Volume 2013, Article ID 656297,15pages http://dx.doi.org/10.1155/2013/656297
Research Article
A Generalized KdV Equation of Neglecting the Highest-Order Infinitesimal Term and Its Exact Traveling Wave Solutions
Xianbin Wu,
1Weiguo Rui,
2and Xiaochun Hong
31Junior College, Zhejiang Wanli University, Ningbo 315100, China
2College of Mathematics, Honghe University, Mengzi, Yunnan 661100, China
3College of Statistics and Mathematics, Yunnan University of Finance and Economics, Kunming, Yunnan 650221, China
Correspondence should be addressed to Weiguo Rui; [email protected] Received 3 November 2012; Accepted 15 January 2013
Academic Editor: Julian L´opez-G´omez
Copyright © 2013 Xianbin Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 software Maple, 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.
1. Introduction
In 1995, based on the physical and asymptotic considerations, Fokas [1] derived the following generalized KdV equation:
𝜂𝑡+ 𝜂𝑥+ 𝛼𝜂𝜂𝑥+ 𝛽𝜂𝑥𝑥𝑥+ 𝜌1𝛼2𝜂2𝜂𝑥+ 𝛼𝛽 (𝜌2𝜂𝜂𝑥𝑥𝑥+ 𝜌3𝜂𝑥𝜂𝑥𝑥) + 𝜌4𝛼3𝜂3𝜂𝑥+ 𝛼2𝛽 (𝜌5𝜂2𝜂𝑥𝑥𝑥+ 𝜌6𝜂𝜂𝑥𝜂𝑥𝑥+ 𝜌7𝜂3𝑥) = 0, (1) which is an important water wave model, where𝛼 = 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
̃𝜌4 to replace the𝜌4𝛼2 (i.e., 𝜌4𝛼2 → ̃𝜌4), (1) becomes the following PDE:
𝑢𝑡+𝑢𝑥+𝛼𝑢𝑢𝑥+𝛽𝑢𝑥𝑥𝑥+𝜌1𝛼2𝑢2𝑢𝑥+𝛼𝛽 (𝜌2𝑢𝑢𝑥𝑥𝑥+𝜌3𝑢𝑥𝑢𝑥𝑥) + ̃𝜌4𝛼𝑢3𝑢𝑥+ 𝛼2𝛽 (𝜌5𝑢2𝑢𝑥𝑥𝑥+ 𝜌6𝑢𝑢𝑥𝑢𝑥𝑥+ 𝜌7𝑢3𝑥) = 0,
(2) 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 meaning-ful.
The motion described by the model (1) or (2) is a 2-dimen- sional, inviscid, incompressible fluid (water) lying above a horizontal fiat bottom located at𝑦 = −ℎ0(ℎ0is 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/6with𝐴 = 𝑎/ℎ0,𝐵 = ℎ20/ℓ2, where𝑎andℓ are two typical values of the amplitude and of the wavelength of the waves. The parameters 𝑎, ℎ0, ℓsatisfy the condition 𝑎 ≪ ℎ0< ℓbecause the system is a model of short amplitude and long wavelength. Therefore the parameters𝛼and𝛽satisfy the condition0 < 𝛼 < 1,0 < 𝛽 < 1.
Assuming that the waves are unidirectional and neglect- ing terms of𝑂(𝛼2, 𝛼3, 𝛼𝛽), (1) can be reduced to the classical KdV equation:
𝜂𝑡+ 𝜂𝑥+ 𝛼𝜂𝜂𝑥+ 𝛽𝜂𝑥𝑥𝑥= 0. (3) In [1], Fokas assumed that𝑂(𝐵)is less than𝑂(𝐴); this implies 𝑂(𝛽) < 𝑂(𝛼). According to this assumption, we easily know
that𝑂(𝛼2𝛽) < 𝑂(𝛼3)and𝑂(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:
𝜂𝑡+ 𝜂𝑥+ 𝛼𝜂𝜂𝑥+ 𝛽𝜂𝑥𝑥𝑥+ 𝜌1𝛼2𝜂2𝜂𝑥
+ 𝛼𝛽 (𝜌2𝜂𝜂𝑥𝑥𝑥+ 𝜌3𝜂𝑥𝜂𝑥𝑥) = 0. (4) Equation (4) is a special case of (1) for𝜌4= 𝜌5= 𝜌6= 𝜌7= 0.
In [1], it was observed that (4) can be reduced by the local transformation of coordinates
𝜂 = 𝑣 − 𝛼𝜌1𝑣2− 𝛽 (3𝜌1+7 4𝜌2−1
2𝜌3) 𝑣𝑥𝑥 (5) to a completely integrable PDE as follows:
𝑣𝑡−3
2𝛽𝜌2𝑣𝑥𝑥𝑡+ 𝛽 (1 −3
2𝜌2) 𝑣𝑥𝑥𝑥+ 𝛼𝑣𝑣𝑥
−1
2𝛼𝛽𝜌2(𝑣𝑣𝑥𝑥𝑥+ 2𝑣𝑥𝑣𝑥𝑥) .
(6)
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 of 𝑂(𝛼2𝛽), then (1) can be reduced to a new generalized KdV equation as follows:
𝜂𝑡+ 𝜂𝑥+ 𝛼𝜂𝜂𝑥+ 𝛽𝜂𝑥𝑥𝑥+ 𝜌1𝛼2𝜂2𝜂𝑥
+ 𝛼𝛽 (𝜌2𝜂𝜂𝑥𝑥𝑥+ 𝜌3𝜂𝑥𝜂𝑥𝑥) + 𝜌4𝛼3𝜂3𝜂𝑥= 0. (7) We call it a generalized KdV equation of neglecting the high- est order infinitesimal term. In fact, (7) is another special case of (1) for𝜌5 = 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 𝜂 = 𝑢 + 𝜆1𝛼𝑢2+ 𝜆2𝛽𝑢𝑥𝑥+ 𝜆3𝛼2𝑢3
+ 𝛼𝛽 (𝜆4𝑢𝑢𝑥𝑥+ 𝜆5𝑢2𝑥) , (8) (1) can be reduced to the following generalized Gardner equation [1]:
𝑢𝑡+ 𝑢𝑥−3
2𝜌2𝛽𝑢𝑥𝑥𝑡+ (1 −3
2𝜌2) 𝛽𝑢𝑥𝑥𝑥+ 𝛼𝑢𝑢𝑥
−1
2𝜌2𝛼𝛽 (𝑢𝑢𝑥𝑥𝑥+ 2𝑢𝑥𝑢𝑥𝑥) + 3𝜇𝛼2𝑢2𝑢𝑥
−3
2𝜇𝜌2𝛼2𝛽 (𝑢2𝑢𝑥𝑥𝑥+ 𝑢3𝑥+ 4𝑢𝑢𝑥𝑢𝑥𝑥) +9
4𝜇𝜌22𝛼2𝛽2(𝑢2𝑥𝑢𝑥𝑥𝑥+ 2𝑢𝑥𝑢2𝑥𝑥) = 0,
(9)
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, 𝑢𝑡+ 𝑢𝑥+ 𝜈𝑢𝑥𝑥𝑡+ ̃𝛽𝑢𝑥𝑥𝑥+ 𝛼𝑢𝑢𝑥+1
3𝜈𝛼 (𝑢𝑢𝑥𝑥𝑥+ 2𝑢𝑥𝑢𝑥𝑥) + 3𝜇𝛼2𝑢2𝑢𝑥+ 𝜈𝜇𝛼2(𝑢2𝑢𝑥𝑥𝑥+ 𝑢3𝑥+ 4𝑢𝑢𝑥𝑢𝑥𝑥) + 𝜈2𝜇𝛼2(𝑢2𝑥𝑢𝑥𝑥𝑥+ 2𝑢𝑥𝑢2𝑥𝑥) = 0.
(10) From the above references and the references cited there- in, 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 conditions𝜌3 = 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) for𝜌1 = 𝜌4 = 𝜌6 = 0, 𝜌3 = 2𝜌2, 𝜌5 = 𝜌7= (1/4)𝜌22. In [11], under different kinds of paramet- ric 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 if𝜌3 = 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 conditions𝜌3 = (𝑝 + 1)𝜌2or𝜌3= 𝑝𝜌2. In [15], under a new ans¨atze, 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 con- centrated 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 the𝜌𝑖 (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. InSection 2, we will derive two-dimensional planar system which is equiv- alent to (7) and give its first integral equation. InSection 3, by using the integral bifurcation method, we will obtain some new traveling wave solutions and discuss their dynamic prop- erties.
2. Two-Dimensional Planar Dynamical System of (7) and Its First Integral and Conservation of Energy
Making a transformation𝜂(𝑡, 𝑥) = 𝜙(𝜉)with𝜉 = 𝑥 − 𝑐𝑡, (7) can be reduced to the following ODE:
− 𝑐𝜙+ 𝜙+ 𝛼𝜙 𝜙+ 𝛽𝜙+ 𝜌1𝛼2𝜙2𝜙
+ 𝛼𝛽 (𝜌2𝜙𝜙+ 𝜌3𝜙𝜙) + 𝜌4𝛼3𝜙3𝜙= 0, (11) where𝑐is wave velocity which moves along the direction of 𝑥-axis and𝑐 ̸= 0. Integrating (11) once and setting the integral constant as zero yields
(1 − 𝑐) 𝜙 + 1
2𝛼𝜙2+ 𝛽𝜙+1
3𝜌1𝛼2𝜙3+ 𝛼𝛽
× [𝜌2𝜙𝜙+1
2(𝜌3− 𝜌2) (𝜙)2] + 1
4𝜌4𝛼3𝜙4 = 0.
(12) Let𝜙= 𝑦. Thus (12) can be reduced to a planar system
𝑑𝜙 𝑑𝜉 = 𝑦, 𝑑𝑦
𝑑𝜉 = − (1
4𝜌4𝛼3𝜙4+1
3𝜌1𝛼2𝜙3+1
2𝛼𝜙2+ (1 − 𝑐) 𝜙 +1
2𝛼𝛽 (𝜌3− 𝜌2) 𝑦2) × (𝛽 (1 + 𝛼𝜌2𝜙))−1 (13)
and a linear equation
𝛽 (1 + 𝛼𝜌2𝜙 ) = 0. (14) Obviously, the solutions of (12) cover the solutions of (13) and (14). We notice that the second equation in (13) is not continuous when𝜙 = −1/𝛼𝜌2; that is, the function𝜙(𝜉)is
not defined by𝜙 = −1/𝛼𝜌2. So (13) is a singular system; the line𝜙 = −1/𝛼𝜌2is called the singular line. Thus, we make the following transformation:
𝑑𝜉 = 12𝛽 (1 + 𝛼𝜌2𝜙) 𝑑𝜏, (15) where𝜏is a free parameter. Under the transformation (15), (13) can be rewritten as the following system
𝑑𝜙
𝑑𝜏 = 12𝛽 (1 + 𝛼𝜌2𝜙 ) 𝑦, 𝑑𝑦
𝑑𝜏= − [3𝜌4𝛼3𝜙4+ 4𝜌1𝛼2𝜙3+ 6𝛼𝜙2+ 12 (1 − 𝑐) 𝜙 +6𝛼𝛽 (𝜌3− 𝜌2) 𝑦2] .
(16)
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 conditions 𝜌3 = 5/3, 𝜌2 = 23/6in the model (1), we easily know that𝜌3≈ 2𝜌2. Second, from [9–11,13], we know that (1) contains many good prop- erties including loop soliton, integrability, and self-adjoint property when𝜌3 = 2𝜌2. In addition, in the condition𝜌3 = 2𝜌2, it becomes easy when we solve (7). Therefore, we only consider the special case of (7) for𝜌3 = 2𝜌2 in this paper.
Thus, by using this assumption, we obtain the first integral of (13) and (16) as follows:
𝑦2= − (3/5) 𝛼3𝜌4𝜙5− 𝛼2𝜌1𝜙4− 2𝛼𝜙3+ 6 (𝑐 − 1) 𝜙2+ 𝐶0
6𝛽 (1 + 𝛼𝜌2𝜙) ,
(17) where 𝐶0 is an integral constant. For the convenience of discussion, taking the integral constant𝐶0= 0in (17) yields
1
2𝑀𝑦2+1
2𝑀𝛼𝜌2𝜙𝑦2+3
5𝛼3𝜌4𝜙5+ 𝛼2𝜌1𝜙4 + 2𝛼𝜙3− 6 (𝑐 − 1) 𝜙2= 0,
(18)
where𝑀 = 12𝛽denotes 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:
𝐸1+ 𝐸2+ 𝑇 = 0, (19) where𝐸1denotes kinetic energy,𝐸2denotes external energy, and𝑇denotes potential energy. In (19), the kinetic energy𝐸1 and potential energy𝑇are not conserved because the external energy𝐸2 ̸= 0, but the global energy (𝐸1, 𝐸2, 𝑇) are subject to the energy conservation.𝐸2 = 0if only if𝜌2 = 0. Obviously, 𝜌3= 0once𝜌2= 0; under this case, (7) becomes the following equation:
𝜂𝑡+ 𝜂𝑥+ 𝛼𝜂𝜂𝑥+ 𝛽𝜂𝑥𝑥𝑥+ 𝜌1𝛼2𝜂2𝜂𝑥+ 𝜌4𝛼3𝜂3𝜂𝑥= 0. (20)
Similarly, (20) can be reduced to the following planar system:
𝑑𝜙 𝑑𝜉 = 𝑦, 𝑑𝑦
𝑑𝜉 = − (1
4𝜌4𝛼3𝜙4+1
3𝜌1𝛼2𝜙3+1
2𝛼𝜙2+ (1 − 𝑐) 𝜙 ) × 𝛽−1. (21) 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:
𝑦2= −1 𝛽[1
10𝛼3𝜌4𝜙5+1
6𝛼2𝜌1𝜙4+1
3𝛼𝜙3− (𝑐 − 1) 𝜙2] . (22) Equation (22) can be rewritten as the following equation of conservation of energy:
𝐸1+ 𝑇 = 0, (23)
where 𝐸1, 𝑇 are given above. The kinetic energy 𝐸1 and potential energy𝑇are 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). When𝐶0= 0, (17) can be reduced to
𝑦=±(√[−3
5𝛼3𝜌4𝜙5−𝛼2𝜌1𝜙4−2𝛼𝜙3+6 (𝑐−1) 𝜙2] (1+𝛼𝜌2𝜙))
× (√6𝛽 (1 + 𝛼𝜌2𝜙))−1.
(24) 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 or𝑐 ̸= 1, 𝜌1 = −2𝜌2, 𝜌4 ̸= − (5/3)𝜌1𝜌2or𝜌4 ̸= (10/3)𝜌22.
Under the parametric conditions of Case 1, (24) can be reduced to
𝑦 = ±𝜙√𝛼4𝜌1𝜌22𝜙4− 𝛼2(𝜌1+ 2𝜌2) 𝜙2+ 2/𝜌2
√6𝛽 (1 + 𝛼𝜌2𝜙) . (25)
Under the parametric conditions of Case 2, (24) can be reduced to
𝑦 = ±𝜙√−2𝛼4𝜌23𝜙4+ 2𝛼 (3𝑐𝜌2− 3𝜌2− 1) 𝜙 + 6 (𝑐 − 1)
√6𝛽 (1 + 𝛼𝜌2𝜙) .
(26) Under the parametric conditions of Case 3, (24) can be reduced to
𝑦 =
±𝜙√− (3/5) 𝛼4𝜌4𝜌2𝜙4+(1/5) 𝛼3(10𝜌22−3𝜌4) 𝜙3+2/𝜌2
√6𝛽 (1 + 𝛼𝜌2𝜙) .
(27) Under the parametric conditions of Case 4, (24) can be reduced to
𝑦 =
±𝜙√− (3/5) 𝛼4𝜌4𝜌2𝜙4+(1/5) 𝛼3(10𝜌22−3𝜌4) 𝜙3−2𝛼𝜙
√6𝛽 (1 + 𝛼𝜌2𝜙) .
(28) Under the the parametric conditions of Case5, (24) can be reduced to
𝑦 =
±𝜙√[−(3/5) 𝛼3𝜌4𝜙3−𝛼2𝜌1𝜙2−2𝛼𝜙+6 (𝑐−1)] (1+𝛼𝜌2𝜙)
√6𝛽 (1 + 𝛼𝜌2𝜙) .
(29)
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 Case 1. Substituting (25) into the left expression (i.e., 𝑑𝜙/𝑑𝜏 = 12𝛽(1 + 𝛼𝜌2𝜙)𝑦) of (16) yields
𝑑𝜙
𝜙√𝛼4𝜌1𝜌22𝜙4−𝛼2(𝜌1+2𝜌2) 𝜙2+2/𝜌2 = ±2√6𝛽 𝑑𝜏. (30) 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 the𝜙 is even function.
Taking “+” in (30), it can be reduced to 𝑑𝜙2
𝜙2√𝛼4𝜌1𝜌22(𝜙2)2− 𝛼2(𝜌1+ 2𝜌2) 𝜙2+ 2/𝜌2 = 4√6𝛽 𝑑𝜏.
(31)
Let𝜓 = 𝜙2. Equation (31) can be rewritten as 𝑑𝜓
𝜓√𝛼4𝜌1𝜌22𝜓2− 𝛼2(𝜌1+ 2𝜌2) 𝜓 + 2/𝜌2 = 4√6𝛽 𝑑𝜏. (32) Taking the(0, 𝜓1,2)as initial values, respectively, integrating (32) we obtain
∫𝜓
𝜓1
𝑑𝜓
𝜓√𝛼4𝜌1𝜌22𝜓2− 𝛼2(𝜌1+ 2𝜌2) 𝜓 + 2/𝜌2
= ∫𝜏
0 4√6𝛽 𝑑𝜏,
∫𝜓
𝜓2
𝑑𝜓
𝜓√𝛼4𝜌1𝜌22𝜓2− 𝛼2(𝜌1+ 2𝜌2) 𝜓 + 2/𝜌2
= ∫𝜏
0 4√6𝛽 𝑑𝜏,
(33)
where𝜓1,2= (𝜌1+ 2𝜌2± √(𝜌1+ 2𝜌2)2− 8𝜌1)/2𝜌1𝜌2𝛼2are two roots of equation𝛼4𝜌1𝜌22𝜓2− 𝛼2(𝜌1+ 2𝜌2)𝜓 + 2/𝜌2= 0.
(i) When𝜌2 > 0, Δ = (𝜌1+ 2𝜌2)2− 8𝜌1 > 0, completing the integrals (33) yields
𝜓 = 4𝑅1
𝑄1− 𝑃1cosh(8√(3𝛽/𝜌2)𝜏),
𝜓 = 4𝑅2
𝑃2cosh(8√(3𝛽/𝜌2)𝜏) − 𝑄2,
(34)
where𝑅1,2 = (𝜌1+ 2𝜌2± √Δ)[(𝜌1+ 2𝜌2)√Δ ± Δ]/2𝜌12𝜌23𝛼2, 𝑄1,2 = (𝜌1+ 2𝜌2)[√Δ ± (𝜌1+ 2𝜌2)][(𝜌1+ 2𝜌2)√Δ ± (𝜌12+ 4𝜌22)]/2𝜌12𝜌22, and𝑃1,2= Δ[√Δ ± (𝜌1+ 2𝜌2)]2/4𝜌12𝜌22withΔ = (𝜌1+ 2𝜌2)2− 8𝜌1.
By using (34) and the transformation𝜙2 = 𝜓, we obtain four exact solutions of (31) as follows:
𝜙 = ± 2√𝑅1
√𝑄1− 𝑃1cosh(8√(3𝛽/𝜌2)𝜏)
, (35)
𝜙 = ± 2√𝑅2
√𝑃2cosh(8√(3𝛽/𝜌2)𝜏) − 𝑄2
, (36)
where the constants𝑅1,2, 𝑃1,2, 𝑄1,2are given above.
Substituting (35) into (15) yields 𝜉 = 12𝛽 ± 𝛼𝜌2𝑔1√3𝜌2𝛽𝑅1
× 𝐹(arcsin√ 𝑄1−𝑃1cosh(8√(3𝛽/𝜌2)𝜏)
𝑄1− 𝑃1 , √Q1−P1 Q1+P1) ,
(37)
where𝐹(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 𝜉 = 12𝛽 ± 𝛼𝜌2𝑔2√3𝜌2𝛽𝑅2
× 𝐹(arcsin√𝑃2(cosh(8√(3𝛽/𝜌2)𝜏)−1)
𝑃2cosh(8√(3𝛽/𝜌2)𝜏)−𝑄2 , √𝑃2+𝑄2 2𝑃2 ) ,
(38) where𝑔2= √2/𝜌2, 𝑃2> 𝑄2> 0, and0 < 𝜏 < +∞.
Combining (35) with (37), we obtain (7)’s two exact solutions of parametric type as follows:
𝜂 = 𝜙 = ± 2√𝑅1
√𝑄1− 𝑃1cosh(8√(3𝛽/𝜌2)𝜏) ,
𝜉 = 12𝛽𝜏 ± 𝛼𝜌2𝑔1√3𝜌2𝛽𝑅1
× 𝐹(arcsin√ 𝑄1−𝑃1cosh(8√(3𝛽/𝜌2)𝜏)
𝑄1− 𝑃1 ,√𝑄1−𝑃1 𝑄1+𝑃1) .
(39) Combining (36) with (38), we obtain (7)’s another two exact solutions of parametric type as follows:
𝜂 = 𝜙 = ± 2√𝑅2
√𝑃2cosh(8√(3𝛽/𝜌2)𝜏) − 𝑄2 ,
𝜉 = 12𝛽𝜏 ± 𝛼𝜌2𝑔2√3𝜌2𝛽𝑅2
× 𝐹(arcsin√𝑃2(cosh(8√(3𝛽/𝜌2)𝜏)−1)
𝑃2cosh(8√(3𝛽/𝜌2)𝜏)−𝑄2 ,√𝑃2+𝑄2 2𝑃2 ) .
(40) (ii) When𝜌2< 0, Δ = (𝜌1+ 2𝜌2)2− 8𝜌1> 0, completing the integrals (33) yields
𝜓 = 4
−𝛼2𝜌2√Δ [−𝛼2(𝜌1+ 2𝜌2) −cos(8√− (3𝛽/𝜌2)𝜏)],
𝜓 = 4
−𝛼2𝜌2√Δ [−𝛼2(𝜌1+ 2𝜌2) +cos(8√− (3𝛽/𝜌2)𝜏)]. (41)
Similarly, by using (41) and the transformation𝜙2 = 𝜓, we obtain four periodic solutions of (31) as follows:
𝜙 = ± 2
√−𝛼2𝜌2√Δ√−𝛼2(𝜌1+ 2𝜌2) −cos(8√− (3𝛽/𝜌2)𝜏) ,
(42)
𝜙 = ± 2
√−𝛼2𝜌2√Δ√−𝛼2(𝜌1+ 2𝜌2) +cos(8√− (3𝛽/𝜌2)𝜏) ,
(43) where𝜌1 < −(2𝜌2+ 1/𝛼2); that is,−𝛼2(𝜌1+ 2𝜌2) > 1 > 0.
Respectively, substituting (42) and (43) into (15) yields
𝜉 = 12𝛽𝜏 ± ̃𝑔𝜌2√3𝛽
√Δ4
× 𝐹(arcsin√[1−cos(8√−(3𝛽/𝜌2)𝜏)] [𝛼2(𝜌1+2𝜌2)−1]
2 [𝛼2(𝜌1+2𝜌2) +cos(8√−(3𝛽/𝜌2)𝜏)] , 𝑘) , (44)
𝜉 = 12𝛽𝜏 ± ̃𝑔𝜌2√3𝛽
√Δ4 𝐹 (4√−3𝛽
𝜌2𝜏, 𝑘) , (45)
where ̃𝑔 = 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:
𝜂 = 𝜙 = ± 2
√−𝛼2𝜌2√Δ√−𝛼2(𝜌1+2𝜌2)−cos(8√− (3𝛽/𝜌2)𝜏) ,
𝜉 = 12𝛽𝜏 ± ̃𝑔𝜌2√3𝛽
√Δ4
× 𝐹(arcsin√[1−cos(8√−(3𝛽/𝜌2)𝜏)] [𝛼2(𝜌1+2𝜌2)−1]
2 [𝛼2(𝜌1+ 2𝜌2)+cos(8√−(3𝛽/𝜌2)𝜏)] , 𝑘) , (46) where0 < 𝜏 ≤ 𝜋√−𝜌2/8√3𝛽.
Combining (43) with (45), we obtain (7)’s another two exact solutions of parametric type as follows:
𝜂 = 𝜙
=± 2
√−𝛼2𝜌2√Δ√−𝛼2(𝜌1+2𝜌2)+cos(8√−(3𝛽/𝜌2)𝜏) ,
𝜉 = 12𝛽𝜏 ± ̃𝑔𝜌2√3𝛽
√Δ4 𝐹 (4√−3𝛽 𝜌2𝜏, 𝑘) .
(47)
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 sin- gular 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.
Figure 1(a)shows a shape of singular solitary wave under the fixed parameters𝛼 = 0.5,𝛽 = 0.4,𝜌1 = 300,𝜌2 = 1.3.
Figure 1(b)shows a shape of semiloop soliton under the fixed parameters𝛼 = 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 parameter𝜌1decreases from1.55to−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 from1.65 to
−6.25, which are shown in Figures2(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].
Figure 2(a)shows a shape of dark soliton under the fixed parameters𝛼 = 0.4, 𝛽 = 0.1, 𝜌2 = −4, and 𝜌1 = 1.55.
Figure 2(b)shows a shape of dark peakon under the fixed parameters𝛼 = 0.4, 𝛽 = 0.1, 𝜌2 = −4, and𝜌1 = 0.479.
Figure 2(c) shows a shape of dark loop soliton under the fixed parameters𝛼 = 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 parameters𝛼 = 0.4,𝛽 = 0.1,𝜌2 = −4, and𝜌1 = −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 parameters𝛼 = 0.4,𝛽 = 0.1, 𝜌2= −4, and𝜌1= 1.65.Figure 2(f)shows a shape of singular compacton under the fixed parameters𝛼 = 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 as𝜉 → ∞.
3.2. The Exact Solutions under the Parametric Conditions of Case2. Notice that the characters of (26)–(29) are quite dif- ferent from (25). Therefore, by using the expression𝑑𝜙/𝑑𝜉 = 𝑦and (26)–(29), we will investigate the implicit solutions of (7) next.
0.3 0.35 0.4 0.45
0 0.2 0.4 0.6 0.8 1 𝜙
−0.4 −0.2
𝜉
(a) Singular solitary wave
0.28 0.29 0.3 0.31 0.32
0 𝜙
−0.4
−0.6 −0.2
𝜉 (b) Semiloop soliton Figure 1: Profiles of solutions (40) and (46) for the fixed parameters.
Substituting (26) into the expression𝑑𝜙/𝑑𝜉 = 𝑦of (13) yields
𝑑𝜙
𝑑𝜉 = ±𝜙√−2𝛼4𝜌23𝜙4+ 2𝛼 (3𝑐𝜌2− 3𝜌2− 1) 𝜙 + 6 (𝑐 − 1)
√6𝛽 (1 + 𝛼𝜌2𝜙) .
(48) Taking “+”, (48) can be reduced to
(1 + 𝛼𝜌2𝜙 ) 𝑑𝜙
𝜙√−2𝛼4𝜌23𝜙4+ 2𝛼 (3𝑐𝜌2− 3𝜌2− 1) 𝜙 + 6 (𝑐 − 1)
= 1
√6𝛽𝑑𝜉.
(49)
(i) When𝜌2< 0, (49) can be reduced to
(1 + 𝛼𝜌2𝜙 ) 𝑑𝜙 (𝜙 √𝜙4−3𝑐𝜌2− 3𝜌2− 1
𝛼3𝜌23 𝜙 −3 (𝑐 − 1) 𝛼4𝜌23 )
−1
= −𝛼2𝜌2√−𝜌2 3𝛽𝑑𝜉.
(50) By the Ferrari method or the Descartes method, it is easy to know that the equation
𝜙4−3𝑐𝜌2− 3𝜌2− 1
𝛼3𝜌23 𝜙 −3 (𝑐 − 1)
𝛼4𝜌23 = 0 (51)
has four roots as follows:
− 1
𝛼𝜌2, (1/6)√Δ3 1− 4/3√Δ3 1+ 1/3
𝛼𝜌2 ,
− (1/12)√Δ3 1+2/3√Δ3 1+1/3 𝛼𝜌2
±𝑖 (√3/2) ((1/6)√Δ3 1+4/3√Δ3 1)
𝛼𝜌2 ,
(52)
whereΔ1 = −28 − 324𝜌2(𝑐 − 1) + 36√Δ2withΔ2 = 81(1 − 𝑐)2𝜌22+ 14(1 − 𝑐)𝜌2+ 1. Especially, when𝑐 = 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 signs𝜈, 𝛾and always assume that the biggest root is denoted by𝜈. We also denote the above two complex roots by the signs𝑠, 𝑠. Thus (50) can be rewritten as
𝑑𝜙
𝜙√(𝜙 − 𝜈) (𝜙 − 𝛾) (𝜙 − 𝑠) (𝜙 − 𝑠)
+ 𝛼𝜌2𝑑𝜙
√(𝜙 − 𝜈) (𝜙 − 𝛾) (𝜙 − 𝑠) (𝜙 − 𝑠)
= −𝛼2𝜌2√−𝜌2 3𝛽𝑑𝜉,
(53)
where𝜈 > 𝛾. Taking(𝜈, 0)as the initial values and integrating (53), we obtain implicit solution of (7) as follows:
( ̃𝐵 − ̃𝐴)
̃𝐵𝜈 + 𝛾̃𝐴[𝜇1𝐹 (𝜑, 𝑘) +𝜇 − 𝜇1
1 − 𝜇2(Π (𝜑, 𝜇2
𝜇2− 1, 𝑘) − 𝜇𝑓1)]
+ 𝑔𝛼𝜌2𝐹 (𝜑, 𝑘) = −𝛼2𝜌2√−𝜌2 3𝛽𝜉,
(54)
0.8 0.9 1 1.1 1.2 1.3
0 𝜙
−3 −2 −1 1 2 3
𝜉 (a) Dark soliton
0.64 0.66 0.68 0.7 0.72
0 0.02 0.04 0.06 𝜙
−0.06−0.04−0.02 𝜉 (b) Dark peakon
0.6 0.62 0.64 0.66
0 0.01 0.02 0.03 𝜙
−0.03 −0.02 −0.01 𝜉 (c) Dark loop soliton
0.32 0.34 0.36 0.38
0 0.1 0.2 0.3 𝜙
−0.3 −0.2 −0.1 𝜉
(d) Broken loop soliton (broken wave of C- form)
−1.4
−1.3
−1.2
−1.1
−1
−0.9
−0.8
𝜙
−4 −2 0 2 4
𝜉 (e) Bright soliton
0 1 2
−0.44
−0.42
−0.4
−0.38
−0.36
𝜙
−2 −1
𝜉
(f) Singular compacton
Figure 2: Different kinds of waveforms on solution (47) under different parametric conditions.
where𝐴̃2= (𝜈 − (𝑠 + 𝑠)/2)2−(𝑠 − 𝑠)2/4,̃𝐵2= (𝛾 − (𝑠 + 𝑠)/2)2− (𝑠 − 𝑠)2/4,𝑔 = 1/√ ̃𝐴 ̃𝐵,𝑘2= (( ̃𝐴 + ̃𝐵)2− (𝜈 − 𝛾)2)/4 ̃𝐴 ̃𝐵,
𝜑 =arccos[( ̃𝐴 − ̃𝐵) 𝜙+ 𝜈 ̃𝐵 − 𝛾 ̃𝐴 ( ̃𝐴 + ̃𝐵) 𝜙− 𝜈 ̃𝐵 − 𝛾 ̃𝐴] ,
𝜇 = 𝛾 ̃𝐴 + 𝜈 ̃𝐵
𝜈 ̃𝐵 − 𝛾 ̃𝐴, 𝜇1= 𝐴 + ̃𝐵̃
̃𝐵 − ̃𝐴,
𝑓1= √ 1 − 𝜇2
𝑘2+ 𝑘2𝜇2arctan[[ [
𝜑√ 𝑘2+ 𝑘2𝜇2 (1 − 𝜇2) (1 − 𝑘2𝜑2)]]
] ,
if 𝜇2 𝜇2− 1 < 𝑘2;
= 𝜑
√1 − 𝑘2𝜑2, if 𝜇2 𝜇2− 1 = 𝑘2;
= 1
2√ 𝜇2− 1 𝑘2+ 𝑘2𝜇2
×ln[[ [
√(𝑘2+ 𝑘2𝜇2) (1 − 𝑘2𝜑2) + 𝜑√𝜇2− 1 (√ 𝑘2+ 𝑘2𝜇2) (1 − 𝑘2𝜑2) − 𝜑√𝜇2− 1
]] ] ,
if 𝜇2 𝜇2− 1 > 𝑘2.
(55) (ii) When𝜌2> 0, (49) can be reduced to
(1 + 𝛼𝜌2𝜙 ) 𝑑𝜙
𝜙√−𝜙4+ ((3𝑐𝜌2− 3𝜌2− 1) / (𝛼3𝜌23)) 𝜙+ 3 (𝑐 − 1) /𝛼4𝜌23
= 𝛼2𝜌2√𝜌2 3𝛽𝑑𝜉.
(56)
Similarly, (56) can be rewritten as 𝑑𝜙
𝜙√(𝜈 − 𝜙) (𝜙 − 𝛾) (𝜙 − 𝑠) (𝜙 − 𝑠)
+ 𝛼𝜌2𝑑𝜙
√(𝜈 − 𝜙) (𝜙 − 𝛾) (𝜙 − 𝑠) (𝜙 − 𝑠)
= 𝛼2𝜌2√𝜌2 3𝛽𝑑𝜉,
(57)
where𝜈, 𝛾𝑠, ̃𝑠are four roots of (51) and𝜈 ≥ 𝜙 > 𝛾. Taking (𝛾, 0)as the initial values then integrating (56), we obtain another implicit solution of (7) as follows:
( ̃𝐴 + ̃𝐵)
𝛾 ̃𝐴 − ̃𝐵𝜈[𝜇2𝐹 ( ̃𝜑, 𝑚) + ̃𝜇 − 𝜇2 1 − ̃𝜇2
× (Π ( ̃𝜑, ̃𝜇2
̃𝜇2− 1, 𝑚) − ̃𝜇𝑓2)]+𝛼𝜌2𝑔𝐹 ( ̃𝜑, 𝑚)
= 𝛼2𝜌2√𝜌2 3𝛽𝜉,
(58) where the constants 𝐴, ̃𝐵, ̃𝑔̃ 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)/4 ̃𝐴 ̃𝐵, ̃𝜇 = (𝛾 ̃𝐴 − 𝜈 ̃𝐵)/(𝛾 ̃𝐴 + 𝜈 ̃𝐵), 𝜇2= ( ̃𝐴 − ̃𝐵)/( ̃𝐴 + ̃𝐵), and
𝑓2= √ 1 − ̃𝜇2
𝑚2+ 𝑚2̃𝜇2 arctan[[ [
̃𝜑√ 𝑚2+ 𝑚2̃𝜇2 (1 − ̃𝜇2) (1 − 𝑚2̃𝜑2)]]
] ,
if ̃𝜇2
̃𝜇2− 1 < 𝑚2;
= ̃𝜑
√1 − 𝑚2̃𝜑2, if ̃𝜇2
̃𝜇2− 1 = 𝑚2;
= 1
2√ ̃𝜇2− 1 𝑚2+ 𝑚2̃𝜇2
×ln[[ [
√(𝑚2+ 𝑚2̃𝜇2) (1 − 𝑚2̃𝜑2) + ̃𝜑√̃𝜇2− 1
√(𝑚2+ 𝑚2̃𝜇2) (1 − 𝑚2̃𝜑2) − ̃𝜑√̃𝜇2− 1 ]] ] ,
if ̃𝜇2
̃𝜇2− 1 > 𝑚2. (59)
3.3. The Exact Solutions under the Parametric Conditions of Case3. As inSection 3.2, substituting (27) into the expres- sion𝑑𝜙/𝑑𝜉 = 𝑦of (13) yields
𝑑𝜙
𝑑𝜉 = ± (𝜙√−3
5𝛼4𝜌4𝜌2𝜙4+1
5𝛼3(10𝜌22− 3𝜌4) 𝜙3+ 2 𝜌2)
× (√6𝛽 (1 + 𝛼𝜌2𝜙))−1.
(60) Taking “+”, (60) can be rewritten as
(1 + 𝛼𝜌2𝜙 ) 𝑑𝜙
𝜙√− (3/5) 𝛼4𝜌4𝜌2𝜙4+ (1/5) 𝛼3(10𝜌22− 3𝜌4) 𝜙3+ 2/𝜌2
= 1
√6𝛽𝑑𝜉.
(61)
When𝜌4𝜌2< 0, (61) can be reduced to (1 + 𝛼𝜌2𝜙 ) 𝑑𝜙
𝜙√𝜙4− ((10𝜌22− 3𝜌4) /3𝛼𝜌4𝜌2) 𝜙3− (10/3𝛼4𝜌4𝜌22)
= 𝛼2√−𝜌4𝜌2 10𝛽 𝑑𝜉.
(62)
When𝜌4 ̸= 10𝜌22/3, the equation 𝜙4−10𝜌22− 3𝜌4
3𝛼𝜌4𝜌2 𝜙3− 10
3𝛼4𝜌4𝜌22 = 0 (63) has four roots as follows:
− 1
𝛼𝜌2, (1/9𝜌2)√Δ3 3+10𝜌2(10𝜌22−9𝜌4) /9√Δ3 3+10𝜌2/9
𝛼𝜌4 ,
((− 1
18𝜌2√Δ3 3−5𝜌2(10𝜌22− 9𝜌2) 9√Δ3 3 +10𝜌2
9 )
±𝑖√3 2 ( 1
9𝜌2√Δ3 3−10𝜌2(10𝜌22− 9𝜌4)
9√Δ3 3 )) × (𝛼𝜌4)−1, (64) whereΔ3 = 𝜌22(−1350𝜌4𝜌22+ 1215𝜌42+ 1000𝜌22+ 135𝜌4√Δ4) withΔ4= −140𝜌4𝜌22+ 100𝜌24+ 81𝜌42. When𝜌4= 10𝜌22/3, (63) has another four roots±1/𝛼𝜌2, ±(1/𝛼𝜌2)𝑖.
(i) Particularly, when𝜌2< 0and𝜌4= 10𝜌22/3, (62) can be reduced to
(1 + 𝛼𝜌2𝜙)
𝜙√𝜙4− (1/𝛼4𝜌24)𝑑𝜙= −𝛼2𝜌2√−𝜌2
3𝛽𝑑𝜉. (65) 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. There- fore, the types of their solutions are the same when𝜌4𝜌2< 0
and𝜌4 ̸= 10𝜌22/3. Thus, we omit these parts of discussions. The especial case of (62) (i.e., (65)) can be reduced to
1 2
𝑑𝜙2
𝜙2√(𝜙2)2−(1/𝛼2𝜌22)2+ 𝛼𝜌2𝑑𝜙
√(𝜙2+1/𝛼2𝜌22) (𝜙2−1/𝛼2𝜌22)
= −𝛼2𝜌2√−𝜌2 3𝛽𝑑𝜉,
(66) where𝜙 > 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:
𝛼2𝜌22𝜙2=sec[2 𝜌2√−𝜌2
3𝛽𝜉−√2𝐹 (arccos( 1
𝛼𝜌2𝜙) ,√2 2 )] .
(67) (ii) Especially, when𝜌2> 0and𝜌4 = 10𝜌22/3, (61) can be reduced to
1 2
𝑑𝜙2
𝜙2√(1/𝛼2𝜌22)2−(𝜙2)2+ 𝛼𝜌2𝑑𝜙
√(𝜙2+1/𝛼2𝜌22) (1/𝛼2𝜌22− 𝜙2)
= 𝛼2𝜌2√𝜌2 3𝛽𝑑𝜉,
(68) where1/𝛼2𝜌22> 𝜙 > 0. Taking the(1/𝛼𝜌2, 0)as initial values and integrating (68), we obtain (7)’s implicit solitary wave solution as follows:
𝛼2𝜌22𝜙2=sech[√2𝐹 (arccos(𝛼𝜌2𝜙 ) ,√2 2 ) − 2
𝜌2√𝜌2 3𝛽𝜉] .
(69) 3.4. The Exact Solutions under the Parametric Conditions of Case4. Substituting (28) into the expression𝑑𝜙/𝑑𝜉 = 𝑦of (13) yields
𝑑𝜙 𝑑𝜉 =
±𝜙√− (3/5) 𝛼4𝜌4𝜌2𝜙4+(1/5) 𝛼3(10𝜌22−3𝜌4) 𝜙3−2𝛼𝜙
√6𝛽 (1 + 𝛼𝜌2𝜙) .
(70) Taking “+”, (70) can be rewritten as
(1 + 𝛼𝜌2𝜙 ) 𝑑𝜙
𝜙√− (3/5) 𝛼4𝜌4𝜌2𝜙4+ (1/5) 𝛼3(10𝜌22− 3𝜌4) 𝜙3− 2𝛼𝜙
= 1
√6𝛽𝑑𝜉.
(71)
(i)When𝜌4𝜌2< 0, (71) can be reduced to (1 + 𝛼𝜌2𝜙 ) 𝑑𝜙
𝜙√𝜙4− ((10𝜌22− 3𝜌4) /3𝛼𝜌4𝜌2) 𝜙3+ (10/3𝛼3𝜌4𝜌2) 𝜙
= 𝛼2√−𝜌4𝜌2 10𝛽 𝑑𝜉.
(72)
Next, we discuss the roots of the following equation:
𝜙4−10𝜌22− 3𝜌4
3𝛼𝜌4𝜌2 𝜙3+ 10
3𝛼3𝜌4𝜌2𝜙 = 0. (73) (a)When𝜌4< 5𝜌22/6, (73) has four real roots as follows:
0, − 1
𝛼𝜌2, 5𝜌2± √25𝜌22− 30𝜌4
3𝛼𝜌4 . (74)
(b)When𝜌4= 5𝜌22/6, (73) has still four real roots as fol- lows:
0, − 1
𝛼𝜌2, 2
𝛼𝜌2, 2
𝛼𝜌2, (75) where2/𝛼𝜌2is a double root.
(c)When𝜌4 > 5𝜌22/6, (73) has two real roots and two complex roots as follows:
0, − 1
𝛼𝜌2, 5𝜌2± 𝑖√30𝜌4− 25𝜌22
3𝛼𝜌4 . (76)
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 conditions0 < 𝜌4< 5𝜌22/6and𝜌2< 0, (72) can be reduced to
𝑑𝜙
𝜙√(𝜙− 𝑎1) (𝜙 − 0) (𝜙 − 𝑐1) (𝜙 − 𝑑1)
+ 𝛼𝜌2𝑑𝜙
√(𝜙− 𝑎1) (𝜙 − 0) (𝜙 − 𝑐1) (𝜙 − 𝑑1)
= 𝛼2√−𝜌4𝜌2 10𝛽𝑑𝜉,
(77)
where𝑎1 = −1/𝛼𝜌2,𝑐1 = (5𝜌2+ √25𝜌22− 30𝜌4)/3𝛼𝜌4,𝑑1 = (5𝜌2−√25𝜌22− 30𝜌4)/3𝛼𝜌4, and𝜙 > 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:
𝑔
𝑎1𝑘2[(𝑘2− 𝜇2) 𝐹 (𝜑, 𝑘) + 𝜇2𝐸 (𝜑, 𝑘)] + 𝑔𝛼𝜌2𝐹 (𝜑, 𝑘)
= 𝛼2√−𝜌4𝜌2 10𝛽𝜉,
(78)
