# Some Further Results on Traveling Wave Solutions for the ZK-BBM(m,n) Equations

Authors: Shaoyong Li; Zhengrong Liu
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

We investigate the traveling wave solutions for the ZK-BBM(m,n) equations ut+ux-aumx+bunxt+kunytx=0 by using bifurcation method of dynamical systems. Firstly, for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave. Secondly, for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. Furthermore, from the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also show that our work extends some previous results.


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## Body

## 1. Introduction

In recent years, many nonlinear wave equations have been derived from solid state physics, plasma physics, chemical physics, fluid mechanics, biology, and other fields. Thus, there has been considerable attention to find exact solutions of these problems. For this purpose, there have been many methods, such as inverse scattering transform method [1], Bäcklund and Darboux transforms [2, 3], Jacobi elliptic function method [4, 5], F-expansion and extended F-expansion method [6, 7], (G′/G)-expansion method [8, 9], and the bifurcation method of dynamical systems [10–14].Zakharov-Kuznetsov (ZK) equation [15]
(1)ut+auux+(uxx+uxy)x=0
is a two-dimensional space generalization of the KdV equation. The nonintegrable ZK equation governs the behavior of weakly nonlinear ion-acoustic waves in a plasma comprising cold ions and hot isothermal electrons in the presence of a uniform magnetic field [16, 17].Benjamin-Bona-Mahony (BBM) equation [18]
(2)ut+ux-a(u2)x-b~uxxt=0
is an alternative model to KdV equation for small-amplitude, surface waves of long wavelength in liquids, acoustic-gravity waves in compressible fluids, hydromagnetic waves in cold plasma, and acoustic waves in anharmonic crystals.Combining the BBM equation with the sense of the ZK equation, Wazwaz [19] considered the following ZK-BBM equation:
(3)ut+ux-a(u2)x-(b~(u)xt+k~(u)yt)x=0,
and its generalized form
(4)ut+ux-a(un)x-(b~(un)xt+k~(un)yt)x=0.
He presented a method called the extended tanh method to seek exact explicit compactons, solitons, solitary patterns, and plane periodic solutions of (3) and (4).Wang and Tang [20] studied the following generalized ZK-BBM equations:
(5)ut+ux-a(um)x+(b(un)xt+k(un)yt)x=0.
By using the bifurcation theory of planar dynamical systems, they gave some exact explicit traveling wave solutions and the sufficient conditions to guarantee the existence of smooth and nonsmooth traveling wave solutions.In the present paper, we continue to study the traveling wave solutions for (5), which we denote by ZK-BBM(m,n) equations for convenience. Our results are as follows: (i) for ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions and point out that the peakon wave is the limit form of the periodic peakon wave; (ii) for ZK-BBM(3, 2) equation, we obtain some elliptic function solutions which include periodic blow-up and periodic wave. From the limit forms of the elliptic function solutions, we obtain some trigonometric and hyperbolic function solutions which include periodic blow-up, blow-up, and smooth solitary wave. We also check the correctness of these solutions by putting them back into the original equation.This paper is organized as follows. In Section2, we state our main results which are included in two propositions. In Sections 3 and 4, we give the derivations for the two propositions, respectively. A brief conclusion is given in Section 5.


## 2. Main Results and Remarks

In this section we list our main results and give some remarks. To begin with, let us recall some symbols. The symbols snu and cn u denote the Jacobian elliptic functions sine amplitude u and cosine amplitude u. cosh u, sinh u, sech u, and csch u are the hyperbolic functions. For the sake of simplification, we only consider the case a>0 (the other case a<0 can be considered similarly). To relate conveniently, for given constant wave speed c, let
(6)ξ=x+y-ct,g1=-2(1-c)29a,g2=80|1-c|3829a.
Via the following two propositions we state our main results.Proposition 1.
Consider ZK-BBM(2, 2) equation(7)ut+ux-a(u2)x+(b(u2)xt+k(u2)yt)x=0
and its traveling wave equation
(8)g+(1-c)φ-aφ2-2c(b+k)(φ′)2-2c(b+k)φφ′′=0.There are the following results.(1)
Whenc(b+k)<0, c≠1, and g=g1, (7) has a peakon wave solution
(9)u1(ξ)=2(1-c)3a(1-e-η1-|ξ|),where(10)η1-=-a4c(b+k).(2)
Whenc(b+k)<0, c≠1, and g1<g<0, (7) has a periodic peakon wave solution
(11)u2(ξ)={αeη1-|ξ|+βe-η1-|ξ|+2(1-c)3a,forc<1,αe-η1-|ξ|+βeη1-|ξ|+2(1-c)3a,forc>1,where(12)ξ∈[(2l-1)T,(2l+1)T),α=c-13a+-g2a,β=c-13a--g2a,l=0,±1,±2,…,T=4c(b+k)-aln|4(1-c)2+18ag3-2ag-2(1-c)|.(3)
Whenc(b+k)>0, c≠1, and g1<g, (7) has two smooth periodic wave solutions
(13)u3(ξ)=2(1-c)3a±γcos(η1+ξ),u4(ξ)=2(1-c)3a±γsin(η1+ξ),where(14)η1+=a4c(b+k),(15)γ=4(1-c)2+18ag3a.Remark 2.
Whenc(b+k)<0, c≠1, and g→g1+0, the periodic peakon wave u2(ξ) becomes the peakon wave u1(ξ); the varying process is displayed in Figure 1.The varying process for the periodic peakon waveu2(ξ) tends to the peakon wave u1(ξ) when g→g1+0, where a=1, c=-1, and b=k=1/2, and (a) g=g1+10-1; (b) g=g1+10-2; (c) g=g1+10-4; (d) g=g1+10-8.
(a)
(b)
(c)
(d)Remark 3.
Whenc(b+k)>0, c≠1, and g→0, the smooth periodic wave u4(ξ) becomes
(16)u4∘(ξ)=2(1-c)3a(1±sin(12ac(b+k)ξ)),
which can be found in [20]; this implies that we extend the previous result.Proposition 4.
Consider ZK-BBM(3, 2) equation(17)ut+ux-a(u3)x+(b(u2)xt+k(u2)yt)x=0
and its traveling wave equation
(18)g+(1-c)φ-aφ3-2c(b+k)(φ′)2-2c(b+k)φφ′′=0.There are the following results.(1°) When c(b+k)>0, c<1, and g<-g2, (17) has two elliptic periodic blow-up solutions
(19)u5(ξ)=φ1+A1-2A11-cn(η3+ξ,k1+),u6(ξ)=φ1+A1-2A11+cn(η3+ξ,k1+),
where
(20)A1=(φ1-φ2)(φ1-φ3),(21)η3+=aA15c(b+k),(22)k1+=2A1+2φ1-φ2-φ34A1,(23)φ1=21003a(c-1)-103Ω2/36aΩ1/3,(24)φ2=53(2a103(1-3i)(1-c)+(1+3i)Ω2/3)643aΩ1/3,(25)φ3=53(2a103(1+3i)(1-c)+(1-3i)Ω2/3)643aΩ1/3,(26)Ω=729g2a4-80(1-c)3a3-27ga2.(2°) When c(b+k)>0, c<1, and -g2<g<g2, (17) has two elliptic periodic blow-up solutions u7(ξ), u8(ξ) and two symmetric elliptic periodic wave solutions u9(ξ), u10(ξ)(27)u7(ξ)=φ3-(φ3-φ1)sn-2(η4+ξ,k2),u8(ξ)=φ1-φ2sn2(η4+ξ,k2)1-sn2(η4+ξ,k2),u9(ξ)=φ3-(φ3-φ2)sn2(η4+ξ,k2),u10(ξ)=φ2-φ1k22sn2(η4+ξ,k2)1-k22sn2(η4+ξ,k2),
where
(28)η4+=a(φ3-φ1)20c(b+k),k2=φ3-φ2φ3-φ1.(3°) When c(b+k)>0, c<1, and g>g2, (17) has two elliptic periodic blow-up solutions
(29)u11(ξ)=φ3+A2-2A21-cn(η5+ξ,k3+),u12(ξ)=φ3+A2-2A21+cn(η5+ξ,k3+),
where
(30)A2=(φ3-φ2)(φ3-φ1),η5+=aA25c(b+k),k3+=2A2+2φ3-φ2-φ14A2.(4°) When c(b+k)<0, c<1, and g>g2, (17) has two elliptic periodic blow-up solutions
(31)u13(ξ)=φ3-A2+2A21-cn(η5-ξ,k3-),u14(ξ)=φ3-A2+2A21+cn(η5-ξ,k3-),
where
(32)η5-=-aA25c(b+k),k3-=2A2-2φ3+φ2+φ14A2.(5°) When c(b+k)<0, c<1, and -g2<g<g2, (17) has two elliptic periodic blow-up solutions u15(ξ), u16(ξ) and two symmetric elliptic periodic wave solutions u17(ξ), u18(ξ)(33)u15(ξ)=φ1+(φ3-φ1)sn-2(η4-ξ,k4),u16(ξ)=φ3-φ2sn2(η4-ξ,k4)1-sn2(η4-ξ,k4),u17(ξ)=φ1+(φ2-φ1)sn2(η4-ξ,k4),u18(ξ)=φ2-φ3k42sn2(η4-ξ,k4)1-k42sn2(η4-ξ,k4),
where
(34)η4-=-a(φ3-φ1)20c(b+k),k4=φ2-φ1φ3-φ1.(6°) When c(b+k)<0, c<1, and g<-g2, (17) has two elliptic periodic blow-up solutions
(35)u19(ξ)=φ1-A1+2A11+cn(η3-ξ,k1-),u20(ξ)=φ1-A1+2A11-cn(η3-ξ,k1-),
where
(36)η3-=-aA15c(b+k),k1-=2A1-2φ1+φ2+φ34A1.Remark 5.
Whenc(b+k)>0, c<1, and g→-g2, the periodic blow-up solutions u5(ξ) (or u7(ξ)) and u6(ξ) (or u8(ξ)) tend to two trigonometric periodic blow-up solutions, respectively,
(37)u21(ξ)=5(1-c)9a(1-3csc2(ac(b+k)1-c80a4ξ)),(38)u22(ξ)=5(1-c)9a(1-3sec2(ac(b+k)1-c80a4ξ)).
The symmetric elliptic periodic wave solutions u9(ξ) and u10(ξ) become a trivial solution u23(ξ)=5(1-c)/9a.Remark 6.
Whenc(b+k)>0, c<1, and g→g2, the periodic blow-up solution u7(ξ) (or u11(ξ)) tends to a hyperbolic blow-up solution
(39)u24(ξ)=-5(1-c)9a(1+3csch2(ac(b+k)1-c80a4ξ)).
The elliptic periodic wave solution u9(ξ) (or the elliptic periodic blow-up solution u12(ξ)) tends to a hyperbolic smooth solitary wave solution
(40)u25(ξ)=-5(1-c)9a(1-3sech2(ac(b+k)1-c80a4ξ)).
For the varying process, see Figures 2, 3, and 4. The elliptic solutions u8(ξ) and u10(ξ) tend to a trivial solution u26(ξ)=-(5(1-c))/9a.The varying process for graphs ofu7(ξ) when g→g2-0, where a=1, c=b=k=1/2, and (a) g=g2-10-1; (b) g=g2-10-2; (c) g=g2-10-4; (d) g=g2-10-6.
(a)
(b)
(c)
(d)The varying process for graphs ofu9(ξ) when g→g2-0, where a=1, c=b=k=1/2, and (a) g=g2-10-1; (b) g=g2-10-2; (c) g=g2-10-4; (d) g=g2-10-6.
(a)
(b)
(c)
(d)The varying process for graphs ofu12(ξ) when g→g2+0, where a=1, c=b=k=1/2, and (a) g=g2+1/5; (b) g=g2+10-2; (c) g=g2+10-4; (d) g=g2+10-6.
(a)
(b)
(c)
(d)Remark 7.
Whenc(b+k)<0, c<1, and g→g2, the periodic blow-up solutions u13(ξ) (or u15(ξ)) and u14(ξ) (or u16(ξ)) tend to two trigonometric periodic blow-up solutions, respectively,
(41)u27(ξ)=-5(1-c)9a(1-3sec2(-ac(b+k)1-c80a4ξ)),u28(ξ)=-5(1-c)9a(1-3csc2(-ac(b+k)1-c80a4ξ)).
The symmetric elliptic periodic wave solutions u17(ξ) and u18(ξ) become the trivial solution u26(ξ).Remark 8.
Whenc(b+k)<0, c<1, and g→-g2, the periodic blow-up solution u15(ξ) (or u20(ξ)) tends to a hyperbolic blow-up solution
(42)u29(ξ)=5(1-c)9a(1+3csch2(-ac(b+k)1-c80a4ξ)).
The elliptic periodic wave solution u17(ξ) (or the elliptic periodic blow-up solution u19(ξ)) tends to a hyperbolic smooth solitary wave solution
(43)u30(ξ)=5(1-c)9a(1-3sech2(-ac(b+k)1-c80a4ξ)).
The varying process is similar to those in Figures 2–4. The elliptic solutions u16(ξ) and u18(ξ) tend to the trivial solution u23(ξ).Remark 9.
Wheng→0, the solutions u10(ξ) and u18(ξ), respectively, become
(44)u10°(ξ)=(5(1-c)3asn2(5(1-c)3a4ξ,22a10c(b+k)×5(1-c)3a4ξ,22))×(2-sn2(a10c(b+k)×5(1-c)3a4ξ,22))-1,forc(b+k)>0,c<1,u18°(ξ)=(5(1-c)3asn2(-a10c(b+k)×5(1-c)3a4ξ,22))×(sn2(-a10c(b+k)×5(1-c)3a4ξ,22)-2)-1,forc(b+k)<0,c<1,
which can be found in [20]; this implies that we extend the previous results.


## 3. The Derivations for Proposition1

In this section, we derive the precise expressions of the traveling wave solutions for ZK-BBM(2, 2) equation. Substitutingu=φ(ξ) with ξ=x+y-ct into (7), it follows that
(45)(1-c)φ′-2aφφ′-2c(b+k)(3φ′φ′′+φφ′′)=0.Integrating (45) once, we have
(46)g+(1-c)φ-aφ2-2c(b+k)(φ′)2+2c(b+k)φφ′′=0,
where g is an integral constant.Lettingψ=φ′, we obtain the following planar system:
(47)dφdξ=ψ,dψdξ=g+(1-c)φ-aφ2-2c(b+k)ψ22c(b+k)φ.
Under the transformation dξ=2c(b+k)φdτ, system (47) becomes
(48)dφdτ=2c(b+k)φψ,dψdτ=g+(1-c)φ-aφ2-2c(b+k)ψ2.Clearly, system (47) and system (48) have the same first integral
(49)2c(b+k)φ2ψ2+a2φ4-2(1-c)3φ3-gφ2=h,
where h is an integral constant. Consequently, these two systems have the same topological phase portraits except for the straight line φ=0. Thus, we can understand the phase portraits of system (47) from those of system (48).When the integral constanth=0, (49) becomes
(50)2c(b+k)ψ2+a2φ2-2(1-c)3φ-g=0.Solving equation(a/2)φ2-(2(1-c)/3)φ-g=0, we get two roots
(51)φ±*=2(1-c)±4(1-c)2+18ag3a,whereg≥-2(1-c)29a.On the other hand, solving equationg+(1-c)φ-aφ2, we obtain
(52)φ±∘=1-c±(1-c)2+4ag2a,whereg≥-(1-c)24a.According to the qualitative theory, we obtain the phase portraits of system (48) as shown in Figure 5.The phase portraits of system (48).
(a)
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0, c<1, g=g1
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0Wheng=g1, on φ-ψ plane the orbit Γ1 has expression
(53)Γ1:ψ={±η1-(2(1-c)3a-φ),0≤φ<2(1-c)3aforc(b+k)<0,c<1,±η1-(φ-2(1-c)3a),2(1-c)3a<φ≤0forc>1,b+k<0.Substituting (53) into dφ/dξ and integrating it along the orbit Γ1, we obtain the peakon wave solution u1(ξ) as (9).Wheng1<g<0, on φ-ψ plane the orbit Γ2 has expression
(54)Γ2:ψ={±η1-(φ+*-φ)(φ-*-φ),0≤φ≤φ-*forc(b+k)<0,c<1,±η1-(φ-φ+*)(φ-φ-*),φ+*≤φ≤0forc>1,b+k<0.Substituting (54) into dφ/dξ and integrating it along the orbit Γ2, we obtain the periodic peakon wave solution u2(ξ) as (11), where
(55)T=1η1-|∫0φ-*‍ds(φ+*-s)(φ-*-s)|=4c(b+k)-aln|4(1-c)2+18ag3-2ag-2(1-c)|.
If g→g1+0, it follows that α→(c-1)/3a, β→0 (or α→0, β→(c-1)/3a), and T→+∞. This implies that the periodic peakon wave solution u2(ξ) tends to the peakon wave solution u1(ξ).Wheng1<g and c(b+k)>0, on φ-ψ plane the orbit Γ3 has expression
(56)Γ3:ψ=±η1+(φ-φ-*)(φ+*-φ),φ-*≤φ≤φ+*.Substituting (56) into dφ/dξ and integrating it along the orbit Γ3, we obtain the smooth periodic wave solutions u3(ξ) and u4(ξ) as (13).Hereto, we have completed the derivations for Proposition1.


## 4. The Derivations for Proposition4

In this section, we derive the explicit elliptic function solutions and their limit forms for ZK-BBM(3, 2) equation. Similar to the derivations in Section3, substituting u=φ(ξ) with ξ=x+y-ct into (17) and integrating it, we have the following planar system:
(57)dφdξ=ψ,dψdξ=g+(1-c)φ-aφ3-2c(b+k)ψ22c(b+k)φ.Similarly, under the transformationdξ=2c(b+k)φdτ, system (57) becomes
(58)dφdτ=2c(b+k)φψ,dψdτ=g+(1-c)φ-aφ3-2c(b+k)ψ2,
which has the first integral
(59)2c(b+k)φ2ψ2+2a5φ5-2(1-c)3φ3-gφ2=h.When the integral constanth=0, (59) becomes
(60)c(b+k)φ2ψ2-φ2(g2+1-c3φ-a5φ3)=0.
Solving equation g/2+((1-c)/3)φ-(a/5)φ3=0, we get three roots φ1, φ2, and φ3 as (23), (24), and (25). On the other hand, solving equations
(61)ψ=0,g+(1-c)φ-aφ3-2c(b+k)ψ2=0,
we get three equilibrium points (φi*,0)(i=1,2,3) of system (58), where
(62)φ1*=2183a(c-1)-123Δ2/36aΔ1/3,φ2*=2a2336(3-3i)(1-c)+4396(1+3i)Δ2/312aΔ1/3,φ3*=2a2336(3+3i)(1-c)+4396(1-3i)Δ2/312aΔ1/3,Δ=81g2a4+12(c-1)3a3-9ga2.According to the qualitative theory, we obtain the phase portraits of system (58) as shown in Figure 6.The phase portraits of system (58).
(a)
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(l)
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0, c<1, g<-g2Now using planar system (57) and the phase portraits in Figure 6, we derive the explicit expressions of solutions for the ZK-BBM(3, 2) equation respectively.Whenc(b+k)>0, c<1, and g<-g2, Γ4 has the expression
(63)Γ4:ψ=±a5c(b+k)(φ1-φ)(φ-φ2)(φ-φ3),φ≤φ1,
where φ2 and φ3 are complex numbers.Substituting (63) into dφ/dξ=ψ and integrating it, we have
(64)∫-∞φ‍ds(φ1-s)(s-φ2)(s-φ3)=a5c(b+k)|ξ|,∫φφ1‍ds(φ1-s)(s-φ2)(s-φ3)=a5c(b+k)|ξ|.
Completing the integrals in the above two equations and noting that u=φ(ξ), we obtain u5(ξ) and u6(ξ) as (19).Whenc(b+k)>0, c<1, and -g2<g<g2, Γ4 and Γ5 have the expressions
(65)Γ4:y=±a5c(b+k)(φ1-φ)(φ2-φ)(φ3-φ),φ≤φ1,Γ5:y=±a5c(b+k)(φ-φ1)(φ-φ2)(φ3-φ),φ2≤φ≤φ3.
Substituting (65) into dφ/dξ=ψ and integrating them, we have
(66)∫-∞φ‍ds(φ1-s)(φ2-s)(φ3-s)=a5c(b+k)|ξ|,∫φφ1‍ds(φ1-s)(φ2-s)(φ3-s)=a5c(b+k)|ξ|,∫φ2φ‍ds(s-φ1)(s-φ2)(φ3-s)=a5c(b+k)|ξ|,∫φφ3‍ds(s-φ1)(s-φ2)(φ3-s)=a5c(b+k)|ξ|.
Completing the integrals in the above four equations and noting that u=φ(ξ), we obtain ui(ξ)(i=7–10) as (27).Whenc(b+k)>0, c<1, and g2<g, Γ6 has the expression
(67)Γ6:y=±a5c(b+k)(φ3-φ)(φ-φ2)(φ-φ1),φ≤φ3,
where φ1 and φ2 are complex numbers.Substituting (67) into dφ/dξ=ψ and integrating it, we have
(68)∫-∞φ‍ds(φ3-s)(s-φ2)(s-φ1)=a5c(b+k)|ξ|,∫φφ3‍ds(φ3-s)(s-φ2)(s-φ1)=a5c(b+k)|ξ|.
Completing the integrals in the above two equations and noting that u=φ(ξ), we obtain u11(ξ) and u12(ξ) as (29).Whenc(b+k)<0, c<1, and g2<g, Γ7 has the expression
(69)Γ7:y=±-a5c(b+k)(φ-φ3)(φ-φ2)(φ-φ1),φ≥φ3,
where φ1 and φ2 are complex numbers.Substituting (69) into dφ/dξ=ψ and integrating it, we have
(70)∫φ3φ‍ds(s-φ3)(s-φ2)(s-φ1)=-a5c(b+k)|ξ|,∫φ+∞‍ds(s-φ3)(s-φ2)(s-φ1)=-a5c(b+k)|ξ|.
Completing the integrals in the above two equations and noting that u=φ(ξ), we obtain u13(ξ) and u14(ξ) as (31).Whenc(b+k)<0, c<1, and -g2<g<g2, Γ7 and Γ8 have the expressions
(71)Γ7:y=±-a5c(b+k)(φ-φ1)(φ-φ2)(φ-φ3),φ≥φ3,Γ8:y=±-a5c(b+k)(φ3-φ)(φ2-φ)(φ-φ1),φ1≤φ≤φ2.Substituting (71) into dφ/dξ=ψ and integrating them, we have
(72)∫φ3φ‍ds(s-φ1)(s-φ2)(s-φ3)=-a5c(b+k)|ξ|,∫φ+∞‍ds(s-φ1)(s-φ2)(s-φ3)=-a5c(b+k)|ξ|,∫φ1φ‍ds(φ3-s)(φ2-s)(s-φ1)=-a5c(b+k)|ξ|,∫φφ2‍ds(φ3-s)(φ2-s)(s-φ1)=-a5c(b+k)|ξ|.
Completing the integrals in the above four equations and noting that u=φ(ξ), we obtain ui(ξ)(i=15–18) as (33).Whenc(b+k)<0, c<1, and g<-g2, Γ9 has the expression
(73)Γ9:y=±-a5c(b+k)(φ-φ1)(φ-φ2)(φ-φ3),φ≥φ1,
where φ2 and φ3 are complex numbers.Substituting (73) into dφ/dξ=ψ and integrating it, we have
(74)∫φ1φ‍ds(s-φ1)(s-φ2)(s-φ3)=-a5c(b+k)|ξ|,∫φ+∞‍ds(s-φ1)(s-φ2)(s-φ3)=-a5c(b+k)|ξ|.
Completing the integrals in the above two equations and noting that u=φ(ξ), we obtain u19(ξ) and u20(ξ) as (35).Hereto, we have finished the derivations for the solutionsui(ξ)(i=5–20). In what follows, we will derive the limit forms of these solutions.Whenc(b+k)>0, c<1, and g→-g2, it follows that
(75)φ1⟶-20(1-c)9a,φ2⟶5(1-c)9a,φ3⟶5(1-c)9a,A1⟶5(1-c)a,η3+⟶ac(b+k)1-c5a4,η4+⟶ac(b+k)1-c80a4,k1+⟶0,k2⟶0,cn(η3+ξ,k1+)⟶cn(ac(b+k)1-c5a4ξ,0)=cos(ac(b+k)1-c5a4ξ),sn(η4+ξ,k2)⟶sn(ac(b+k)1-c80a4ξ,0)=sin(ac(b+k)1-c80a4ξ).
Thus we have
(76)u5(ξ)⟶-20(1-c)9a+5(1-c)a-25(1-c)/a1-cn(a/c(b+k)(1-c)/5a4ξ,0)=5(1-c)9a-25(1-c)/a1-cos(a/c(b+k)(1-c)/5a4ξ)=5(1-c)9a-5(1-c)a×csc2(ac(b+k)1-c80a4ξ)=u21(ξ)(see(37)),u6(ξ)⟶-20(1-c)9a+5(1-c)a-25(1-c)/a1+cn(a/c(b+k)((1-c)/5a)4ξ,0)=5(1-c)9a-25(1-c)/a1+cos(a/c(b+k)((1-c)/5a)4ξ)=5(1-c)9a-5(1-c)asec2(ac(b+k)1-c80a4ξ)=u22(ξ)(see(38)),u7(ξ)⟶5(1-c)9a-5(1-c)a×sn-2(ac(b+k)1-c80a4ξ,0)=5(1-c)9a-5(1-c)a×sin-2(ac(b+k)1-c80a4ξ)=5(1-c)9a-5(1-c)a×csc2(ac(b+k)1-c80a4ξ)=u21(ξ)(see(37)),u8(ξ)⟶(-20(1-c)9a-5(1-c)9a×sn2(ac(b+k)1-c80a4ξ,0)20(1-c)9a-5(1-c)9a)×(1-sn2(ac(b+k)1-c80a4ξ,0))-1=(-20(1-c)9a-5(1-c)9a×sin2(ac(b+k)1-c80a4ξ)20(1-c)9a-5(1-c)9a)×(1-sin2(ac(b+k)1-c80a4ξ))-1=(-5(1-c)a+5(1-c)9a×cos2(ac(b+k)1-c80a4ξ)-5(1-c)a+5(1-c)9a)×(cos2(ac(b+k)1-c80a4ξ))-1=5(1-c)9a-5(1-c)a×sec2(ac(b+k)1-c80a4ξ)=u22(ξ)(see(38)).
When c(b+k)>0, c<1, and g→g2, it follows that
(77)φ1⟶-5(1-c)9a,φ2⟶-5(1-c)9a,φ3⟶20(1-c)9a,A2⟶5(1-c)a,η4+⟶ac(b+k)1-c80a4,η5+⟶ac(b+k)1-c5a4,k2⟶1,k3+⟶1,sn(η4+ξ,k2)⟶sn(ac(b+k)1-c80a4ξ,1)=tanh(ac(b+k)1-c80a4ξ),cn(η5+ξ,k3+)⟶cn(ac(b+k)1-c5a4ξ,1)=sech(ac(b+k)1-c5a4ξ).
Thus we have
(78)u7(ξ)⟶20(1-c)9a-5(1-c)a×sn-2(ac(b+k)1-c80a4ξ,1)=20(1-c)9a-5(1-c)a×tanh-2(ac(b+k)1-c80a4ξ)=-5(1-c)9a-5(1-c)a×csch2(ac(b+k)1-c80a4ξ)=u24(ξ)(see(39)),u9(ξ)⟶20(1-c)9a-5(1-c)a×sn2(ac(b+k)1-c80a4ξ,1)=20(1-c)9a-5(1-c)a×tanh2(ac(b+k)1-c80a4ξ)=-5(1-c)9a+5(1-c)a×sech2(ac(b+k)1-c80a4ξ)=u25(ξ)(see(40)),u11(ξ)⟶20(1-c)9a+5(1-c)a-25(1-c)/a1-cn(a/c(b+k)((1-c)/5a)4ξ,1)=55(1-c)9a-(25(1-c)a)×(1-sech(ac(b+k)1-c5a4ξ))-1=55(1-c)9a-(25(1-c)a+45(1-c)a×sinh2(ac(b+k)1-c80a4ξ)5(1-c)a+45(1-c)a)×2sinh2(ac(b+k)1-c80a4ξ)-1=-5(1-c)9a-5(1-c)a×csch2(ac(b+k)1-c80a4ξ)=u24(ξ)(see(39)),u12(ξ)⟶20(1-c)9a+5(1-c)a-25(1-c)/a1+cn(a/(c(b+k))((1-c)/5a)4ξ,1)=55(1-c)9a-(25(1-c)a)×(1+sech(ac(b+k)(1-c)5a4ξ))-1=55(1-c)9a-(-25(1-c)a+45(1-c)a×cosh2(ac(b+k)1-c80a4ξ)25(1-c)a+45(1-c)a)×(2cosh2(ac(b+k)1-c80a4ξ))-1=-5(1-c)9a+5(1-c)a×sech2(ac(b+k)1-c80a4ξ)=u25(ξ)(see(40)).The limit forms of the other solutions can be derived similarly, so here we omit them. Hereto, we have completed the derivations for Proposition4.


## 5. Conclusion

In this paper, we have investigated ZK-BBM(m,2)(m=2,3) equations. For ZK-BBM(2, 2) equation, we obtain peakon wave, periodic peakon wave, and smooth periodic wave solutions (see ui(ξ)(i=1–4)). For ZK-BBM(3, 2) equation, we obtain some elliptic function solutions (see ui(ξ)(i=5–20)). Furthermore, from the limit forms of these solutions, we obtain some trigonometric and hyperbolic function solutions (see Remarks 5–8 and the corresponding derivations). We also showed that some previous results are our special cases (see Remarks 3 and 9). We would like to study the ZK-BBM(m, n) equations further.



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*Source: 156139-2013-11-02.xml*