# A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions

Authors: Xianbin Wu; Weiguo Rui; Xiaochun Hong
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We study a generalized KdV equation of neglecting the highest order infinitesimal term, which is an important water wave model. Some exact traveling wave solutions such as singular solitary wave solutions, semiloop soliton solutions, dark soliton solutions, dark peakon solutions, dark loop-soliton solutions, broken loop-soliton solutions, broken wave solutions of U-form and C-form, periodic wave solutions of singular type, and broken wave solution of semiparabola form are obtained. By using mathematical softwareMaple, we show their profiles and discuss their dynamic properties. Investigating these properties, we find that the waveforms of some traveling wave solutions vary with changes of certain parameters.


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## Body

## 1. Introduction

In 1995, based on the physical and asymptotic considerations, Fokas [1] derived the following generalized KdV equation:
(1)ηt+ηx+αηηx+βηxxx+ρ1α2η2ηx+αβ(ρ2ηηxxx+ρ3ηxηxx)+ρ4α3η3ηx+α2β(ρ5η2ηxxx+ρ6ηηxηxx+ρ7ηx3)=0,
which is an important water wave model, where α=3A/2, β=B/6, ρ1=-1/6, ρ2=5/3, ρ3=23/6, ρ4=1/8, ρ5=7/18, ρ6=79/36, ρ7=45/36. Regarding the ρ1,ρ2,ρ3,ρ4,ρ5,ρ6,ρ7 as free parameters and using the ρ~4 to replace the ρ4α2 (i.e., ρ4α2↦ρ~4), (1) becomes the following PDE:
(2)ut+ux+αuux+βuxxx+ρ1α2u2ux+αβ(ρ2uuxxx+ρ3uxuxx)+ρ~4αu3ux+α2β(ρ5u2uxxx+ρ6uuxuxx+ρ7ux3)=0,
which is given by Tzirtzilakis et al. in [2]; they called it high-order wave equation of KdV type. Just as Tzirtzilakis [2] said these two equations are both water wave equations of KdV type, which are more physically and practically meaningful. The motion described by the model (1) or (2) is a 2-dimensional, inviscid, incompressible fluid (water) lying above a horizontal fiat bottom located at y-=-h-0 (h-0 is a constant), and letting the air above the water. It turns out that, for such a system if the vorticity is zero initially, it remains zero. The fluids (water) analyzed by Fokas are only irrotational flows. In addition, this system is characterized by two parameters α=3A/2, β=B/6 with A=a/h-0, B=h-02/ℓ2, where a and ℓ are two typical values of the amplitude and of the wavelength of the waves. The parameters a, h-0, ℓ satisfy the condition a≪h-0<ℓ because the system is a model of short amplitude and long wavelength. Therefore the parameters α and β satisfy the condition  0<α<1, 0<β<1.Assuming that the waves are unidirectional and neglecting terms ofO(α2,α3,αβ), (1) can be reduced to the classical KdV equation:
(3)ηt+ηx+αηηx+βηxxx=0.
In [1], Fokas assumed that O(B) is less than O(A); this implies O(β)<O(α). According to this assumption, we easily know that O(α2β)<O(α3) and O(α2β)<O(αβ). Neglecting two high-order infinitesimal terms of O(α3,α2β), (1) can be reduced to another high-order wave equation of KdV type [2–5] as follows:
(4)ηt+ηx+αηηx+βηxxx+ρ1α2η2ηx+αβ(ρ2ηηxxx+ρ3ηxηxx)=0.
Equation (4) is a special case of (1) for ρ4=ρ5=ρ6=ρ7=0. In [1], it was observed that (4) can be reduced by the local transformation of coordinates
(5)η=v-αρ1v2-β(3ρ1+74ρ2-12ρ3)vxx
to a completely integrable PDE as follows:
(6)vt-32βρ2vxxt+β(1-32ρ2)vxxx+αvvx-12αβρ2(vvxxx+2vxvxx).
Equation (6) was first derived in [6] by using the method of bi-Hamiltonian systems, which based on the physical considerations, its Lax Pair was also given in [7].If only neglecting the highest order infinitesimal term ofO(α2β), then (1) can be reduced to a new generalized KdV equation as follows:
(7)ηt+ηx+αηηx+βηxxx+ρ1α2η2ηx+αβ(ρ2ηηxxx+ρ3ηxηxx)+ρ4α3η3ηx=0.
We call it a generalized KdV equation of neglecting the highest order infinitesimal term. In fact, (7) is another special case of (1) for ρ5=ρ6=ρ7=0; it is also third-order approximate equation of KdV type. Of course, on describing dynamical behaviors of water waves, (4) is only a rough approximative model of (1) compared with (7); that is, the precision of model (7) is better than that of model (4) on describing dynamical behaviors of water waves. In other words, model (7) exhibits much richer phenomenology than model (4). Therefore, the investigation of exact traveling wave solutions for (7) are more practically meaningful than that of (4).On the other hand, under the local transformation(8)η=u+λ1αu2+λ2βuxx+λ3α2u3+αβ(λ4uuxx+λ5ux2),
(1) can be reduced to the following generalized Gardner equation [1]:
(9)ut+ux-32ρ2βuxxt+(1-32ρ2)βuxxx+αuux-12ρ2αβ(uuxxx+2uxuxx)+3μα2u2ux-32μρ2α2β(u2uxxx+ux3+4uuxuxx)+94μρ22α2β2(ux2uxxx+2uxuxx2)=0,
where λi(i=1,2,3,4,5) are certain expressions of  ρi. Clearly, (9) is equivalent to the following generalization form of the modified KdV equation [1, 8] under the transformation -(3/2)ρ2β↦ν,(1-(3/2)ρ2)β↦β~,
(10)ut+ux+νuxxt+β~uxxx+αuux+13να(uuxxx+2uxuxx)+3μα2u2ux+νμα2(u2uxxx+ux3+4uuxuxx)+ν2μα2(ux2uxxx+2uxuxx2)=0.From the above references and the references cited therein, we can know that (1) and (2) are very important water wave models. However, (1) and (2) are too complex to obtain their exact solution under universal conditions. Only under some special parametric conditions, their exact solutions were obtained in the existing literature. Next, let us briefly review the research backgrounds for the above equations.In [2], Tzirtzilakis et al. only obtained two soliton-like solutions of (2) in the two groups of special conditions ρ3=2(ρ1-ρ2), ρ4=0, ρ5=2ρ1(ρ2-2ρ1), ρ6=6ρ1(2ρ1-ρ2), ρ7=3ρ1(ρ2-2ρ1) and ρ2=2ρ1, ρ3=-2ρ1, ρ3=3ρ4, ρ6=-6ρ4, ρ7=3ρ4. In [9], by using the planar bifurcation method of dynamical systems, under the four groups of special conditions (A1)ρ22>4ρ5>0, ρ3=2ρ2, ρ6=4ρ5, ρ7=ρ5; (A2)ρ3=2ρ2, ρ5=ρ7=(1/4)ρ22, ρ6=(1/2)ρ1ρ2; (B1)ρ3=ρ2, ρ6=2ρ5, ρ7=0; (B2)ρ3=ρ2, ρ5=(1/4)ρ22, ρ6=2ρ5=(1/2)ρ22, ρ7=0, Li et al. studied (2); the existence of all kinds of traveling wave solutions were discussed completely, but its exact solutions were not investigated although some results of numerical simulation were obtained in this literature. In [10], in order to answer what is the dynamical behavior of one-loop soliton solution, Li studied the special case of (1) for ρ1=ρ4=ρ6=0,ρ3=2ρ2,ρ5=ρ7=(1/4)ρ22. In [11], under different kinds of parametric conditions, Marinakis discussed two integrable cases for the third-order approximation model (1). In [12], Marinakis proved that (1) and some its of special cases are integrable. In [13], Gandarias and Bruzon proved that (1) is self-adjoint if and only if ρ3=2ρ2,ρ7=ρ6-3ρ5. In [14], by using the method as in [9], Li et al. studied (10); the existence of solitary wave, kink, and antikink wave solutions and uncountably infinite many smooth and nonsmooth periodic wave solutions was discussed except exact solutions. In [8], Bi also obtained some results of numerical simulation of (10); it is a pity that the exact traveling wave solutions of (10) were not obtained yet. In [4, 5], we obtained some exact traveling wave solutions of (4) under the parametric conditions ρ3=(p+1)ρ2 or ρ3=pρ2. In [15], under a new ansätze, Khuri studied (4); some exact solitary wave solutions and periodic wave solutions were obtained. In [16], by using method of planar dynamical system, Long et al. studied (6); the existence of smooth solitary wave and uncountably infinite many smooth and nonsmooth periodic wave solutions was proved in this literature.From the above research backgrounds of (1) and (2), we can see that their exact solutions in universal conditions are hard to obtain because they are highly nonlinear equations and most probably they are not integrable equations in general. Thus, large numbers of research results are still concentrated in the classical KdV equation and some other high-order equations with KdV type, such as KdV-Burgers equation [17, 18] and KdV-Burgers-Kuramoto equation [19], at present. Therefore, the investigation of the more exact solutions for (1) is very important and necessary. However, by using the current methods, we can not obtain exact solutions of (1) in universal conditions; the next best thing is the investigation of exact solutions of (7). In this paper, still regarding the ρi(i=1,2,3,4) as free parameters and by using the integral bifurcation method [20, 21], we will investigate exact traveling wave solutions and their properties of (7).The rest of this paper is organized as follows. In Section2, we will derive two-dimensional planar system which is equivalent to (7) and give its first integral equation. In Section 3, by using the integral bifurcation method, we will obtain some new traveling wave solutions and discuss their dynamic properties.


## 2. Two-Dimensional Planar Dynamical System of (7) and Its First Integral and Conservation of Energy

Making a transformationη(t,x)=ϕ(ξ) with ξ=x-ct, (7) can be reduced to the following ODE:
(11)-cϕ′+ϕ′+αϕϕ′+βϕ′′′+ρ1α2ϕ2ϕ′+αβ(ρ2ϕϕ′′′+ρ3ϕ′ϕ′′)+ρ4α3ϕ3ϕ′=0,
where c is wave velocity which moves along the direction of x-axis and c≠0. Integrating (11) once and setting the integral constant as zero yields
(12)(1-c)ϕ+12αϕ2+βϕ′′+13ρ1α2ϕ3+αβ×[ρ2ϕϕ′′+12(ρ3-ρ2)(ϕ′)2]+14ρ4α3ϕ4=0.
Let ϕ′=y. Thus (12) can be reduced to a planar system
(13)dϕdξ=y,dydξ=-(14ρ4α3ϕ4+13ρ1α2ϕ3+12αϕ2+(1-c)ϕ+12αβ(ρ3-ρ2)y2)×(β(1+αρ2ϕ))-1
and a linear equation
(14)β(1+αρ2ϕ)=0.
Obviously, the solutions of (12) cover the solutions of (13) and (14). We notice that the second equation in (13) is not continuous when ϕ=-1/αρ2; that is, the function ϕ′′(ξ) is not defined by ϕ=-1/αρ2. So (13) is a singular system; the line ϕ=-1/αρ2 is called the singular line. Thus, we make the following transformation:
(15)dξ=12β(1+αρ2ϕ)dτ,
where τ is a free parameter. Under the transformation (15), (13) can be rewritten as the following system
(16)dϕdτ=12β(1+αρ2ϕ)y,dydτ=-[3ρ4α3ϕ4+4ρ1α2ϕ3+6αϕ2+12(1-c)ϕ+6αβ(ρ3-ρ2)y2].
Clearly, (16) is equivalent to (12). Except for the singular line ϕ=-1/αρ2, (13) and (16) have the same first integral.First, from the parametric conditionsρ3=5/3,ρ2=23/6 in the model (1), we easily know that ρ3≈2ρ2. Second, from [9–11, 13], we know that (1) contains many good properties including loop soliton, integrability, and self-adjoint property when ρ3=2ρ2. In addition, in the condition ρ3=2ρ2, it becomes easy when we solve (7). Therefore, we only consider the special case of (7) for ρ3=2ρ2 in this paper. Thus, by using this assumption, we obtain the first integral of (13) and (16) as follows:
(17)y2=-(3/5)α3ρ4ϕ5-α2ρ1ϕ4-2αϕ3+6(c-1)ϕ2+C06β(1+αρ2ϕ),
where C0 is an integral constant. For the convenience of discussion, taking the integral constant C0=0 in (17) yields
(18)12My2+12Mαρ2ϕy2+35α3ρ4ϕ5+α2ρ1ϕ4+2αϕ3-6(c-1)ϕ2=0,
where M=12β denotes the particle's mass of system. Let E1=(1/2)My2=(1/2)M(ϕ′)2, E2=(1/2)Mαρ2ϕy2=(1/2)Mαρ2ϕ(ϕ′)2, T=(3/5)α3ρ4ϕ5+α2ρ1ϕ4+2αϕ3-6(c-1)ϕ2. Thus, (18) can be rewritten as the following equation of conservation of energy:
(19)E1+E2+T=0,
where E1 denotes kinetic energy, E2 denotes external energy, and T denotes potential energy. In (19), the kinetic energy E1 and potential energy T are not conserved because the external energy E2≠0, but the global energy (E1,E2,T) are subject to the energy conservation.  E2=0 if only if  ρ2=0. Obviously, ρ3=0 once ρ2=0; under this case, (7) becomes the following equation:
(20)ηt+ηx+αηηx+βηxxx+ρ1α2η2ηx+ρ4α3η3ηx=0.
Similarly,  (20) can be reduced to the following planar system:
(21)dϕdξ=y,dydξ=-(14ρ4α3ϕ4+13ρ1α2ϕ3+12αϕ2+(1-c)ϕ)×β-1.
Obviously, the system (21) is a regular system; it has no singular line. Taking the integral constant as zero, we obtain (21)'s first integral as follows:
(22)y2=-1β[110α3ρ4ϕ5+16α2ρ1ϕ4+13αϕ3-(c-1)ϕ2].
Equation (22) can be rewritten as the following equation of conservation of energy:
(23)E1+T=0,
where E1,T are given above. The kinetic energy E1 and potential energy T are conserved in (23); in other words, the particle motion satisfies the conservation of kinetic energy and potential energy, which converts the kinetic energy from the potential energy then it converts the potential energy from the kinetic energy and go round and round. Therefore, (20) only has nonsingular traveling wave solutions, and all solutions are smooth; this is very different from (7).In order to discuss singular or nonsingular traveling wave solutions of (7), we will consider (17). When  C0=0, (17) can be reduced to
(24)y=±([-35α3ρ4ϕ5-α2ρ1ϕ4-2αϕ3+6(c-1)ϕ2](1+αρ2ϕ))×(6β(1+αρ2ϕ))-1.
Equation (24) can be further simplified under the following parametric conditions.Case 1.
One hasc=1+1/3ρ2,ρ4=-(5/3)ρ1ρ2.Case 2.
One hasρ1=-2ρ2,ρ4=(10/3)ρ22.Case 3.
One hasc=1+1/3ρ2,ρ1=-2ρ2.Case 4.
One hasc=1,ρ1=-2ρ2.Case 5.
One hasc≠1+1/3ρ2 or c≠1,ρ1=-2ρ2,ρ4≠-(5/3)ρ1ρ2 or ρ4≠(10/3)ρ22.
Under the parametric conditions of Case1, (24) can be reduced to
(25)y=±ϕα4ρ1ρ22ϕ4-α2(ρ1+2ρ2)ϕ2+2/ρ26β(1+αρ2ϕ).
Under the parametric conditions of Case 2, (24) can be reduced to
(26)y=±ϕ-2α4ρ23ϕ4+2α(3cρ2-3ρ2-1)ϕ+6(c-1)6β(1+αρ2ϕ).
Under the parametric conditions of Case 3, (24) can be reduced to
(27)y=±ϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3+2/ρ26β(1+αρ2ϕ).
Under the parametric conditions of Case 4, (24) can be reduced to
(28)y=±ϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3-2αϕ6β(1+αρ2ϕ).
Under the the parametric conditions of Case 5, (24) can be reduced to
(29)y=±ϕ[-(3/5)α3ρ4ϕ3-α2ρ1ϕ2-2αϕ+6(c-1)](1+αρ2ϕ)6β(1+αρ2ϕ).


## 3. Exact Traveling Wave Solutions of (7) and Their Dynamic Properties

In this section, by using (25), (26), (27), (28), and (29), we will derive many exact traveling wave solutions and discuss their dynamic properties.


### 3.1. The Exact Solutions under the Parametric Conditions of Case1

Substituting (25) into the left expression (i.e., dϕ/dτ=12β(1+αρ2ϕ)y) of (16) yields
(30)dϕϕα4ρ1ρ22ϕ4-α2(ρ1+2ρ2)ϕ2+2/ρ2=±26βdτ.
For the sake of convenience, in all the following discussions, we only discuss the case where the right of equation is “+” sign; the case of “−” sign can be similarly discussed. By the way, the solutions obtained by taking “+” sign and the solutions obtained by taking “−” sign are same when the ϕ is even function.Taking “+” in (30), it can be reduced to
(31)dϕ2ϕ2α4ρ1ρ22(ϕ2)2-α2(ρ1+2ρ2)ϕ2+2/ρ2=46βdτ.
Let ψ=ϕ2. Equation (31) can be rewritten as
(32)dψψα4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=46βdτ.
Taking the (0,ψ1,2) as initial values, respectively, integrating (32) we obtain
(33)∫ψ1ψ‍dψψα4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=∫0τ‍46βdτ,∫ψ2ψ‍dψψα4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=∫0τ‍46βdτ,
where ψ1,2=(ρ1+2ρ2±(ρ1+2ρ2)2-8ρ1)/2ρ1ρ2α2 are two roots of equation α4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=0.(i) Whenρ2>0,Δ=(ρ1+2ρ2)2-8ρ1>0, completing the integrals (33) yields
(34)ψ=4R1Q1-P1cosh(8(3β/ρ2)τ),ψ=4R2P2cosh(8(3β/ρ2)τ)-Q2,
where R1,2=(ρ1+2ρ2±Δ)[(ρ1+2ρ2)Δ±Δ]/2ρ12ρ23α2,Q1,2=(ρ1+2ρ2)[Δ±(ρ1+2ρ2)][(ρ1+2ρ2)Δ±(ρ12+4ρ22)]/2ρ12ρ22, and P1,2=Δ[Δ±(ρ1+2ρ2)]2/4ρ12ρ22 with Δ=(ρ1+2ρ2)2-8ρ1.By using (34) and the transformation ϕ2=ψ, we obtain four exact solutions of (31) as follows:
(35)ϕ=±2R1Q1-P1cosh(8(3β/ρ2)τ),(36)ϕ=±2R2P2cosh(8(3β/ρ2)τ)-Q2,
where the constants R1,2,P1,2,Q1,2 are given above.Substituting (35) into (15) yields
(37)ξ=12β±αρ2g13ρ2βR1×F(arcsinQ1-P1cosh(8(3β/ρ2)τ)Q1-P1,Q1-P1Q1+P1),
where F(φ,k) is a normal elliptic integral of the first kind and g1=2/Q1+P1,Q1>P1>0,0<τ<(1/8)(ρ2/3β)cosh-1(Q1/P1).Substituting (36) into (15) yields
(38)ξ=12β±αρ2g23ρ2βR2×F(arcsinP2(cosh(8(3β/ρ2)τ)-1)P2cosh(8(3β/ρ2)τ)-Q2,P2+Q22P2),
where g2=2/ρ2,P2>Q2>0, and 0<τ<+∞.Combining (35) with (37), we obtain (7)'s two exact solutions of parametric type as follows:
(39)η=ϕ=±2R1Q1-P1cosh(8(3β/ρ2)τ),ξ=12βτ±αρ2g13ρ2βR1×F(arcsinQ1-P1cosh(8(3β/ρ2)τ)Q1-P1,Q1-P1Q1+P1).Combining (36) with (38), we obtain (7)'s another two exact solutions of parametric type as follows:(40)η=ϕ=±2R2P2cosh(8(3β/ρ2)τ)-Q2,ξ=12βτ±αρ2g23ρ2βR2×F(arcsinP2(cosh(8(3β/ρ2)τ)-1)P2cosh(8(3β/ρ2)τ)-Q2,P2+Q22P2).(ii) Whenρ2<0,Δ=(ρ1+2ρ2)2-8ρ1>0, completing the integrals (33) yields
(41)ψ=4-α2ρ2Δ[-α2(ρ1+2ρ2)-cos(8-(3β/ρ2)τ)],ψ=4-α2ρ2Δ[-α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ)].
Similarly, by using (41) and the transformation ϕ2=ψ, we obtain four periodic solutions of (31) as follows:
(42)ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)-cos(8-(3β/ρ2)τ),(43)ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ),
where ρ1<-(2ρ2+1/α2); that is, -α2(ρ1+2ρ2)>1>0. Respectively, substituting (42) and (43) into (15) yields(44)ξ=12βτ±g~ρ23βΔ4×F(arcsin[1-cos(8-(3β/ρ2)τ)][α2(ρ1+2ρ2)-1]2[α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ)],k),(45)ξ=12βτ±g~ρ23βΔ4F(4-3βρ2τ,k),whereg~=2/1-α2(ρ1+2ρ2),k=2/(1-α2(ρ1+2ρ2)).Combining (42) with (44), we obtain (7)'s two exact solutions of parametric type as follows:(46)η=ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)-cos(8-(3β/ρ2)τ),ξ=12βτ±g~ρ23βΔ4×F(arcsin[1-cos(8-(3β/ρ2)τ)][α2(ρ1+2ρ2)-1]2[α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ)],k),where0<τ≤π-ρ2/83β.Combining (43) with (45), we obtain (7)'s another two exact solutions of parametric type as follows:
(47)η=ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ),ξ=12βτ±g~ρ23βΔ4F(4-3βρ2τ,k).Among the above traveling wave solutions, it is very worthy to mention solutions (40), (46), and (47); they have some peculiar dynamical properties and their traveling wave phenomena are very interesting. Solution (40) denotes a singular solitary wave; its shape is very similar to bright soliton, but it is not a normal soliton; the left parts of waveform degenerate into a crook. Solution (46) denotes a bounded wave with level asymptote; its shape is a half of a whole loop soliton; we call it semiloop soliton. In order to describe the dynamic properties of these two traveling wave solutions intuitively, we, respectively, plot profile of solutions (40) and (46) by using software Maple, which are shown in Figure 1.Profiles of solutions (40) and (46) for the fixed parameters.
(a)
Singular solitary wave
(b)
Semiloop solitonFigure1(a) shows a shape of singular solitary wave under the fixed parameters α=0.5, β=0.4, ρ1=300, ρ2=1.3. Figure 1(b) shows a shape of semiloop soliton under the fixed parameters α=0.5, β=0.4, ρ1=0.5, ρ2=-4.It is very interesting that the solution (47) has six kinds of waveforms. Solution (47) with “+”, respectively, denotes a dark peakon, a dark loop soliton, and a broken loop soliton when the parameter ρ1 decreases from 1.55 to -8.25, which are shown in Figures 2(a)–2(d). Solution (47) with “−”, respectively, denotes a bright soliton and a singular compacton when the parameter ρ1 decreases from 1.65 to -6.25, which are shown in Figures 2(e) and 2(f). In other words, its waveforms depend on parameter ρ1 extremity. Similar traveling wave phenomena that one solution contains multiwaveform first appeared in the investigation of the Degasperis-Procesi equation [22].Different kinds of waveforms on solution (47) under different parametric conditions.
(a)
Dark soliton
(b)
Dark peakon
(c)
Dark loop soliton
(d)
Broken loop soliton (broken wave of C-form)
(e)
Bright soliton
(f)
Singular compactonFigure2(a) shows a shape of dark soliton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=1.55. Figure 2(b) shows a shape of dark peakon under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=0.479. Figure 2(c) shows a shape of dark loop soliton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=0.05. Similar traveling wave phenomena also appear in [5], in “Concluding remarks” of this literature; by using these phenomena, we successfully explain the movement of water waves. Figure 2(d) shows a shape of broken wave of C-form (upward hatch C) under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=-8.25; we also call it broken loop soliton or open-orbicular wave. Figure 2(e) shows a shape of bright soliton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=1.65. Figure 2(f) shows a shape of singular compacton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=-6.25; we call it singular compacton because it is not a normal compacton. Specifically, both sides of compacton wave become upward crook; it very much likes solitary waves but it is not a solitary wave after all because the left and right parts of waveform do not extend as ξ→∞.


### 3.2. The Exact Solutions under the Parametric Conditions of Case2

Notice that the characters of (26)–(29) are quite different from (25). Therefore, by using the expression dϕ/dξ=y and (26)–(29), we will investigate the implicit solutions of (7) next.Substituting (26) into the expression dϕ/dξ=y of (13) yields
(48)dϕdξ=±ϕ-2α4ρ23ϕ4+2α(3cρ2-3ρ2-1)ϕ+6(c-1)6β(1+αρ2ϕ).
Taking “+”,  (48) can be reduced to
(49)(1+αρ2ϕ)dϕϕ-2α4ρ23ϕ4+2α(3cρ2-3ρ2-1)ϕ+6(c-1)=16βdξ.(i) Whenρ2<0, (49) can be reduced to
(50)(1+αρ2ϕ)dϕ(ϕϕ4-3cρ2-3ρ2-1α3ρ23ϕ-3(c-1)α4ρ23)-1=-α2ρ2-ρ23βdξ.
By the Ferrari method or the Descartes method, it is easy to know that the equation
(51)ϕ4-3cρ2-3ρ2-1α3ρ23ϕ-3(c-1)α4ρ23=0
has four roots as follows:
(52)-1αρ2,(1/6)Δ13-4/3Δ13+1/3αρ2,-(1/12)Δ13+2/3Δ13+1/3αρ2±i(3/2)((1/6)Δ13+4/3Δ13)αρ2,
where Δ1=-28-324ρ2(c-1)+36Δ2 with Δ2=81(1-c)2ρ22+14(1-c)ρ2+1. Especially, when c=1, these four roots can be reduced to -1/αρ2,0,(1±3i)/2αρ2. For the convenience of discussion, we denote the above two real roots by the signs ν,γ and always assume that the biggest root is denoted by ν. We also denote the above two complex roots by the signs s,s-. Thus (50) can be rewritten as
(53)dϕϕ(ϕ-ν)(ϕ-γ)(ϕ-s)(ϕ-s-)+αρ2dϕ(ϕ-ν)(ϕ-γ)(ϕ-s)(ϕ-s-)=-α2ρ2-ρ23βdξ,
where ν>γ. Taking (ν,0) as the initial values and integrating (53), we obtain implicit solution of (7) as follows:
(54)(B~-A~)B~ν+γA~[μ1F(φ,k)+μ-μ11-μ2(Π(φ,μ2μ2-1,k)-μf1)]+gαρ2F(φ,k)=-α2ρ2-ρ23βξ,
where A~2=(ν-(s+s-)/2)2-(s-s-)2/4, B~2=(γ-(s+s-)/2)2-(s-s-)2/4, g=1/A~B~, k2=((A~+B~)2-(ν-γ)2)/4A~B~,
(55)φ=arccos[(A~-B~)ϕ+νB~-γA~(A~+B~)ϕ-νB~-γA~],μ=γA~+νB~νB~-γA~,μ1=A~+B~B~-A~,f1=1-μ2k2+k′2μ2arctan[φk2+k′2μ2(1-μ2)(1-k2φ2)],ifμ2μ2-1<k2;=φ1-k2φ2,ifμ2μ2-1=k2;=12μ2-1k2+k′2μ2×ln[(k2+k′2μ2)(1-k2φ2)+φμ2-1(k2+k′2μ2)(1-k2φ2)-φμ2-1],ifμ2μ2-1>k2.
(ii)  When ρ2>0,   (49) can be reduced to
(56)(1+αρ2ϕ)dϕϕ-ϕ4+((3cρ2-3ρ2-1)/(α3ρ23))ϕ+3(c-1)/α4ρ23=α2ρ2ρ23βdξ.
Similarly, (56) can be rewritten as
(57)dϕϕ(ν-ϕ)(ϕ-γ)(ϕ-s)(ϕ-s-)+αρ2dϕ(ν-ϕ)(ϕ-γ)(ϕ-s)(ϕ-s-)=α2ρ2ρ23βdξ,
where ν,γs,s~ are four roots of (51) and ν≥ϕ>γ. Taking (γ,0) as the initial values then integrating  (56), we obtain another implicit solution of (7) as follows:
(58)(A~+B~)γA~-B~ν[μ2F(φ~,m)+μ~-μ21-μ~2×(Π(φ~,μ~2μ~2-1,m)-μ~f2)]+αρ2gF(φ~,m)=α2ρ2ρ23βξ,
where the constants A~,B~,g~ are given above, the φ~=arccos[(B~(ν-ϕ)-A~(ϕ-γ))/(B~(ν-ϕ)+A~(ϕ-γ))],m2=((ν-γ)2-(A~-B~)2)/4A~B~,μ~=(γA~-νB~)/(γA~+νB~),μ2=(A~-B~)/(A~+B~), and
(59)f2=1-μ~2m2+m′2μ~2arctan[φ~m2+m′2μ~2(1-μ~2)(1-m2φ~2)],ifμ~2μ~2-1<m2;=φ~1-m2φ~2,ifμ~2μ~2-1=m2;=12μ~2-1m2+m′2μ~2×ln[(m2+m′2μ~2)(1-m2φ~2)+φ~μ~2-1(m2+m′2μ~2)(1-m2φ~2)-φ~μ~2-1],ifμ~2μ~2-1>m2.


### 3.3. The Exact Solutions under the Parametric Conditions of Case3

As in Section3.2, substituting (27) into the expression dϕ/dξ=y of (13) yields
(60)dϕdξ=±(ϕ-35α4ρ4ρ2ϕ4+15α3(10ρ22-3ρ4)ϕ3+2ρ2)×(6β(1+αρ2ϕ))-1.
Taking “+”, (60) can be rewritten as
(61)(1+αρ2ϕ)dϕϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3+2/ρ2=16βdξ.
When ρ4ρ2<0, (61) can be reduced to
(62)(1+αρ2ϕ)dϕϕϕ4-((10ρ22-3ρ4)/3αρ4ρ2)ϕ3-(10/3α4ρ4ρ22)=α2-ρ4ρ210βdξ.
When ρ4≠10ρ22/3, the equation
(63)ϕ4-10ρ22-3ρ43αρ4ρ2ϕ3-103α4ρ4ρ22=0
has four roots as follows:(64)-1αρ2,(1/9ρ2)Δ33+10ρ2(10ρ22-9ρ4)/9Δ33+10ρ2/9αρ4,((-118ρ2Δ33-5ρ2(10ρ22-9ρ2)9Δ33+10ρ29)±i32(19ρ2Δ33-10ρ2(10ρ22-9ρ4)9Δ33))×(αρ4)-1,whereΔ3=ρ22(-1350ρ4ρ22+1215ρ42+1000ρ22+135ρ4Δ4) with Δ4=-140ρ4ρ22+100ρ24+81ρ42. When ρ4=10ρ22/3, (63) has another four roots ±1/αρ2,±(1/αρ2)i.(i) Particularly, whenρ2<0 and ρ4=10ρ22/3, (62) can be reduced to
(65)(1+αρ2ϕ)ϕϕ4-(1/α4ρ24)dϕ=-α2ρ2-ρ23βdξ.According to the above cases, we find that (63) still has two real roots and two complex roots; this case is very similar to that in (51) except their especial cases are different. Therefore, the types of their solutions are the same when ρ4ρ2<0 and ρ4≠10ρ22/3. Thus, we omit these parts of discussions. The especial case of (62) (i.e., (65)) can be reduced to
(66)12dϕ2ϕ2(ϕ2)2-(1/α2ρ22)2+αρ2dϕ(ϕ2+1/α2ρ22)(ϕ2-1/α2ρ22)=-α2ρ2-ρ23βdξ,
where ϕ>1/α2ρ22>0. Taking the (1/αρ2,0) as initial values and integrating (66), we obtain (7)'s   periodic wave solution of implicit function as follows:
(67)α2ρ22ϕ2=sec[2ρ2-ρ23βξ-2F(arccos(1αρ2ϕ),22)].(ii) Especially, whenρ2>0 and ρ4=10ρ22/3, (61) can be reduced to
(68)12dϕ2ϕ2(1/α2ρ22)2-(ϕ2)2+αρ2dϕ(ϕ2+1/α2ρ22)(1/α2ρ22-ϕ2)=α2ρ2ρ23βdξ,
where 1/α2ρ22>ϕ>0. Taking the (1/αρ2,0) as initial values and integrating (68), we obtain (7)'s implicit solitary wave solution as follows:
(69)α2ρ22ϕ2=sech[2F(arccos(αρ2ϕ),22)-2ρ2ρ23βξ].


### 3.4. The Exact Solutions under the Parametric Conditions of Case4

Substituting (28) into the expression dϕ/dξ=y of (13) yields
(70)dϕdξ=±ϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3-2αϕ6β(1+αρ2ϕ).
Taking “+”, (70) can be rewritten as
(71)(1+αρ2ϕ)dϕϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3-2αϕ=16βdξ.(
i
) When ρ4ρ2<0, (71) can be reduced to
(72)(1+αρ2ϕ)dϕϕϕ4-((10ρ22-3ρ4)/3αρ4ρ2)ϕ3+(10/3α3ρ4ρ2)ϕ=α2-ρ4ρ210βdξ.
Next, we discuss the roots of the following equation:
(73)ϕ4-10ρ22-3ρ43αρ4ρ2ϕ3+103α3ρ4ρ2ϕ=0.(
a
) When  ρ4<5ρ22/6,   (73) has four real roots as follows:
(74)0,-1αρ2,5ρ2±25ρ22-30ρ43αρ4.(
b
) When ρ4=5ρ22/6, (73) has still four real roots as follows:
(75)0,-1αρ2,2αρ2,2αρ2,
where 2/αρ2 is a double root.(
c
) When ρ4>5ρ22/6, (73) has two real roots and two complex roots as follows:
(76)0,-1αρ2,5ρ2±i30ρ4-25ρ223αρ4.
Thus, by using the above parametric conditions, we can obtain three kinds of exact solutions of implicit function type; see the following discussions.(
1
) Under the conditions 0<ρ4<5ρ22/6 and ρ2<0, (72) can be reduced to
(77)dϕϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-d1)+αρ2dϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-d1)=α2-ρ4ρ210βdξ,
where a1=-1/αρ2, c1=(5ρ2+25ρ22-30ρ4)/3αρ4, d1=(5ρ2-25ρ22-30ρ4)/3αρ4, and ϕ>a1>0>c1>d1. Taking (a1,0) as initial values and integrating (77), we obtain (7)'s   periodic wave solution of implicit function type as follows:
(78)ga1k2[(k2-μ2)F(φ,k)+μ2E(φ,k)]+gαρ2F(φ,k)=α2-ρ4ρ210βξ,
where φ=arcsind1(a1-ϕ)/(a1-d1)ϕ,μ=1-a1/d1,g=2/d1(c1-a1),k=c1(d1-a1)/d1(c1-a1).(
2
) Under the conditions ρ4=5ρ22/6 and ρ2<0, (72) can be reduced to
(79)(1+αρ2ϕ)dϕϕ(ϕ-2/αρ2)ϕ(ϕ+1/αρ2)=-α2ρ2-ρ212βdξ.
Equation (79) can be rewritten as
(80)-αρ2dϕ2ϕϕ(ϕ+1/αρ2)+3αρ2dϕ2(ϕ-2/αρ2)ϕ(ϕ+1/αρ2)=-α2ρ2-ρ212βdξ.
Taking  (-1/αρ2,0) as initial values and integrating (80), we obtain (7)'s solitary wave solution of implicit function type as follows: (81)ρ2ϕ+1/αρ2ϕ-3ρ226×[ln|2/αρ2+5ϕ+26ϕ(ϕ+1/αρ2)αρ2ϕ-2|+ln|αρ2|]=--ρ212βξ.(
3
) Under the conditions ρ4>5ρ22/6 and ρ2<0, (72) can be reduced to
(82)dϕϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-c-1)+αρ2dϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-c-1)=α2-ρ4ρ210βdξ,
where a1=-1/αρ2,c1=(5ρ2+i30ρ4-25ρ22)/3αρ4,c-1=(5ρ2-i30ρ4-25ρ22)/3αρ4, and 0<a1<ϕ<+∞. Taking  (a1,0) as initial values and integrating (82), we obtain (7)'s   periodic wave solution of implicit function type as follows:
(83)(B^-A^)g^B^a1×[F(φ,k^)+(1-α2)E(φ,k^)+(1-α2)φ1-k^2φ1+1-φ]+gαρ2F(φ,k^)=α2-ρ4ρ210βξ,
where φ=arccos[((A^-B^)ϕ+a1B^)/((A^+B^)ϕ-a1B^)], A^=(ρ4+5ρ22)2/α2ρ22ρ42+(30ρ4-25ρ22)/6αρ4, B^=25ρ22/9α2ρ42+(30ρ4-25ρ22)/6αρ4, g^=1/A^B^, k^=((A^+B^)2-a12)/4A^B^.(ii) Whenρ4ρ2>0,  (72) can be rewritten as
(84)(1+αρ2ϕ)dϕϕ-ϕ4+((10ρ22+3ρ4)/3αρ4ρ2)ϕ3-(10/3α3ρ4ρ2)ϕ=α2ρ4ρ210βdξ.(
1
) Under the conditions 0<ρ4<5ρ22/6 and ρ2>0, (84) can be reduced to
(85)dϕϕ(a2-ϕ)(ϕ-b2)(ϕ-0)(ϕ-d2)+αρ2dϕ(a2-ϕ)(ϕ-b2)(ϕ-0)(ϕ-d2)=α2ρ4ρ210βdξ,
where a2=(5ρ2+25ρ22-30ρ4)/3αρ4,b2=(5ρ2-25ρ22-30ρ4)/3αρ4,d2=-1/αρ2, and a2>ϕ>b2>0>d2. Taking (b2,0) as initial values and integrating (85), we obtain   (7)'s   periodic wave solution of implicit function type as follows:
(86)g2b2[b2d2F(φ,k2)+(1-b2d2)E(φ,k2)]+g2αρ2F(φ,k2)=α2ρ4ρ210βξ,
where φ=arcsina2(ϕ-b2)/(a2-b2)ϕ,g2=2/a2(b2-d2),k2=-d2(a2-b2)/a2(b2-d2).(
2
) Under the conditions ρ4=5ρ22/6 and ρ2>0, (84) can be reduced to
(87)-αρ2dϕ2ϕ-ϕ(ϕ+1/αρ2)+3αρ2dϕ2(ϕ-2/αρ2)-ϕ(ϕ+1/αρ2)=α2ρ2ρ212βdξ.
Respectively, taking  (-1/αρ2,0) and (0,0) as initial values, integrating (87), we obtain (7)'s   periodic wave solution of implicit function type as follows:
(88)2+5αρ2ϕ2-ρ2ϕ=cos[αρ2-6ϕ(ϕ+1/αρ2)ϕ+αρ22βξ].(
3
) Under the conditions ρ4>5ρ22/6 and ρ2>0, (84) can be reduced to
(89)dϕϕ(0-ϕ)(ϕ-b1)(ϕ-c1)(ϕ-c-1)+αρ2dϕ(0-ϕ)(ϕ-b1)(ϕ-c1)(ϕ-c-1)=α2ρ4ρ210βdξ,
where b1=-1/αρ2,c1=(5ρ2+i30ρ4-25ρ22)/3αρ4,c-1=(5ρ2-i30ρ4-25ρ22)/3αρ4, and a1<ϕ<0. Taking  (a1,0) as initial values and integrating (89), we obtain (7)'s periodic wave solution of implicit function type as follows:
(90)(A~+B~)b1A~[μ2F(φ~,m)+μ~-μ21-μ~2(Π(φ~,μ~2μ~2-1,m)-μ~f2)]+αρ2gF(φ~,m)=α2ρ4ρ210βξ,
where constants A~,B~,g~ are given above, the φ~=arccos[(B~(ν-ϕ)-A~(ϕ-γ))/(B~(ν-ϕ)+A~(ϕ-γ))],m2=((ν-γ)2-(A~-B~)2)/4A~B~,μ~=(γA~-νB~)/(γA~+νB~),μ2=(A~-B~)/(A~+B~), and
(91)f2=1-μ~2m2+m′2μ~2arctan[φ~m2+m′2μ~2(1-μ~2)(1-m2φ~2)],ifμ~2μ~2-1<m2;=φ~1-m2φ~2,ifμ~2μ~2-1=m2;=12μ~2-1m2+m′2μ~2×ln[(m2+m′2μ~2)(1-m2φ~2)+φ~μ~2-1(m2+m′2μ~2)(1-m2φ~2)-φ~μ~2-1],ifμ~2μ~2-1>m2.


### 3.5. The Exact Solutions under the Parametric Conditions of Case5

Under different parameters, (29) can be reduced different types. As an example, here we only discuss one reduced to type under the parametric conditions ρ1>0,ρ2>0,ρ4<0 for   (29); the others can be similarly discussed.Especially, whenρ1>0,ρ2>0 and ρ4<0, (29) can be reduced to(92)y=±α2-ρ2ρ410β×(ϕ[ϕ3+5ρ13αρ4ϕ2+103α2ρ4ϕ-10(c-1)α3ρ4][ϕ-(-1αρ2)])×(1+αρ2ϕ)-1.Also, we only consider one case where the equationϕ3+(5ρ1/3αρ4)ϕ2+(10/3α2ρ4)ϕ-(10(c-1)/α3ρ4)=0 has three real roots here. The other two cases (i.e., two real roots and one real root) can be similarly discussed. Because the expressions of these three roots are very complex, we use the symbols ϕ1,ϕ2,ϕ3  instead of them. In fact, once the parameters are fixed, these three roots can be solved by the above equation. For example, if α=0.5,β=0.6,c=2,ρ1=6,ρ4=-2, then ϕ1≐1.843,ϕ2≐10.270,ϕ3≐-2.113. In addition, if ρ2=5, then -1/αρ2=-0.4>ϕ3; if ρ2=1/5, then -1/αρ2=-10<ϕ3. Thus, under the above parametric conditions, (92) can be rewritten as
(93)y=±α2-ρ2ρ410βϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-ϕ3)(ϕ-δ)1+αρ2ϕ(ϕ1>ϕ2>δ>ϕ3),
or
(94)y=±α2-ρ2ρ410βϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-δ)(ϕ-ϕ3)1+αρ2ϕ,(ϕ3<δ<ϕ2<ϕ1),
where δ=-1/αρ2.Respectively, substituting (93) and (94) into the equation dϕ/dξ=y of (13) yields
(95)(1+αρ2ϕ)dϕϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-ϕ3)(ϕ-δ)=±α2-ρ2ρ410βdξ,(1+αρ2ϕ)dϕϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-δ)(ϕ-ϕ3)=±α2-ρ2ρ410βdξ.
Taking   (ϕ1,0) as initial values and integrating (95), respectively, we obtain (7)'s two groups of exact solutions of implicit function type as follows:
(96)g1α12ϕ2[(α12ϕ2ϕ1-α12)Π(ϑ1,α12,κ1)+α12F(ϑ1,κ1)]+g1αρ2F(ϑ1,κ1)=±α2-ρ2ρ410βξ,(97)g2α22ϕ2[(α22ϕ2ϕ1-α22)Π(ϑ2,α22,κ2)+α22F(ϑ2,κ2)]+g2αρ2F(ϑ2,κ2)=±α2-ρ2ρ410βξ,
where g1=2/(ϕ1-ϕ3)(ϕ2-δ), g2=2/(ϕ1-δ)(ϕ2-ϕ3), α12=(ϕ1-δ)/(ϕ2-δ)>1, α22=(ϕ1-ϕ3)/(ϕ2-ϕ3)>1, ϑ1=arcsin(ϕ2-δ)(ϕ-ϕ1)/(ϕ1-δ)(ϕ-ϕ2), ϑ2=arcsin(ϕ2-ϕ3)(ϕ-ϕ1)/(ϕ1-ϕ3)(ϕ-ϕ2), κ1=(ϕ2-ϕ3)(ϕ1-δ)/(ϕ1-ϕ3)(ϕ2-δ), κ2=(ϕ2-δ)(ϕ1-ϕ3)/(ϕ1-δ)(ϕ2-ϕ3).Among the above traveling wave solutions, it is also worthy to mention solutions (81), (88), (96), and (97); they have some peculiar dynamical properties, too. Solution (81) denotes a bounded wave with level asymptote under the fixed parameters α=0.7, β=0.4, ρ2=-2. Solution (88) denotes a periodic wave of singular type under the fixed parameters α=0.8, β=0.6, ρ2=4. Solution (96) denotes a broken wave of U-form (leftward fallen U) under the fixed parameters α=0.5, β=0.4, ρ2=5, c=2, ρ1=6, ρ4=-2, ϕ1≈1026989443, ϕ2≈1.843205997, ϕ3≈-2.113100425, δ=-0.40. Solution (97) denotes a broken wave of semi-parabola form under the fixed parameters α=0.5, β=0.4, ρ2=0.2, c=2, ρ1=6, ρ4=-2, ϕ1≈1026989443, ϕ2≈1.843205997, ϕ3≈-2.113100425, δ=-10. In order to describe the dynamic properties of the above traveling wave solutions intuitively, similarly, we, respectively, draw profile of solutions (81), (88), (96), and (97) by using software Maple, which are shown in Figure 3.The profiles of solutions (81), (88), (96), and (97) for the fixed parameters.
(a)
Bounded wave
(b)
Periodic wave of singular type
(c)
Broken wave of U-form
(d)
Broken wave of semiparabola form


## 3.1. The Exact Solutions under the Parametric Conditions of Case1

Substituting (25) into the left expression (i.e., dϕ/dτ=12β(1+αρ2ϕ)y) of (16) yields
(30)dϕϕα4ρ1ρ22ϕ4-α2(ρ1+2ρ2)ϕ2+2/ρ2=±26βdτ.
For the sake of convenience, in all the following discussions, we only discuss the case where the right of equation is “+” sign; the case of “−” sign can be similarly discussed. By the way, the solutions obtained by taking “+” sign and the solutions obtained by taking “−” sign are same when the ϕ is even function.Taking “+” in (30), it can be reduced to
(31)dϕ2ϕ2α4ρ1ρ22(ϕ2)2-α2(ρ1+2ρ2)ϕ2+2/ρ2=46βdτ.
Let ψ=ϕ2. Equation (31) can be rewritten as
(32)dψψα4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=46βdτ.
Taking the (0,ψ1,2) as initial values, respectively, integrating (32) we obtain
(33)∫ψ1ψ‍dψψα4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=∫0τ‍46βdτ,∫ψ2ψ‍dψψα4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=∫0τ‍46βdτ,
where ψ1,2=(ρ1+2ρ2±(ρ1+2ρ2)2-8ρ1)/2ρ1ρ2α2 are two roots of equation α4ρ1ρ22ψ2-α2(ρ1+2ρ2)ψ+2/ρ2=0.(i) Whenρ2>0,Δ=(ρ1+2ρ2)2-8ρ1>0, completing the integrals (33) yields
(34)ψ=4R1Q1-P1cosh(8(3β/ρ2)τ),ψ=4R2P2cosh(8(3β/ρ2)τ)-Q2,
where R1,2=(ρ1+2ρ2±Δ)[(ρ1+2ρ2)Δ±Δ]/2ρ12ρ23α2,Q1,2=(ρ1+2ρ2)[Δ±(ρ1+2ρ2)][(ρ1+2ρ2)Δ±(ρ12+4ρ22)]/2ρ12ρ22, and P1,2=Δ[Δ±(ρ1+2ρ2)]2/4ρ12ρ22 with Δ=(ρ1+2ρ2)2-8ρ1.By using (34) and the transformation ϕ2=ψ, we obtain four exact solutions of (31) as follows:
(35)ϕ=±2R1Q1-P1cosh(8(3β/ρ2)τ),(36)ϕ=±2R2P2cosh(8(3β/ρ2)τ)-Q2,
where the constants R1,2,P1,2,Q1,2 are given above.Substituting (35) into (15) yields
(37)ξ=12β±αρ2g13ρ2βR1×F(arcsinQ1-P1cosh(8(3β/ρ2)τ)Q1-P1,Q1-P1Q1+P1),
where F(φ,k) is a normal elliptic integral of the first kind and g1=2/Q1+P1,Q1>P1>0,0<τ<(1/8)(ρ2/3β)cosh-1(Q1/P1).Substituting (36) into (15) yields
(38)ξ=12β±αρ2g23ρ2βR2×F(arcsinP2(cosh(8(3β/ρ2)τ)-1)P2cosh(8(3β/ρ2)τ)-Q2,P2+Q22P2),
where g2=2/ρ2,P2>Q2>0, and 0<τ<+∞.Combining (35) with (37), we obtain (7)'s two exact solutions of parametric type as follows:
(39)η=ϕ=±2R1Q1-P1cosh(8(3β/ρ2)τ),ξ=12βτ±αρ2g13ρ2βR1×F(arcsinQ1-P1cosh(8(3β/ρ2)τ)Q1-P1,Q1-P1Q1+P1).Combining (36) with (38), we obtain (7)'s another two exact solutions of parametric type as follows:(40)η=ϕ=±2R2P2cosh(8(3β/ρ2)τ)-Q2,ξ=12βτ±αρ2g23ρ2βR2×F(arcsinP2(cosh(8(3β/ρ2)τ)-1)P2cosh(8(3β/ρ2)τ)-Q2,P2+Q22P2).(ii) Whenρ2<0,Δ=(ρ1+2ρ2)2-8ρ1>0, completing the integrals (33) yields
(41)ψ=4-α2ρ2Δ[-α2(ρ1+2ρ2)-cos(8-(3β/ρ2)τ)],ψ=4-α2ρ2Δ[-α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ)].
Similarly, by using (41) and the transformation ϕ2=ψ, we obtain four periodic solutions of (31) as follows:
(42)ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)-cos(8-(3β/ρ2)τ),(43)ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ),
where ρ1<-(2ρ2+1/α2); that is, -α2(ρ1+2ρ2)>1>0. Respectively, substituting (42) and (43) into (15) yields(44)ξ=12βτ±g~ρ23βΔ4×F(arcsin[1-cos(8-(3β/ρ2)τ)][α2(ρ1+2ρ2)-1]2[α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ)],k),(45)ξ=12βτ±g~ρ23βΔ4F(4-3βρ2τ,k),whereg~=2/1-α2(ρ1+2ρ2),k=2/(1-α2(ρ1+2ρ2)).Combining (42) with (44), we obtain (7)'s two exact solutions of parametric type as follows:(46)η=ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)-cos(8-(3β/ρ2)τ),ξ=12βτ±g~ρ23βΔ4×F(arcsin[1-cos(8-(3β/ρ2)τ)][α2(ρ1+2ρ2)-1]2[α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ)],k),where0<τ≤π-ρ2/83β.Combining (43) with (45), we obtain (7)'s another two exact solutions of parametric type as follows:
(47)η=ϕ=±2-α2ρ2Δ-α2(ρ1+2ρ2)+cos(8-(3β/ρ2)τ),ξ=12βτ±g~ρ23βΔ4F(4-3βρ2τ,k).Among the above traveling wave solutions, it is very worthy to mention solutions (40), (46), and (47); they have some peculiar dynamical properties and their traveling wave phenomena are very interesting. Solution (40) denotes a singular solitary wave; its shape is very similar to bright soliton, but it is not a normal soliton; the left parts of waveform degenerate into a crook. Solution (46) denotes a bounded wave with level asymptote; its shape is a half of a whole loop soliton; we call it semiloop soliton. In order to describe the dynamic properties of these two traveling wave solutions intuitively, we, respectively, plot profile of solutions (40) and (46) by using software Maple, which are shown in Figure 1.Profiles of solutions (40) and (46) for the fixed parameters.
(a)
Singular solitary wave
(b)
Semiloop solitonFigure1(a) shows a shape of singular solitary wave under the fixed parameters α=0.5, β=0.4, ρ1=300, ρ2=1.3. Figure 1(b) shows a shape of semiloop soliton under the fixed parameters α=0.5, β=0.4, ρ1=0.5, ρ2=-4.It is very interesting that the solution (47) has six kinds of waveforms. Solution (47) with “+”, respectively, denotes a dark peakon, a dark loop soliton, and a broken loop soliton when the parameter ρ1 decreases from 1.55 to -8.25, which are shown in Figures 2(a)–2(d). Solution (47) with “−”, respectively, denotes a bright soliton and a singular compacton when the parameter ρ1 decreases from 1.65 to -6.25, which are shown in Figures 2(e) and 2(f). In other words, its waveforms depend on parameter ρ1 extremity. Similar traveling wave phenomena that one solution contains multiwaveform first appeared in the investigation of the Degasperis-Procesi equation [22].Different kinds of waveforms on solution (47) under different parametric conditions.
(a)
Dark soliton
(b)
Dark peakon
(c)
Dark loop soliton
(d)
Broken loop soliton (broken wave of C-form)
(e)
Bright soliton
(f)
Singular compactonFigure2(a) shows a shape of dark soliton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=1.55. Figure 2(b) shows a shape of dark peakon under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=0.479. Figure 2(c) shows a shape of dark loop soliton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=0.05. Similar traveling wave phenomena also appear in [5], in “Concluding remarks” of this literature; by using these phenomena, we successfully explain the movement of water waves. Figure 2(d) shows a shape of broken wave of C-form (upward hatch C) under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=-8.25; we also call it broken loop soliton or open-orbicular wave. Figure 2(e) shows a shape of bright soliton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=1.65. Figure 2(f) shows a shape of singular compacton under the fixed parameters α=0.4, β=0.1, ρ2=-4, and ρ1=-6.25; we call it singular compacton because it is not a normal compacton. Specifically, both sides of compacton wave become upward crook; it very much likes solitary waves but it is not a solitary wave after all because the left and right parts of waveform do not extend as ξ→∞.


## 3.2. The Exact Solutions under the Parametric Conditions of Case2

Notice that the characters of (26)–(29) are quite different from (25). Therefore, by using the expression dϕ/dξ=y and (26)–(29), we will investigate the implicit solutions of (7) next.Substituting (26) into the expression dϕ/dξ=y of (13) yields
(48)dϕdξ=±ϕ-2α4ρ23ϕ4+2α(3cρ2-3ρ2-1)ϕ+6(c-1)6β(1+αρ2ϕ).
Taking “+”,  (48) can be reduced to
(49)(1+αρ2ϕ)dϕϕ-2α4ρ23ϕ4+2α(3cρ2-3ρ2-1)ϕ+6(c-1)=16βdξ.(i) Whenρ2<0, (49) can be reduced to
(50)(1+αρ2ϕ)dϕ(ϕϕ4-3cρ2-3ρ2-1α3ρ23ϕ-3(c-1)α4ρ23)-1=-α2ρ2-ρ23βdξ.
By the Ferrari method or the Descartes method, it is easy to know that the equation
(51)ϕ4-3cρ2-3ρ2-1α3ρ23ϕ-3(c-1)α4ρ23=0
has four roots as follows:
(52)-1αρ2,(1/6)Δ13-4/3Δ13+1/3αρ2,-(1/12)Δ13+2/3Δ13+1/3αρ2±i(3/2)((1/6)Δ13+4/3Δ13)αρ2,
where Δ1=-28-324ρ2(c-1)+36Δ2 with Δ2=81(1-c)2ρ22+14(1-c)ρ2+1. Especially, when c=1, these four roots can be reduced to -1/αρ2,0,(1±3i)/2αρ2. For the convenience of discussion, we denote the above two real roots by the signs ν,γ and always assume that the biggest root is denoted by ν. We also denote the above two complex roots by the signs s,s-. Thus (50) can be rewritten as
(53)dϕϕ(ϕ-ν)(ϕ-γ)(ϕ-s)(ϕ-s-)+αρ2dϕ(ϕ-ν)(ϕ-γ)(ϕ-s)(ϕ-s-)=-α2ρ2-ρ23βdξ,
where ν>γ. Taking (ν,0) as the initial values and integrating (53), we obtain implicit solution of (7) as follows:
(54)(B~-A~)B~ν+γA~[μ1F(φ,k)+μ-μ11-μ2(Π(φ,μ2μ2-1,k)-μf1)]+gαρ2F(φ,k)=-α2ρ2-ρ23βξ,
where A~2=(ν-(s+s-)/2)2-(s-s-)2/4, B~2=(γ-(s+s-)/2)2-(s-s-)2/4, g=1/A~B~, k2=((A~+B~)2-(ν-γ)2)/4A~B~,
(55)φ=arccos[(A~-B~)ϕ+νB~-γA~(A~+B~)ϕ-νB~-γA~],μ=γA~+νB~νB~-γA~,μ1=A~+B~B~-A~,f1=1-μ2k2+k′2μ2arctan[φk2+k′2μ2(1-μ2)(1-k2φ2)],ifμ2μ2-1<k2;=φ1-k2φ2,ifμ2μ2-1=k2;=12μ2-1k2+k′2μ2×ln[(k2+k′2μ2)(1-k2φ2)+φμ2-1(k2+k′2μ2)(1-k2φ2)-φμ2-1],ifμ2μ2-1>k2.
(ii)  When ρ2>0,   (49) can be reduced to
(56)(1+αρ2ϕ)dϕϕ-ϕ4+((3cρ2-3ρ2-1)/(α3ρ23))ϕ+3(c-1)/α4ρ23=α2ρ2ρ23βdξ.
Similarly, (56) can be rewritten as
(57)dϕϕ(ν-ϕ)(ϕ-γ)(ϕ-s)(ϕ-s-)+αρ2dϕ(ν-ϕ)(ϕ-γ)(ϕ-s)(ϕ-s-)=α2ρ2ρ23βdξ,
where ν,γs,s~ are four roots of (51) and ν≥ϕ>γ. Taking (γ,0) as the initial values then integrating  (56), we obtain another implicit solution of (7) as follows:
(58)(A~+B~)γA~-B~ν[μ2F(φ~,m)+μ~-μ21-μ~2×(Π(φ~,μ~2μ~2-1,m)-μ~f2)]+αρ2gF(φ~,m)=α2ρ2ρ23βξ,
where the constants A~,B~,g~ are given above, the φ~=arccos[(B~(ν-ϕ)-A~(ϕ-γ))/(B~(ν-ϕ)+A~(ϕ-γ))],m2=((ν-γ)2-(A~-B~)2)/4A~B~,μ~=(γA~-νB~)/(γA~+νB~),μ2=(A~-B~)/(A~+B~), and
(59)f2=1-μ~2m2+m′2μ~2arctan[φ~m2+m′2μ~2(1-μ~2)(1-m2φ~2)],ifμ~2μ~2-1<m2;=φ~1-m2φ~2,ifμ~2μ~2-1=m2;=12μ~2-1m2+m′2μ~2×ln[(m2+m′2μ~2)(1-m2φ~2)+φ~μ~2-1(m2+m′2μ~2)(1-m2φ~2)-φ~μ~2-1],ifμ~2μ~2-1>m2.


## 3.3. The Exact Solutions under the Parametric Conditions of Case3

As in Section3.2, substituting (27) into the expression dϕ/dξ=y of (13) yields
(60)dϕdξ=±(ϕ-35α4ρ4ρ2ϕ4+15α3(10ρ22-3ρ4)ϕ3+2ρ2)×(6β(1+αρ2ϕ))-1.
Taking “+”, (60) can be rewritten as
(61)(1+αρ2ϕ)dϕϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3+2/ρ2=16βdξ.
When ρ4ρ2<0, (61) can be reduced to
(62)(1+αρ2ϕ)dϕϕϕ4-((10ρ22-3ρ4)/3αρ4ρ2)ϕ3-(10/3α4ρ4ρ22)=α2-ρ4ρ210βdξ.
When ρ4≠10ρ22/3, the equation
(63)ϕ4-10ρ22-3ρ43αρ4ρ2ϕ3-103α4ρ4ρ22=0
has four roots as follows:(64)-1αρ2,(1/9ρ2)Δ33+10ρ2(10ρ22-9ρ4)/9Δ33+10ρ2/9αρ4,((-118ρ2Δ33-5ρ2(10ρ22-9ρ2)9Δ33+10ρ29)±i32(19ρ2Δ33-10ρ2(10ρ22-9ρ4)9Δ33))×(αρ4)-1,whereΔ3=ρ22(-1350ρ4ρ22+1215ρ42+1000ρ22+135ρ4Δ4) with Δ4=-140ρ4ρ22+100ρ24+81ρ42. When ρ4=10ρ22/3, (63) has another four roots ±1/αρ2,±(1/αρ2)i.(i) Particularly, whenρ2<0 and ρ4=10ρ22/3, (62) can be reduced to
(65)(1+αρ2ϕ)ϕϕ4-(1/α4ρ24)dϕ=-α2ρ2-ρ23βdξ.According to the above cases, we find that (63) still has two real roots and two complex roots; this case is very similar to that in (51) except their especial cases are different. Therefore, the types of their solutions are the same when ρ4ρ2<0 and ρ4≠10ρ22/3. Thus, we omit these parts of discussions. The especial case of (62) (i.e., (65)) can be reduced to
(66)12dϕ2ϕ2(ϕ2)2-(1/α2ρ22)2+αρ2dϕ(ϕ2+1/α2ρ22)(ϕ2-1/α2ρ22)=-α2ρ2-ρ23βdξ,
where ϕ>1/α2ρ22>0. Taking the (1/αρ2,0) as initial values and integrating (66), we obtain (7)'s   periodic wave solution of implicit function as follows:
(67)α2ρ22ϕ2=sec[2ρ2-ρ23βξ-2F(arccos(1αρ2ϕ),22)].(ii) Especially, whenρ2>0 and ρ4=10ρ22/3, (61) can be reduced to
(68)12dϕ2ϕ2(1/α2ρ22)2-(ϕ2)2+αρ2dϕ(ϕ2+1/α2ρ22)(1/α2ρ22-ϕ2)=α2ρ2ρ23βdξ,
where 1/α2ρ22>ϕ>0. Taking the (1/αρ2,0) as initial values and integrating (68), we obtain (7)'s implicit solitary wave solution as follows:
(69)α2ρ22ϕ2=sech[2F(arccos(αρ2ϕ),22)-2ρ2ρ23βξ].


## 3.4. The Exact Solutions under the Parametric Conditions of Case4

Substituting (28) into the expression dϕ/dξ=y of (13) yields
(70)dϕdξ=±ϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3-2αϕ6β(1+αρ2ϕ).
Taking “+”, (70) can be rewritten as
(71)(1+αρ2ϕ)dϕϕ-(3/5)α4ρ4ρ2ϕ4+(1/5)α3(10ρ22-3ρ4)ϕ3-2αϕ=16βdξ.(
i
) When ρ4ρ2<0, (71) can be reduced to
(72)(1+αρ2ϕ)dϕϕϕ4-((10ρ22-3ρ4)/3αρ4ρ2)ϕ3+(10/3α3ρ4ρ2)ϕ=α2-ρ4ρ210βdξ.
Next, we discuss the roots of the following equation:
(73)ϕ4-10ρ22-3ρ43αρ4ρ2ϕ3+103α3ρ4ρ2ϕ=0.(
a
) When  ρ4<5ρ22/6,   (73) has four real roots as follows:
(74)0,-1αρ2,5ρ2±25ρ22-30ρ43αρ4.(
b
) When ρ4=5ρ22/6, (73) has still four real roots as follows:
(75)0,-1αρ2,2αρ2,2αρ2,
where 2/αρ2 is a double root.(
c
) When ρ4>5ρ22/6, (73) has two real roots and two complex roots as follows:
(76)0,-1αρ2,5ρ2±i30ρ4-25ρ223αρ4.
Thus, by using the above parametric conditions, we can obtain three kinds of exact solutions of implicit function type; see the following discussions.(
1
) Under the conditions 0<ρ4<5ρ22/6 and ρ2<0, (72) can be reduced to
(77)dϕϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-d1)+αρ2dϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-d1)=α2-ρ4ρ210βdξ,
where a1=-1/αρ2, c1=(5ρ2+25ρ22-30ρ4)/3αρ4, d1=(5ρ2-25ρ22-30ρ4)/3αρ4, and ϕ>a1>0>c1>d1. Taking (a1,0) as initial values and integrating (77), we obtain (7)'s   periodic wave solution of implicit function type as follows:
(78)ga1k2[(k2-μ2)F(φ,k)+μ2E(φ,k)]+gαρ2F(φ,k)=α2-ρ4ρ210βξ,
where φ=arcsind1(a1-ϕ)/(a1-d1)ϕ,μ=1-a1/d1,g=2/d1(c1-a1),k=c1(d1-a1)/d1(c1-a1).(
2
) Under the conditions ρ4=5ρ22/6 and ρ2<0, (72) can be reduced to
(79)(1+αρ2ϕ)dϕϕ(ϕ-2/αρ2)ϕ(ϕ+1/αρ2)=-α2ρ2-ρ212βdξ.
Equation (79) can be rewritten as
(80)-αρ2dϕ2ϕϕ(ϕ+1/αρ2)+3αρ2dϕ2(ϕ-2/αρ2)ϕ(ϕ+1/αρ2)=-α2ρ2-ρ212βdξ.
Taking  (-1/αρ2,0) as initial values and integrating (80), we obtain (7)'s solitary wave solution of implicit function type as follows: (81)ρ2ϕ+1/αρ2ϕ-3ρ226×[ln|2/αρ2+5ϕ+26ϕ(ϕ+1/αρ2)αρ2ϕ-2|+ln|αρ2|]=--ρ212βξ.(
3
) Under the conditions ρ4>5ρ22/6 and ρ2<0, (72) can be reduced to
(82)dϕϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-c-1)+αρ2dϕ(ϕ-a1)(ϕ-0)(ϕ-c1)(ϕ-c-1)=α2-ρ4ρ210βdξ,
where a1=-1/αρ2,c1=(5ρ2+i30ρ4-25ρ22)/3αρ4,c-1=(5ρ2-i30ρ4-25ρ22)/3αρ4, and 0<a1<ϕ<+∞. Taking  (a1,0) as initial values and integrating (82), we obtain (7)'s   periodic wave solution of implicit function type as follows:
(83)(B^-A^)g^B^a1×[F(φ,k^)+(1-α2)E(φ,k^)+(1-α2)φ1-k^2φ1+1-φ]+gαρ2F(φ,k^)=α2-ρ4ρ210βξ,
where φ=arccos[((A^-B^)ϕ+a1B^)/((A^+B^)ϕ-a1B^)], A^=(ρ4+5ρ22)2/α2ρ22ρ42+(30ρ4-25ρ22)/6αρ4, B^=25ρ22/9α2ρ42+(30ρ4-25ρ22)/6αρ4, g^=1/A^B^, k^=((A^+B^)2-a12)/4A^B^.(ii) Whenρ4ρ2>0,  (72) can be rewritten as
(84)(1+αρ2ϕ)dϕϕ-ϕ4+((10ρ22+3ρ4)/3αρ4ρ2)ϕ3-(10/3α3ρ4ρ2)ϕ=α2ρ4ρ210βdξ.(
1
) Under the conditions 0<ρ4<5ρ22/6 and ρ2>0, (84) can be reduced to
(85)dϕϕ(a2-ϕ)(ϕ-b2)(ϕ-0)(ϕ-d2)+αρ2dϕ(a2-ϕ)(ϕ-b2)(ϕ-0)(ϕ-d2)=α2ρ4ρ210βdξ,
where a2=(5ρ2+25ρ22-30ρ4)/3αρ4,b2=(5ρ2-25ρ22-30ρ4)/3αρ4,d2=-1/αρ2, and a2>ϕ>b2>0>d2. Taking (b2,0) as initial values and integrating (85), we obtain   (7)'s   periodic wave solution of implicit function type as follows:
(86)g2b2[b2d2F(φ,k2)+(1-b2d2)E(φ,k2)]+g2αρ2F(φ,k2)=α2ρ4ρ210βξ,
where φ=arcsina2(ϕ-b2)/(a2-b2)ϕ,g2=2/a2(b2-d2),k2=-d2(a2-b2)/a2(b2-d2).(
2
) Under the conditions ρ4=5ρ22/6 and ρ2>0, (84) can be reduced to
(87)-αρ2dϕ2ϕ-ϕ(ϕ+1/αρ2)+3αρ2dϕ2(ϕ-2/αρ2)-ϕ(ϕ+1/αρ2)=α2ρ2ρ212βdξ.
Respectively, taking  (-1/αρ2,0) and (0,0) as initial values, integrating (87), we obtain (7)'s   periodic wave solution of implicit function type as follows:
(88)2+5αρ2ϕ2-ρ2ϕ=cos[αρ2-6ϕ(ϕ+1/αρ2)ϕ+αρ22βξ].(
3
) Under the conditions ρ4>5ρ22/6 and ρ2>0, (84) can be reduced to
(89)dϕϕ(0-ϕ)(ϕ-b1)(ϕ-c1)(ϕ-c-1)+αρ2dϕ(0-ϕ)(ϕ-b1)(ϕ-c1)(ϕ-c-1)=α2ρ4ρ210βdξ,
where b1=-1/αρ2,c1=(5ρ2+i30ρ4-25ρ22)/3αρ4,c-1=(5ρ2-i30ρ4-25ρ22)/3αρ4, and a1<ϕ<0. Taking  (a1,0) as initial values and integrating (89), we obtain (7)'s periodic wave solution of implicit function type as follows:
(90)(A~+B~)b1A~[μ2F(φ~,m)+μ~-μ21-μ~2(Π(φ~,μ~2μ~2-1,m)-μ~f2)]+αρ2gF(φ~,m)=α2ρ4ρ210βξ,
where constants A~,B~,g~ are given above, the φ~=arccos[(B~(ν-ϕ)-A~(ϕ-γ))/(B~(ν-ϕ)+A~(ϕ-γ))],m2=((ν-γ)2-(A~-B~)2)/4A~B~,μ~=(γA~-νB~)/(γA~+νB~),μ2=(A~-B~)/(A~+B~), and
(91)f2=1-μ~2m2+m′2μ~2arctan[φ~m2+m′2μ~2(1-μ~2)(1-m2φ~2)],ifμ~2μ~2-1<m2;=φ~1-m2φ~2,ifμ~2μ~2-1=m2;=12μ~2-1m2+m′2μ~2×ln[(m2+m′2μ~2)(1-m2φ~2)+φ~μ~2-1(m2+m′2μ~2)(1-m2φ~2)-φ~μ~2-1],ifμ~2μ~2-1>m2.


## 3.5. The Exact Solutions under the Parametric Conditions of Case5

Under different parameters, (29) can be reduced different types. As an example, here we only discuss one reduced to type under the parametric conditions ρ1>0,ρ2>0,ρ4<0 for   (29); the others can be similarly discussed.Especially, whenρ1>0,ρ2>0 and ρ4<0, (29) can be reduced to(92)y=±α2-ρ2ρ410β×(ϕ[ϕ3+5ρ13αρ4ϕ2+103α2ρ4ϕ-10(c-1)α3ρ4][ϕ-(-1αρ2)])×(1+αρ2ϕ)-1.Also, we only consider one case where the equationϕ3+(5ρ1/3αρ4)ϕ2+(10/3α2ρ4)ϕ-(10(c-1)/α3ρ4)=0 has three real roots here. The other two cases (i.e., two real roots and one real root) can be similarly discussed. Because the expressions of these three roots are very complex, we use the symbols ϕ1,ϕ2,ϕ3  instead of them. In fact, once the parameters are fixed, these three roots can be solved by the above equation. For example, if α=0.5,β=0.6,c=2,ρ1=6,ρ4=-2, then ϕ1≐1.843,ϕ2≐10.270,ϕ3≐-2.113. In addition, if ρ2=5, then -1/αρ2=-0.4>ϕ3; if ρ2=1/5, then -1/αρ2=-10<ϕ3. Thus, under the above parametric conditions, (92) can be rewritten as
(93)y=±α2-ρ2ρ410βϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-ϕ3)(ϕ-δ)1+αρ2ϕ(ϕ1>ϕ2>δ>ϕ3),
or
(94)y=±α2-ρ2ρ410βϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-δ)(ϕ-ϕ3)1+αρ2ϕ,(ϕ3<δ<ϕ2<ϕ1),
where δ=-1/αρ2.Respectively, substituting (93) and (94) into the equation dϕ/dξ=y of (13) yields
(95)(1+αρ2ϕ)dϕϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-ϕ3)(ϕ-δ)=±α2-ρ2ρ410βdξ,(1+αρ2ϕ)dϕϕ(ϕ-ϕ1)(ϕ-ϕ2)(ϕ-δ)(ϕ-ϕ3)=±α2-ρ2ρ410βdξ.
Taking   (ϕ1,0) as initial values and integrating (95), respectively, we obtain (7)'s two groups of exact solutions of implicit function type as follows:
(96)g1α12ϕ2[(α12ϕ2ϕ1-α12)Π(ϑ1,α12,κ1)+α12F(ϑ1,κ1)]+g1αρ2F(ϑ1,κ1)=±α2-ρ2ρ410βξ,(97)g2α22ϕ2[(α22ϕ2ϕ1-α22)Π(ϑ2,α22,κ2)+α22F(ϑ2,κ2)]+g2αρ2F(ϑ2,κ2)=±α2-ρ2ρ410βξ,
where g1=2/(ϕ1-ϕ3)(ϕ2-δ), g2=2/(ϕ1-δ)(ϕ2-ϕ3), α12=(ϕ1-δ)/(ϕ2-δ)>1, α22=(ϕ1-ϕ3)/(ϕ2-ϕ3)>1, ϑ1=arcsin(ϕ2-δ)(ϕ-ϕ1)/(ϕ1-δ)(ϕ-ϕ2), ϑ2=arcsin(ϕ2-ϕ3)(ϕ-ϕ1)/(ϕ1-ϕ3)(ϕ-ϕ2), κ1=(ϕ2-ϕ3)(ϕ1-δ)/(ϕ1-ϕ3)(ϕ2-δ), κ2=(ϕ2-δ)(ϕ1-ϕ3)/(ϕ1-δ)(ϕ2-ϕ3).Among the above traveling wave solutions, it is also worthy to mention solutions (81), (88), (96), and (97); they have some peculiar dynamical properties, too. Solution (81) denotes a bounded wave with level asymptote under the fixed parameters α=0.7, β=0.4, ρ2=-2. Solution (88) denotes a periodic wave of singular type under the fixed parameters α=0.8, β=0.6, ρ2=4. Solution (96) denotes a broken wave of U-form (leftward fallen U) under the fixed parameters α=0.5, β=0.4, ρ2=5, c=2, ρ1=6, ρ4=-2, ϕ1≈1026989443, ϕ2≈1.843205997, ϕ3≈-2.113100425, δ=-0.40. Solution (97) denotes a broken wave of semi-parabola form under the fixed parameters α=0.5, β=0.4, ρ2=0.2, c=2, ρ1=6, ρ4=-2, ϕ1≈1026989443, ϕ2≈1.843205997, ϕ3≈-2.113100425, δ=-10. In order to describe the dynamic properties of the above traveling wave solutions intuitively, similarly, we, respectively, draw profile of solutions (81), (88), (96), and (97) by using software Maple, which are shown in Figure 3.The profiles of solutions (81), (88), (96), and (97) for the fixed parameters.
(a)
Bounded wave
(b)
Periodic wave of singular type
(c)
Broken wave of U-form
(d)
Broken wave of semiparabola form


## 4.   Conclusion

In this work, by using the integral bifurcation method, we studied a generalized KdV equation of neglecting the highest order infinitesimal term. Some singular and nonsingular traveling wave solutions including singular solitary wave solutions, semiloop soliton solutions, dark soliton solutions, dark peakon solutions, dark loop-soliton solutions, broken wave solutions of C-form and U-form, periodic wave solutions of singular type, and broken wave solution of semi-parabola form were obtained. Among these traveling wave solutions, the solutions such as (40), (46), (47), (81), (88), (96), and (97) have some peculiar dynamical properties and very interesting traveling wave phenomena. It is very worthy to mention   solution (47); it denotes a dark peakon, a dark loop soliton, a broken loop soliton, a bright soliton, and a singular compacton when the parameter ρ1 varies. This course of changes of waveform shows that solution (47) is a continuous solution. Indeed, the loop soliton which contains (47) is also continuous; this case has proved again that the loop soliton solution of some nonlinear traveling wave equation is one continuous solution, not three breaking solutions; see the arguments in [23, 24]. However, the phenomenon of half loop soliton exists in troth. For example, the semiloop soliton and broken loop soliton show that their waveforms are only a part of a whole loop soliton. Maybe, a loop soliton can be divided into two parts or three parts like the semiloop soliton or three breaking waves which appeared in [24], but every part of them is still continuous, and then joining them together, they come into being a continuous loop soliton again. Indeed, in 2006, Vakhnenko and Parkes's work [25] successfully explained similar phenomena. In [25], the graphical interpretation of the solution for gDPE is presented. In this analysis, the 3D-spiral (whether one loop from a spiral or a half loop of a spiral) has the different projections that is the essence of the possible solutions.The results obtained in this work are very different from those in other literatures because this special case of   (1) for ρ5=ρ6=ρ7=0,ρ3=2ρ2 is not studied in other references. Especially, the results of (7) are different from those of (4) because of ρ4≠0. In addition, we hope more and more researchers pay attention to investigations of exact solutions for (7) under general conditions in the future work.



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*Source: 656297-2013-03-21.xml*