μν-μβ-ν/18να2ν-μ-ν2-μν+ν-μ+ν2-μν3αν2-μνu+μ+μνν2-μνu2+μ2dudξ2μν-μβ-ν/18να2ν-μ-ν2-μν=0.
Equation (17) can be reduced to the following two ordinary differential equations:
(18)μν-μβ-ν/18νμ2ν-μ+ν2-μν+αμ-ν+ν2-μν3μ2ν2-μνu+α2μ1-ν2-μννu2+α2dudξ2=0
or
(19)μν-μβ-ν/18να2ν-μ-ν2-μν+ν-μ+ν2-μν3αν2-μνu+μ+μνν2-μνu2+μ2dudξ2μν-μβ-ν/18να2ν-μ-ν2-μν=0.
Solving (18), we obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows:
(20)u=±μν2α2ν-ν2-μν×αΩ13μνsinh⁡ω1ξ∓αμ-ν+ν2-μν3μ2ν2-μν,(21)u=∓μν2α2ν-ν2-μν×αΩ23μνsinω2ξ±αμ-ν+ν2-μν3μ2ν2-μν,
where the Ω1,Ω2,ω1,ω2, and ξ are defined by(22)Ω1=36μβ-ν+νν2-μν+36μνν-β+ν2-μνμν-μ+ν2-μν,ω1=ν2-μν-νμν,Ω2=-[36μ(β-ν)+ν]ν2-μν+[36μν(ν-β)+ν2-μν]μ(ν-μ+ν2-μν),ω2=ν-ν2-μνμν,ξ=x-1-118μt.Similarly solving (19), we also obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows:
(23)u=±ν2μν+ν2-μν×μΩ33ανsinh⁡ω3ξ∓ν-μ+ν2-μν3αν2-μν,(24)u=∓ν2μν+ν2-μν×μΩ43ανsinω4ξ±ν-μ+ν2-μν3αν2-μν,
where the Ω3,Ω4,ω3,ω4, and ξ are defined by
(25)Ω3=[36(ν-β)-ν]ν2-μν+[36ν(ν-β)-ν2/μ-ν]ν-μ-ν2-μν,ω3=1μ-μ+μνν2-μν,Ω4=-[36(ν-β)-ν]ν2-μν+[36ν(ν-β)-ν2/μ-ν]ν-μ-ν2-μν,ω4=1μμ+μνν2-μν,ξ=x-1-118μt.


### 3.3. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants g≠0 and h=0

Whenν2-μν>0 and g=(ν+18μβ-ν)/108αμ2ν2-μν≠0, h=0, c=(ν+18ν(β-μ)+18ν2-μν)/18ν(ν-μ+ν2-μν), (8) can be decomposed in the following form:
(26)18μ(ν-β)-ν9νμ2ν-μ+ν2-μν+αμ-ν+ν2-μν3μ2ν2-μνu+α2ν-ν2-μννμu2+α2dudξ2×ν-μ+ν2-μν3αν2-μνu+1+μνν2-μνu2+μ2dudξ2ν-μ+ν2-μν3αν2-μν=0.
Similarly, solving (26) we obtain four hyperbolic function solutions and four periodic wave solutions of (1) as follows:
(27)u=±μν2α2ν-ν2-μν×αΩ53μνsinh⁡ω5ξ±αν-μ-ν2-μν3μ2ν2-μν,(28)u=∓μν2α2ν-ν2-μν×αΩ63μνsinω6ξ∓αν-μ-ν2-μν3μ2ν2-μν,
where the Ω5,Ω6,ω5,ω6, and ξ are defined by(29)Ω5=72μνν-β-3ν2-μν+72μβ-ν+3νν2-μνμν-μ+ν2-μν,ω5=ν-ν2-μνμν,Ω6=-72μν(ν-β)-3ν2-μν+72μβ-ν+3νν2-μνμν-μ+ν2-μν,ω6=ν2-μν-νμν,ξ=x-ν+18νβ-μ+18ν2-μν18νν-μ+ν2-μνt,(30)u=±ν2-μν+νν2-μν6αμν+ν2-μνν2-μν×sinh⁡-ν+ν2-μνμνξ∓1,(31)u=∓ν2-μν+νν2-μν6αμν+ν2-μνν2-μν×sinν+ν2-μνμνξ±1,where ξ=x-((ν+18ν(β-μ)+18ν2-μν)/18ν(ν-μ+ν2-μν))t.


### 3.4. Hyperbolic Function Solutions, Periodic Wave Solutions, and Rational Function Solution of (1) as the Two Integral Constants g=h=0

Whenμ=ν, β>ν and g=h=0, c=β/ν, (8) can be decomposed in the following form:
(32)α3ν21+1+18β-νu+α2νu2+α2dudξ2×1-1+18β-ν3αu+νu2+μ2dudξ2=0.
Solving (32) we obtain four hyperbolic function solutions and four periodic wave solutions of (1) as follows:
(33)u=±1+1+18β-ν6ανsinh⁡ξ-ν∓1,ν<0<β,(34)u=∓1+1+18β-ν6ανsinξν±1,β>ν>0,(35)u=∓1+1+18β-ν6ανsinh⁡-νξμ±1,ν<0<β,(36)u=±1+1+18β-ν6ανsinνξμ∓1,β>ν>0,
where ξ=x-β/νt.Whenμ=-ν/18(β-ν), β>ν and g=h=0, c=β/ν, (8) can be decomposed in the following form:
(37)Au-ανA6β-νu2+α2dudξ2×Bu-νB6β-νu2+μ2dudξ2=0,
where A=6αβ-ν18β-ν-336(β-ν)2+2(β-ν)/ν2, B=(18(β-ν)+336(β-ν)2+2(β-ν))/54α(β-ν).Solving (37) we obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows:
(38)u=6β-νανcos⁡212νA6αν-βξ,0<β<ν,A>0,(39)u=6β-νανcosh⁡212νA6αβ-νξ,ν<0<β,A<0,(40)u=6(β-ν)νcos⁡212μνB6(ν-β)ξ,0<β<ν,B>0,(41)u=6β-ννcosh⁡212μνB6β-νξ,β>ν>0,B>0,
where ξ=x-β/νt and A,B have been given above.Whenμ=0 and g=h=0, c=β/ν, (8) can be decomposed in the following form:
(42)3αν-βν2u+α2νu2+α2dudξ2×23αu+μ2dudξ2=0.
Solving (42), we obtain a hyperbolic function solution, a periodic wave solution, and a rational function solution as follows:
(43)u=3β-νανcos⁡212νξ,ν>0,(44)u=3β-νανcosh⁡212-νξ,ν<0,(45)u=-16αμ2ξ2,
where ξ=x-β/νt and A, B have been given above.All the above exact solutions which were obtained by us are smooth travelling wave solutions including smooth periodic wave solutions and smooth hyperbolic function solutions. In order to show the dynamical profiles of periodic wave solutions, as examples, we plot the graphs of solutions (14) and (38) for α=0.5,β=0.8,μ=1.2,t=0.1, which are shown in Figures 1(a) and 1(b).The graphs of profiles for the smooth periodic wave solutions defined by (14).
(a)
Periodic wave defined by (14)
(b)
Periodic wave defined by (38)


### 3.5. Peakon Solutions under Some Special Parametric Condition

The expression in the right side of (8) cannot be reduced to a form of au2-bdu/dξ22=0 by using the factoring technique because this equation contains the terms -4h/ν-4g/νu+2α/3νu3 + 2α/3udu/dξ2. Thus, the peakon solutions of (1) such as Cammasa-Holm’s form re-δ|ξ| cannot be obtained by direct integral method together with factoring technique as in [36]. However, the research works given by Li et al. in [45] show that the peakon solutions of (1) exist though they did not obtain exact peakon solutions of this equation. Indeed, the existence of peakon solution of (1) is proved by Li via analysis of phase portraits in this paper. We notice that the terms u3 and udu/dξ2 are kindred terms when u=re-δ|ξ|. Thus we assume that (1) has peakon solutions of the form u=re-δ|ξ| or u=s+re-δ|ξ|.Wheng≠0 and h≠0, we suppose that (8) has a peakon solution as the following form:
(46)u=s+re-δξ,
where the parameters s,r, and δ can be determined further in the below discussions. We define s+re-δ|ξ|=s+rexp⁡(-δξ2) when ξ≠0; particularly u=s+r=(constant) when ξ=0, and obviously u=s+r is a constant solution which satisfies (1).Whenξ≠0, substituting (46) (i.e. u=s+rexp⁡(-δξ2)) into (8) we obtain
(47)C0+C1exp⁡-δξ2+C2exp⁡-2δξ2+C3exp⁡-3δξ2+C4exp⁡-4δξ2=0,
where coefficients C0,C1,C2,C3, and C4 satisfy
(48)C0=-4hν-4gsν+2αs33ν+21-cs2ν+μα2s4ν,C1=41-crsν+2αrsν-4grν+4μα2rs3ν,C2=2(1-c)r2ν+2αr2sν+6μα2r2s2ν+2μα2r2s2δ2+2β-cνr2δ2ν+2αr2sδ23,C3=μα2r4ν+2μα2r4δ2+μ2α2r4δ4,C4=μα2r4+2μα2r4δ2+μ2α2r4δ4ν.
Let the coefficients of every terms of exp-function (including the term of constant) in (47) as zero; it follows
(49)C0=0,C1=0,C2=0,C3=0,C4=0.Solving the above group of equations yields(50)c=(ν2-3μν+36μ2ν-36βμν)+(ν-36βμ)ν2-μν36μν(μ-ν-ν2-μν),g=-ν2+μν-36μν2+36βμν+ν+36βμ-36μνν2-μν216μ2νμ-ν-ν2-μν,h=-3ν2+μν-72μν2+72βμν+3ν+72βμ-72μνν2-μν5184μ3να2μ-ν-ν2-μν,r=rfreeparameter,s=-16αμ,δ=ν2-μν-νμν,where ν<0<μ. Thus, when the constants c,g, and h satisfy the above conditions, (1) has a peakon solution as follows:
(51)u=-16αμ+rexp⁡-ν2-μν-νμνx-ct,
where r is an arbitrary nonzero constant and c is given above.Wheng=h=0 and β can be regarded as a free parameter, we suppose that (8) has a peakon solution as the following form:
(52)u=r~e-δ~ξ,
where the parameters r~,δ~ can be determined further in the below discussions.Whenξ≠0, substituting (52) (i.e., u=r~e-δ~ξ2) into (8) we obtain
(53)Aexp⁡-2δξ2+Bexp⁡-3δξ2+Cexp⁡-4δξ2=0,
where coefficients A, B, and C satisfy
(54)A=21-cr~2ν+2β-cνr~2δ~2ν,B=2αr~33ν+23αr~3δ~2,C=μα2r~4ν+2μα2r~4δ~2+μ2α2r~4δ~4.
Let the coefficients of every terms of exp-function in (53) as zero; it follows
(55)A=0,B=0,C=0.
Solving the above group of equations yields
(56)β=μ=ν<0,δ~=1-ν,r~=r~,c=c.
Thus, when the constants g=h=0 and β=μ=ν, δ~=1/-ν, (1) has a peakon solution as follows:
(57)u=r~exp⁡-1-νx-ct,
where r~ and c are arbitrary nonzero constants.In order to show the dynamical profiles of peakon solutions (51) and (57), we plot the graphs of them for α=0.5,β=0.8,μ=0.2,r=5,c=4,t=0.1, which are shown in Figures 2(a) and 2(b).The graphs of profiles for the nonsmooth peakon solutions defined by (51) and (57) under different parametric values: (a) ν=1.4; (b) ν=-1.4.
(a)
Peakon wave defined by (51)
(b)
Peakon wave defined by (57)


## 3.1. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants g≠0 and h≠0

Whenν(ν-μ)>0 and h=(324νμ2(c-1)2 + μ648cν2-648νβ+324β2+36cν-36ν-324c2ν2 + ν + 36νβ-cν)/1296ν(μ-ν)α2μ2≠0, g=(18μ(c-1)+1)/108αμ2≠0, (8) can be decomposed in the following form:
(9)r0+r1u+r2u2+μ2dudξ2×s0+s1u+s2u2+α2dudξ2=0,
where the coefficients r0,r1,r2,s0,s1, and s2 are defined by the following expressions:
(10)r0=νcν-β+μ1-c-1/18-cν-βνν-μνα2νν-μ,r1=μ-ν+νν-μ3ανν-μ,r2=1-μννν-μ,s0=-νcν-β+μ1-c-1/18+cν-βνν-μνμ2νν-μ,s1=αν-μ+νν-μ3μ2νν-μ,s2=α2μ-α2μννν-μ.
Thus (9) can be reduced to the following two ordinary differential equations:
(11)r0+r1u+r2u2+μ2dudξ2=0,
or
(12)s0+s1u+s2u2+α2dudξ2=0,
where the coefficients r0,r1,r2,s0,s1, and s2 are given by (10).Solving (11) under the conditions r2<0 and q=4r0r2-r12>0, we obtain two hyperbolic function solutions of (1) as follows:
(13)u=±12r2qsinh⁡-r2μ(x-ct)∓r1.Solving (11) under the conditions r2>0 and q=4r0r2-r12<0, we obtain two periodic wave solutions of (1) as follows:
(14)u=∓12r2-qsinr2μx-ct±r1.Similarly, solving (12) under the conditions s2<0 and q~=4s0s2-s12>0, we obtain two hyperbolic function solutions of (1) as follows:
(15)u=±12s2q~sinh⁡-s2αx-ct∓s1.Solving (12) under the conditions s2>0 and q~=4s0s2-s12<0, we obtain two periodic wave solutions of (1) as follows:
(16)u=∓12s2-q~sins2αx-ct±s1.


## 3.2.  Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants g=0 and h≠0

Whenν2-μν>0 and g=0,h=(18μν-18μβ-ν)2/1296να2μ3(ν-μ)≠0, c=1-1/18μ, (8) can be decomposed in the following form:
(17)μν-μβ-ν/18νμ2ν-μ+ν2-μν+αμ-ν+ν2-μν3μ2ν2-μνu+α2μ1-ν2-μννu2+α2dudξ2×μν-μβ-ν/18να2ν-μ-ν2-μν+ν-μ+ν2-μν3αν2-μνu+μ+μνν2-μνu2+μ2dudξ2μν-μβ-ν/18να2ν-μ-ν2-μν=0.
Equation (17) can be reduced to the following two ordinary differential equations:
(18)μν-μβ-ν/18νμ2ν-μ+ν2-μν+αμ-ν+ν2-μν3μ2ν2-μνu+α2μ1-ν2-μννu2+α2dudξ2=0
or
(19)μν-μβ-ν/18να2ν-μ-ν2-μν+ν-μ+ν2-μν3αν2-μνu+μ+μνν2-μνu2+μ2dudξ2μν-μβ-ν/18να2ν-μ-ν2-μν=0.
Solving (18), we obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows:
(20)u=±μν2α2ν-ν2-μν×αΩ13μνsinh⁡ω1ξ∓αμ-ν+ν2-μν3μ2ν2-μν,(21)u=∓μν2α2ν-ν2-μν×αΩ23μνsinω2ξ±αμ-ν+ν2-μν3μ2ν2-μν,
where the Ω1,Ω2,ω1,ω2, and ξ are defined by(22)Ω1=36μβ-ν+νν2-μν+36μνν-β+ν2-μνμν-μ+ν2-μν,ω1=ν2-μν-νμν,Ω2=-[36μ(β-ν)+ν]ν2-μν+[36μν(ν-β)+ν2-μν]μ(ν-μ+ν2-μν),ω2=ν-ν2-μνμν,ξ=x-1-118μt.Similarly solving (19), we also obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows:
(23)u=±ν2μν+ν2-μν×μΩ33ανsinh⁡ω3ξ∓ν-μ+ν2-μν3αν2-μν,(24)u=∓ν2μν+ν2-μν×μΩ43ανsinω4ξ±ν-μ+ν2-μν3αν2-μν,
where the Ω3,Ω4,ω3,ω4, and ξ are defined by
(25)Ω3=[36(ν-β)-ν]ν2-μν+[36ν(ν-β)-ν2/μ-ν]ν-μ-ν2-μν,ω3=1μ-μ+μνν2-μν,Ω4=-[36(ν-β)-ν]ν2-μν+[36ν(ν-β)-ν2/μ-ν]ν-μ-ν2-μν,ω4=1μμ+μνν2-μν,ξ=x-1-118μt.


## 3.3. Hyperbolic Function Solutions and Periodic Wave Solutions of (1) as the Two Integral Constants g≠0 and h=0

Whenν2-μν>0 and g=(ν+18μβ-ν)/108αμ2ν2-μν≠0, h=0, c=(ν+18ν(β-μ)+18ν2-μν)/18ν(ν-μ+ν2-μν), (8) can be decomposed in the following form:
(26)18μ(ν-β)-ν9νμ2ν-μ+ν2-μν+αμ-ν+ν2-μν3μ2ν2-μνu+α2ν-ν2-μννμu2+α2dudξ2×ν-μ+ν2-μν3αν2-μνu+1+μνν2-μνu2+μ2dudξ2ν-μ+ν2-μν3αν2-μν=0.
Similarly, solving (26) we obtain four hyperbolic function solutions and four periodic wave solutions of (1) as follows:
(27)u=±μν2α2ν-ν2-μν×αΩ53μνsinh⁡ω5ξ±αν-μ-ν2-μν3μ2ν2-μν,(28)u=∓μν2α2ν-ν2-μν×αΩ63μνsinω6ξ∓αν-μ-ν2-μν3μ2ν2-μν,
where the Ω5,Ω6,ω5,ω6, and ξ are defined by(29)Ω5=72μνν-β-3ν2-μν+72μβ-ν+3νν2-μνμν-μ+ν2-μν,ω5=ν-ν2-μνμν,Ω6=-72μν(ν-β)-3ν2-μν+72μβ-ν+3νν2-μνμν-μ+ν2-μν,ω6=ν2-μν-νμν,ξ=x-ν+18νβ-μ+18ν2-μν18νν-μ+ν2-μνt,(30)u=±ν2-μν+νν2-μν6αμν+ν2-μνν2-μν×sinh⁡-ν+ν2-μνμνξ∓1,(31)u=∓ν2-μν+νν2-μν6αμν+ν2-μνν2-μν×sinν+ν2-μνμνξ±1,where ξ=x-((ν+18ν(β-μ)+18ν2-μν)/18ν(ν-μ+ν2-μν))t.


## 3.4. Hyperbolic Function Solutions, Periodic Wave Solutions, and Rational Function Solution of (1) as the Two Integral Constants g=h=0

Whenμ=ν, β>ν and g=h=0, c=β/ν, (8) can be decomposed in the following form:
(32)α3ν21+1+18β-νu+α2νu2+α2dudξ2×1-1+18β-ν3αu+νu2+μ2dudξ2=0.
Solving (32) we obtain four hyperbolic function solutions and four periodic wave solutions of (1) as follows:
(33)u=±1+1+18β-ν6ανsinh⁡ξ-ν∓1,ν<0<β,(34)u=∓1+1+18β-ν6ανsinξν±1,β>ν>0,(35)u=∓1+1+18β-ν6ανsinh⁡-νξμ±1,ν<0<β,(36)u=±1+1+18β-ν6ανsinνξμ∓1,β>ν>0,
where ξ=x-β/νt.Whenμ=-ν/18(β-ν), β>ν and g=h=0, c=β/ν, (8) can be decomposed in the following form:
(37)Au-ανA6β-νu2+α2dudξ2×Bu-νB6β-νu2+μ2dudξ2=0,
where A=6αβ-ν18β-ν-336(β-ν)2+2(β-ν)/ν2, B=(18(β-ν)+336(β-ν)2+2(β-ν))/54α(β-ν).Solving (37) we obtain two hyperbolic function solutions and two periodic wave solutions of (1) as follows:
(38)u=6β-νανcos⁡212νA6αν-βξ,0<β<ν,A>0,(39)u=6β-νανcosh⁡212νA6αβ-νξ,ν<0<β,A<0,(40)u=6(β-ν)νcos⁡212μνB6(ν-β)ξ,0<β<ν,B>0,(41)u=6β-ννcosh⁡212μνB6β-νξ,β>ν>0,B>0,
where ξ=x-β/νt and A,B have been given above.Whenμ=0 and g=h=0, c=β/ν, (8) can be decomposed in the following form:
(42)3αν-βν2u+α2νu2+α2dudξ2×23αu+μ2dudξ2=0.
Solving (42), we obtain a hyperbolic function solution, a periodic wave solution, and a rational function solution as follows:
(43)u=3β-νανcos⁡212νξ,ν>0,(44)u=3β-νανcosh⁡212-νξ,ν<0,(45)u=-16αμ2ξ2,
where ξ=x-β/νt and A, B have been given above.All the above exact solutions which were obtained by us are smooth travelling wave solutions including smooth periodic wave solutions and smooth hyperbolic function solutions. In order to show the dynamical profiles of periodic wave solutions, as examples, we plot the graphs of solutions (14) and (38) for α=0.5,β=0.8,μ=1.2,t=0.1, which are shown in Figures 1(a) and 1(b).The graphs of profiles for the smooth periodic wave solutions defined by (14).
(a)
Periodic wave defined by (14)
(b)
Periodic wave defined by (38)


## 3.5. Peakon Solutions under Some Special Parametric Condition

The expression in the right side of (8) cannot be reduced to a form of au2-bdu/dξ22=0 by using the factoring technique because this equation contains the terms -4h/ν-4g/νu+2α/3νu3 + 2α/3udu/dξ2. Thus, the peakon solutions of (1) such as Cammasa-Holm’s form re-δ|ξ| cannot be obtained by direct integral method together with factoring technique as in [36]. However, the research works given by Li et al. in [45] show that the peakon solutions of (1) exist though they did not obtain exact peakon solutions of this equation. Indeed, the existence of peakon solution of (1) is proved by Li via analysis of phase portraits in this paper. We notice that the terms u3 and udu/dξ2 are kindred terms when u=re-δ|ξ|. Thus we assume that (1) has peakon solutions of the form u=re-δ|ξ| or u=s+re-δ|ξ|.Wheng≠0 and h≠0, we suppose that (8) has a peakon solution as the following form:
(46)u=s+re-δξ,
where the parameters s,r, and δ can be determined further in the below discussions. We define s+re-δ|ξ|=s+rexp⁡(-δξ2) when ξ≠0; particularly u=s+r=(constant) when ξ=0, and obviously u=s+r is a constant solution which satisfies (1).Whenξ≠0, substituting (46) (i.e. u=s+rexp⁡(-δξ2)) into (8) we obtain
(47)C0+C1exp⁡-δξ2+C2exp⁡-2δξ2+C3exp⁡-3δξ2+C4exp⁡-4δξ2=0,
where coefficients C0,C1,C2,C3, and C4 satisfy
(48)C0=-4hν-4gsν+2αs33ν+21-cs2ν+μα2s4ν,C1=41-crsν+2αrsν-4grν+4μα2rs3ν,C2=2(1-c)r2ν+2αr2sν+6μα2r2s2ν+2μα2r2s2δ2+2β-cνr2δ2ν+2αr2sδ23,C3=μα2r4ν+2μα2r4δ2+μ2α2r4δ4,C4=μα2r4+2μα2r4δ2+μ2α2r4δ4ν.
Let the coefficients of every terms of exp-function (including the term of constant) in (47) as zero; it follows
(49)C0=0,C1=0,C2=0,C3=0,C4=0.Solving the above group of equations yields(50)c=(ν2-3μν+36μ2ν-36βμν)+(ν-36βμ)ν2-μν36μν(μ-ν-ν2-μν),g=-ν2+μν-36μν2+36βμν+ν+36βμ-36μνν2-μν216μ2νμ-ν-ν2-μν,h=-3ν2+μν-72μν2+72βμν+3ν+72βμ-72μνν2-μν5184μ3να2μ-ν-ν2-μν,r=rfreeparameter,s=-16αμ,δ=ν2-μν-νμν,where ν<0<μ. Thus, when the constants c,g, and h satisfy the above conditions, (1) has a peakon solution as follows:
(51)u=-16αμ+rexp⁡-ν2-μν-νμνx-ct,
where r is an arbitrary nonzero constant and c is given above.Wheng=h=0 and β can be regarded as a free parameter, we suppose that (8) has a peakon solution as the following form:
(52)u=r~e-δ~ξ,
where the parameters r~,δ~ can be determined further in the below discussions.Whenξ≠0, substituting (52) (i.e., u=r~e-δ~ξ2) into (8) we obtain
(53)Aexp⁡-2δξ2+Bexp⁡-3δξ2+Cexp⁡-4δξ2=0,
where coefficients A, B, and C satisfy
(54)A=21-cr~2ν+2β-cνr~2δ~2ν,B=2αr~33ν+23αr~3δ~2,C=μα2r~4ν+2μα2r~4δ~2+μ2α2r~4δ~4.
Let the coefficients of every terms of exp-function in (53) as zero; it follows
(55)A=0,B=0,C=0.
Solving the above group of equations yields
(56)β=μ=ν<0,δ~=1-ν,r~=r~,c=c.
Thus, when the constants g=h=0 and β=μ=ν, δ~=1/-ν, (1) has a peakon solution as follows:
(57)u=r~exp⁡-1-νx-ct,
where r~ and c are arbitrary nonzero constants.In order to show the dynamical profiles of peakon solutions (51) and (57), we plot the graphs of them for α=0.5,β=0.8,μ=0.2,r=5,c=4,t=0.1, which are shown in Figures 2(a) and 2(b).The graphs of profiles for the nonsmooth peakon solutions defined by (51) and (57) under different parametric values: (a) ν=1.4; (b) ν=-1.4.
(a)
Peakon wave defined by (51)
(b)
Peakon wave defined by (57)


## 4. Different Kinds of Exact Solutions under the Special Caseν=0

Under the special case of parameterν=0, (7) can be rewritten as
(58)-h-gu+121-cu2+16αu3+14μα2u4+12βdudξ2=0.
Solving (58) in different kinds of parametric conditions, we obtain different kinds of exact traveling wave solutions including solitary wave solutions and kink wave solutions; see the below discussions.


### 4.1. Different Kinds of Exact Traveling Wave Solutions for the Constantsg=h=0

When two integral constantsg,h are both zero (i.e., g=h=0), (58) can be reduced to
(59)dudξ2=c-1βu2-α3βu3-μα22βu4.
Under different parametric conditions solving (59), we obtain a series of exact traveling wave solutions as follows:
(60)u=6c-1sech21/2(c-1)/βξ2α+9αμc-11±tanh1/2(c-1)/βξ2,
where c>1, β>0, and ξ=x-ct:
(61)u=6c-1α1-18μ1-c-1sinh⁡(c-1)/βξ,
where c>1, β>0, μ<-1/18(c-1), and ξ=x-ct:
(62)u=6(c-1)α1+18μ(c-1)+1sin(1-c)/βξ,
where c<1, β>0, μ<-1/18(c-1), and ξ=x-ct:
(63)u=6(c-1)α1-18μ(c-1)+1cosh⁡(c-1)/βξ,
where c>1, β>0, μ>-1/18(c-1), and ξ=x-ct:
(64)u=6(c-1)α1-18μ(c-1)+1cos⁡(1-c)/βξ,
where c<1, β>0, μ>-1/18(c-1), and ξ=x-ct:
(65)u=3(c-1)×αcosh⁡212c-1βξ-3μ1-c2sinh⁡c-1βξ-1,
where c>1, β>0, μ<0, and ξ=x-ct:
(66)u=3c-1×αcos⁡212c-1βξ+3μc-12sin1-cβξ-1,
where c<1,β>0,μ<0, and ξ=x-ct:(67)u=-16αμ1±tanh112-2βμξ
or
(68)u=-16αμ1±coth⁡112-2βμξ,
where β>0,μ<0, and ξ=x-(1-1/18μ)t.Among these exact travelling wave solutions obtained in Section4.1, it is worth pointing that the solutions (60) and (63) describe smooth solitary waves and the solution (67) describes kink wave and antikink wave (Figure 4). In order to show the dynamic profiles of solutions (60) and (63), we plot their graphs for α=0.5,β=0.8,c=5, and t=0.1, which are shown in Figures 3(a) and 3(b).The graphs of profiles for smooth solitary wave solutions defined by (60) and (63) under different parametric values: (a) μ=0.7; (b) μ=2.
(a)
Bright solitary wave defined by (60)
(b)
Dark solitary wave defined by (63)The graphs of profiles for kink wave and antikink wave solutions defined by (67) with “+” and “−”.
(a)
Kink wave
(b)
Antikink waveIn order to show the dynamic profiles of solution (67), we plot its graphs for α=0.5,β=0.8, μ=-0.2, and t=0.1, which are shown in Figures 3(a) and 3(b).


### 4.2. Exact Traveling Wave Solutions for the Constantsg≠0 and h=0

Wheng≠0 and h=0, (58) can be reduced to
(69)-gu+121-cu2+16αu3+14μα2u4+12βdudξ2=0.Ifβμ<0, then (69) can be rewritten as
(70)duuu3+2/3αμu2-2(c-1)/μα2u-4g/μα2=±α-μ2βdξ.
By using factoring method, (70) can be rewritten in the following two forms:
(71)duu-0u-z0u-z1u-z1¯=±α-μ2βdξ,z0<0
or
(72)duu-z0u-0u-z1u-z1¯=±α-μ2βdξ,z0>0,
where z0,z1, and z1¯ are roots of the equation u3+2/3αμu2-2(c-1)/μα2u-4g/μα2=0 and the z0 is real root; the  z1,z1¯ are conjugate complex roots. z0,z1, and z1¯ can be expressed by the parameters α,μ,c, and g, and here we omit their expressions because they are very complex. But we can always obtain their values by using computer once the parameters α,μ,c, and g are fixed on concrete values. For example, taking α=0.8,c=5,μ=-5,g=-4, we get z0=1.204421920, z1=-0.6855442934+1.918696998i, and z1¯=-0.6855442934-1.918696998i; taking α=0.8,c=5,μ=-5,g=4, we get z0=-1.276547333, z1=0.5549403329+1.899699037i, and z1¯=0.5549403329+1.899699037i.Solving (71) and (72), we obtain two periodic wave solutions of (1) as follows:
(73)u=z0z1cnω~1ξ,m1∓1×z02-2z0·Rez1+z12z1cnω~1ξ,m1∓1+z02-2z0·Rez1+z12cnω~1ξ,m1±1-1,
where (74)ω~1=α-μz1z02-2z0·Rez1+z122β,m1=122z12-2z0·Rez1+2z1z02-2z0·Rez1+z12z1z02-2z0·Rez1+z12.If βμ>0, then (69) can be rewritten as
(75)du-uu3+2/3αμu2-2c-1/μα2u-4g/μα2=±αμ2βdξ.
By using factoring method, (75) can be rewritten in the following two forms:
(76)du0-uu-z0u-z1u-z1¯=±αμ2βdξ,z0<0
or
(77)du(z0-u)(u-0)(u-z1)(u-z1¯)=±αμ2βdξ,z0>0,
where z0,z1, and z1¯ are same as the above case.Solving (76) and (77), we obtain two periodic wave solutions of (1) as follows:
(78)u=z0z11+cnω~2ξ,m2×z02-2z0·Rez1+z12z11+cnω~2ξ,m2+z02-2z0·Rez1+z121-cnω~2ξ,m2-1,(79)u=z0z02-2z0·Rez1+z121+cnω~2ξ,m2×z11-cnω~2ξ,m2+z02-2z0·Rez1+z12×1+cnω~2ξ,m2-1,
where(80)ω~2=αμz1z02-2z0·Rez1+z122β,m2=122z0·Rez1-2z12+2z1z02-2z0·Rez1+z12z1z02-2z0·Rez1+z12.


### 4.3. Exact Traveling Wave Solutions for the Constantsg≠0 and h≠0

Wheng≠0 and h≠0, (58) can be rewritten as
(81)du-μα22βu4+23μαu3-2(c-1)μα2u2-4gμα2u-4hμα2-1/2=±dξ.
The types of solutions of (81) contain many cases due to the roots of the quartic equation u4+2/3μαu3-2(c-1)/μα2u2-4g/μα2u-4h/μα2=0 which is in (81) and have many possibilities, but all solutions of (1) under different kinds of cases are periodic solutions of Jacobian elliptic function type such as the forms of solutions (73), (78), and (79). For convenience to discuss, here we only consider one case which this quartic equation has four real roots. Obviously, we can always obtain the real roots of this quartic equation by using computer once the parameters α,β,μ,c,g, and h are fixed on concrete values. Supposing ϕ1,ϕ2,ϕ3, and ϕ4 are four real roots of this quartic equation, we will obtain different kinds of periodic wave solutions of Jacobian elliptic function type for (1); see the following discussions.Case 1.
Underβμ<0, respectively, taking different root of the ϕ1,ϕ2,ϕ3, and ϕ4 as initial value to integrate (81) yields
(82)∫ϕ1u‍du~u~-ϕ1u~-ϕ2u~-ϕ3u~-ϕ4=±α-μ2β∫0ξ‍dξ~,
where u>ϕ1>ϕ2>ϕ3>ϕ4 and
(83)∫uϕ2‍du~ϕ1-u~ϕ2-u~u~-ϕ3u~-ϕ4=±α-μ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>u≥ϕ3>ϕ4 and
(84)∫ϕ3u‍du~ϕ1-u~ϕ2-u~u~-ϕ3u~-ϕ4=±α-μ2β∫0ξ‍dξ~,
where ϕ1>ϕ2≥u>ϕ3>ϕ4 and
(85)∫uϕ4‍du~ϕ1-u~ϕ2-u~ϕ3-u~ϕ4-u~=±α-μ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>u>ϕ3>ϕ4>u.
Respectively, completing the above integrals in (82), (83), (84), and (85), we obtain four periodic wave solutions of Jacobian elliptic function type of (1) as follows:
(86)u=(ϕ1ϕ2-ϕ2ϕ4)sn2(Ω1ξ,k1)+(ϕ1ϕ4-ϕ1ϕ2)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ1ϕ2-ϕ1ϕ3)sn2(Ω1ξ,k1)+(ϕ2ϕ3-ϕ1ϕ2)(ϕ2-ϕ3)sn2(Ω1ξ,k1)+(ϕ3-ϕ1),u=(ϕ1ϕ2-ϕ2ϕ4)sn2(Ω1ξ,k1)+(ϕ1ϕ4-ϕ1ϕ2)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ2ϕ4-ϕ3ϕ4)sn2(Ω1ξ,k1)+(ϕ3ϕ4-ϕ2ϕ3)(ϕ2-ϕ3)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ1ϕ2-ϕ2ϕ4)sn2(Ω1ξ,k1)+(ϕ1ϕ4-ϕ1ϕ2)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ1ϕ3-ϕ3ϕ4)sn2(Ω1ξ,k1)+(ϕ3ϕ4-ϕ1ϕ4)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ3-ϕ1),
where Ω1=±α/2-μ(ϕ1-ϕ3)(ϕ2-ϕ4)/2β and k1=(ϕ2-ϕ3)(ϕ1-ϕ4)/(ϕ1-ϕ3)(ϕ2-ϕ4).Case 2.
Underβμ>0, respectively, taking different root of the ϕ1,ϕ2,ϕ3, and ϕ4 as initial value to integrate (81) yields
(87)∫uϕ1‍du~ϕ1-u~u~-ϕ2u~-ϕ3u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1>u≥ϕ2>ϕ3>ϕ4 and
(88)∫ϕ2u‍du~ϕ1-u~u~-ϕ2u~-ϕ3u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1≥u>ϕ2>ϕ3>ϕ4 and
(89)∫uϕ3‍du~ϕ1-u~ϕ2-u~ϕ3-u~u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>ϕ3>u≥ϕ4 and
(90)∫ϕ4u‍du~ϕ1-u~ϕ2-u~ϕ3-u~u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>ϕ3≥u>ϕ4.
Similarly completing the above integrals in (87), (88), (89), and (90), we obtain another four periodic wave solutions of Jacobian elliptic function type of (1) as follows:
(91)u=(ϕ1ϕ4-ϕ2ϕ4)sn2(Ω2ξ,k2)+(ϕ1ϕ2-ϕ1ϕ4)(ϕ1-ϕ2)sn2(Ω2ξ,k2)+(ϕ2-ϕ4),u=(ϕ1ϕ3-ϕ2ϕ3)sn2(Ω2ξ,k2)+(ϕ2ϕ3-ϕ1ϕ2)(ϕ1-ϕ2)sn2(Ω2ξ,k2)+(ϕ3-ϕ1),u=(ϕ2ϕ3-ϕ2ϕ4)sn2(Ω2ξ,k2)+(ϕ3ϕ4-ϕ2ϕ3)(ϕ3-ϕ4)sn2(Ω2ξ,k2)+(ϕ4-ϕ2),(92)u=(ϕ1ϕ3-ϕ1ϕ4)sn2(Ω2ξ,k2)+(ϕ1ϕ4-ϕ3ϕ4)(ϕ3-ϕ4)sn2(Ω2ξ,k2)+(ϕ3-ϕ1),
where Ω2=±α/2μ(ϕ1-ϕ3)(ϕ2-ϕ4)/2β and k2=(ϕ1-ϕ2)(ϕ3-ϕ4)/(ϕ1-ϕ3)(ϕ2-ϕ4).In Sections4.2 and 4.3, all the exact solutions obtained by us are periodic solutions of Jacobian elliptic function types. As an example, we plot the graphs of profiles of the solutions (78) and (79) for α=0.3,β=0.8,c=1,μ=2, and t=0.1, which are shown in Figures 5(a) and 5(b).The graphs of profiles for periodic wave solutions of Jacobin elliptic function types defined by (78) and (79) under different parametric values: (a) g=4; (b) g=-4.
(a)
Periodic wave defined by (78)
(b)
Periodic wave defined by (79)


## 4.1. Different Kinds of Exact Traveling Wave Solutions for the Constantsg=h=0

When two integral constantsg,h are both zero (i.e., g=h=0), (58) can be reduced to
(59)dudξ2=c-1βu2-α3βu3-μα22βu4.
Under different parametric conditions solving (59), we obtain a series of exact traveling wave solutions as follows:
(60)u=6c-1sech21/2(c-1)/βξ2α+9αμc-11±tanh1/2(c-1)/βξ2,
where c>1, β>0, and ξ=x-ct:
(61)u=6c-1α1-18μ1-c-1sinh⁡(c-1)/βξ,
where c>1, β>0, μ<-1/18(c-1), and ξ=x-ct:
(62)u=6(c-1)α1+18μ(c-1)+1sin(1-c)/βξ,
where c<1, β>0, μ<-1/18(c-1), and ξ=x-ct:
(63)u=6(c-1)α1-18μ(c-1)+1cosh⁡(c-1)/βξ,
where c>1, β>0, μ>-1/18(c-1), and ξ=x-ct:
(64)u=6(c-1)α1-18μ(c-1)+1cos⁡(1-c)/βξ,
where c<1, β>0, μ>-1/18(c-1), and ξ=x-ct:
(65)u=3(c-1)×αcosh⁡212c-1βξ-3μ1-c2sinh⁡c-1βξ-1,
where c>1, β>0, μ<0, and ξ=x-ct:
(66)u=3c-1×αcos⁡212c-1βξ+3μc-12sin1-cβξ-1,
where c<1,β>0,μ<0, and ξ=x-ct:(67)u=-16αμ1±tanh112-2βμξ
or
(68)u=-16αμ1±coth⁡112-2βμξ,
where β>0,μ<0, and ξ=x-(1-1/18μ)t.Among these exact travelling wave solutions obtained in Section4.1, it is worth pointing that the solutions (60) and (63) describe smooth solitary waves and the solution (67) describes kink wave and antikink wave (Figure 4). In order to show the dynamic profiles of solutions (60) and (63), we plot their graphs for α=0.5,β=0.8,c=5, and t=0.1, which are shown in Figures 3(a) and 3(b).The graphs of profiles for smooth solitary wave solutions defined by (60) and (63) under different parametric values: (a) μ=0.7; (b) μ=2.
(a)
Bright solitary wave defined by (60)
(b)
Dark solitary wave defined by (63)The graphs of profiles for kink wave and antikink wave solutions defined by (67) with “+” and “−”.
(a)
Kink wave
(b)
Antikink waveIn order to show the dynamic profiles of solution (67), we plot its graphs for α=0.5,β=0.8, μ=-0.2, and t=0.1, which are shown in Figures 3(a) and 3(b).


## 4.2. Exact Traveling Wave Solutions for the Constantsg≠0 and h=0

Wheng≠0 and h=0, (58) can be reduced to
(69)-gu+121-cu2+16αu3+14μα2u4+12βdudξ2=0.Ifβμ<0, then (69) can be rewritten as
(70)duuu3+2/3αμu2-2(c-1)/μα2u-4g/μα2=±α-μ2βdξ.
By using factoring method, (70) can be rewritten in the following two forms:
(71)duu-0u-z0u-z1u-z1¯=±α-μ2βdξ,z0<0
or
(72)duu-z0u-0u-z1u-z1¯=±α-μ2βdξ,z0>0,
where z0,z1, and z1¯ are roots of the equation u3+2/3αμu2-2(c-1)/μα2u-4g/μα2=0 and the z0 is real root; the  z1,z1¯ are conjugate complex roots. z0,z1, and z1¯ can be expressed by the parameters α,μ,c, and g, and here we omit their expressions because they are very complex. But we can always obtain their values by using computer once the parameters α,μ,c, and g are fixed on concrete values. For example, taking α=0.8,c=5,μ=-5,g=-4, we get z0=1.204421920, z1=-0.6855442934+1.918696998i, and z1¯=-0.6855442934-1.918696998i; taking α=0.8,c=5,μ=-5,g=4, we get z0=-1.276547333, z1=0.5549403329+1.899699037i, and z1¯=0.5549403329+1.899699037i.Solving (71) and (72), we obtain two periodic wave solutions of (1) as follows:
(73)u=z0z1cnω~1ξ,m1∓1×z02-2z0·Rez1+z12z1cnω~1ξ,m1∓1+z02-2z0·Rez1+z12cnω~1ξ,m1±1-1,
where (74)ω~1=α-μz1z02-2z0·Rez1+z122β,m1=122z12-2z0·Rez1+2z1z02-2z0·Rez1+z12z1z02-2z0·Rez1+z12.If βμ>0, then (69) can be rewritten as
(75)du-uu3+2/3αμu2-2c-1/μα2u-4g/μα2=±αμ2βdξ.
By using factoring method, (75) can be rewritten in the following two forms:
(76)du0-uu-z0u-z1u-z1¯=±αμ2βdξ,z0<0
or
(77)du(z0-u)(u-0)(u-z1)(u-z1¯)=±αμ2βdξ,z0>0,
where z0,z1, and z1¯ are same as the above case.Solving (76) and (77), we obtain two periodic wave solutions of (1) as follows:
(78)u=z0z11+cnω~2ξ,m2×z02-2z0·Rez1+z12z11+cnω~2ξ,m2+z02-2z0·Rez1+z121-cnω~2ξ,m2-1,(79)u=z0z02-2z0·Rez1+z121+cnω~2ξ,m2×z11-cnω~2ξ,m2+z02-2z0·Rez1+z12×1+cnω~2ξ,m2-1,
where(80)ω~2=αμz1z02-2z0·Rez1+z122β,m2=122z0·Rez1-2z12+2z1z02-2z0·Rez1+z12z1z02-2z0·Rez1+z12.


## 4.3. Exact Traveling Wave Solutions for the Constantsg≠0 and h≠0

Wheng≠0 and h≠0, (58) can be rewritten as
(81)du-μα22βu4+23μαu3-2(c-1)μα2u2-4gμα2u-4hμα2-1/2=±dξ.
The types of solutions of (81) contain many cases due to the roots of the quartic equation u4+2/3μαu3-2(c-1)/μα2u2-4g/μα2u-4h/μα2=0 which is in (81) and have many possibilities, but all solutions of (1) under different kinds of cases are periodic solutions of Jacobian elliptic function type such as the forms of solutions (73), (78), and (79). For convenience to discuss, here we only consider one case which this quartic equation has four real roots. Obviously, we can always obtain the real roots of this quartic equation by using computer once the parameters α,β,μ,c,g, and h are fixed on concrete values. Supposing ϕ1,ϕ2,ϕ3, and ϕ4 are four real roots of this quartic equation, we will obtain different kinds of periodic wave solutions of Jacobian elliptic function type for (1); see the following discussions.Case 1.
Underβμ<0, respectively, taking different root of the ϕ1,ϕ2,ϕ3, and ϕ4 as initial value to integrate (81) yields
(82)∫ϕ1u‍du~u~-ϕ1u~-ϕ2u~-ϕ3u~-ϕ4=±α-μ2β∫0ξ‍dξ~,
where u>ϕ1>ϕ2>ϕ3>ϕ4 and
(83)∫uϕ2‍du~ϕ1-u~ϕ2-u~u~-ϕ3u~-ϕ4=±α-μ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>u≥ϕ3>ϕ4 and
(84)∫ϕ3u‍du~ϕ1-u~ϕ2-u~u~-ϕ3u~-ϕ4=±α-μ2β∫0ξ‍dξ~,
where ϕ1>ϕ2≥u>ϕ3>ϕ4 and
(85)∫uϕ4‍du~ϕ1-u~ϕ2-u~ϕ3-u~ϕ4-u~=±α-μ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>u>ϕ3>ϕ4>u.
Respectively, completing the above integrals in (82), (83), (84), and (85), we obtain four periodic wave solutions of Jacobian elliptic function type of (1) as follows:
(86)u=(ϕ1ϕ2-ϕ2ϕ4)sn2(Ω1ξ,k1)+(ϕ1ϕ4-ϕ1ϕ2)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ1ϕ2-ϕ1ϕ3)sn2(Ω1ξ,k1)+(ϕ2ϕ3-ϕ1ϕ2)(ϕ2-ϕ3)sn2(Ω1ξ,k1)+(ϕ3-ϕ1),u=(ϕ1ϕ2-ϕ2ϕ4)sn2(Ω1ξ,k1)+(ϕ1ϕ4-ϕ1ϕ2)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ2ϕ4-ϕ3ϕ4)sn2(Ω1ξ,k1)+(ϕ3ϕ4-ϕ2ϕ3)(ϕ2-ϕ3)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ1ϕ2-ϕ2ϕ4)sn2(Ω1ξ,k1)+(ϕ1ϕ4-ϕ1ϕ2)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ4-ϕ2),u=(ϕ1ϕ3-ϕ3ϕ4)sn2(Ω1ξ,k1)+(ϕ3ϕ4-ϕ1ϕ4)(ϕ1-ϕ4)sn2(Ω1ξ,k1)+(ϕ3-ϕ1),
where Ω1=±α/2-μ(ϕ1-ϕ3)(ϕ2-ϕ4)/2β and k1=(ϕ2-ϕ3)(ϕ1-ϕ4)/(ϕ1-ϕ3)(ϕ2-ϕ4).Case 2.
Underβμ>0, respectively, taking different root of the ϕ1,ϕ2,ϕ3, and ϕ4 as initial value to integrate (81) yields
(87)∫uϕ1‍du~ϕ1-u~u~-ϕ2u~-ϕ3u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1>u≥ϕ2>ϕ3>ϕ4 and
(88)∫ϕ2u‍du~ϕ1-u~u~-ϕ2u~-ϕ3u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1≥u>ϕ2>ϕ3>ϕ4 and
(89)∫uϕ3‍du~ϕ1-u~ϕ2-u~ϕ3-u~u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>ϕ3>u≥ϕ4 and
(90)∫ϕ4u‍du~ϕ1-u~ϕ2-u~ϕ3-u~u~-ϕ4=±αμ2β∫0ξ‍dξ~,
where ϕ1>ϕ2>ϕ3≥u>ϕ4.
Similarly completing the above integrals in (87), (88), (89), and (90), we obtain another four periodic wave solutions of Jacobian elliptic function type of (1) as follows:
(91)u=(ϕ1ϕ4-ϕ2ϕ4)sn2(Ω2ξ,k2)+(ϕ1ϕ2-ϕ1ϕ4)(ϕ1-ϕ2)sn2(Ω2ξ,k2)+(ϕ2-ϕ4),u=(ϕ1ϕ3-ϕ2ϕ3)sn2(Ω2ξ,k2)+(ϕ2ϕ3-ϕ1ϕ2)(ϕ1-ϕ2)sn2(Ω2ξ,k2)+(ϕ3-ϕ1),u=(ϕ2ϕ3-ϕ2ϕ4)sn2(Ω2ξ,k2)+(ϕ3ϕ4-ϕ2ϕ3)(ϕ3-ϕ4)sn2(Ω2ξ,k2)+(ϕ4-ϕ2),(92)u=(ϕ1ϕ3-ϕ1ϕ4)sn2(Ω2ξ,k2)+(ϕ1ϕ4-ϕ3ϕ4)(ϕ3-ϕ4)sn2(Ω2ξ,k2)+(ϕ3-ϕ1),
where Ω2=±α/2μ(ϕ1-ϕ3)(ϕ2-ϕ4)/2β and k2=(ϕ1-ϕ2)(ϕ3-ϕ4)/(ϕ1-ϕ3)(ϕ2-ϕ4).In Sections4.2 and 4.3, all the exact solutions obtained by us are periodic solutions of Jacobian elliptic function types. As an example, we plot the graphs of profiles of the solutions (78) and (79) for α=0.3,β=0.8,c=1,μ=2, and t=0.1, which are shown in Figures 5(a) and 5(b).The graphs of profiles for periodic wave solutions of Jacobin elliptic function types defined by (78) and (79) under different parametric values: (a) g=4; (b) g=-4.
(a)
Periodic wave defined by (78)
(b)
Periodic wave defined by (79)


## 5. Conclusions

Though Frobenius’ idea is a well-known general method, it can solve some very complex PDE models with highly nonlinear terms and high order terms such as (1) when it combines with the integral bifurcation method. In this work, by using Frobenius’ idea together with integral bifurcation method, we study the third order nonlinear water wave model (1). Under different kinds of parametric conditions, we obtain eight types of exact travelling wave solutions including the smooth solitary wave solutions (60), (63), and (65), the nonsmooth peakon wave solutions (51) and (57), the kink wave and antikink wave solutions (67) and (68), the smooth periodic wave solutions of trigonometric function type (14), (16), (21), (24), (28), (31), (34), (36), (38), (40), and (43), the nonsmooth periodic wave solutions of trigonometric function type (62), (64), and (66), the periodic wave solutions of Jacobian elliptic function type (78), (86), (91), and (92), the hyperbolic function solutions (13), (15), (20), (23), (27), (30), (33), (35), (44), and (61), and the rational function solution (45). Though the types of these solutions obtained in this work are not new and they are familiar types, the results of (1) obtained by us in this paper did not appear in any existing literatures. Particularly, compared with reference [45], all results obtained in this paper are new. Among these solutions obtained in this paper, some of them have direct physical applications. For example, using the smooth solitary wave solutions, nonsmooth peakon wave solutions, and kink and antikink wave solutions, we can explain lots of motion phenomena for water wave; indeed (1) is just a very important water wave model.



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*Source: 641308-2015-02-22.xml*