# The Traveling Wave Solutions and Their Bifurcations for the BBM-LikeB(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 and their bifurcations for the BBM-likeB(m,n) equations ut+αux+β(um)x−γ(un)xxt=0 by using bifurcation method and numerical simulation approach of dynamical systems. Firstly, for BBM-like B(3,2) equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. Furthermore, we reveal the relationships among these solutions theoretically. Secondly, for BBM-like B(4,2) equation, we construct two periodic wave solutions and two blow-up solutions. In order to confirm the correctness of these solutions, we also check them by software Mathematica.


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## Body

## 1. Introduction

In recent years, the nonlinear phenomena exist in all fields including either the scientific work or engineering fields, such as fluid mechanics, plasma physics, optical fibers, biology, solid-state physics, chemical kinematics, and chemical physics. Many nonlinear evolution equations are playing important roles in the analysis of the phenomena.In order to find the traveling wave solutions of these nonlinear evolution equations, there have been many methods, such as Jacobi elliptic function method [1, 2], F-expansion and extended F-expansion method [3, 4], (G′/G)-expansion method [5, 6], and the bifurcation method of dynamical systems [7–11].BBM equation or regularized long-wave equation (RLW equation)(1)ut+uux-uxxt=0
was derived by Peregine [12, 13] and Benjamin et al. [14] as an alternative model to Korteweg-de Vries equation for small-amplitude, long wavelength surface water waves.There are various generalized form related to (1). Shang [15] introduced a family of BBM-like equations with nonlinear dispersion
(2)ut+(um)x-(un)xxt=0,m,n>1,
which were called BBM-like B(m,n) equations as alternative model to the nonlinear dispersive K(m,n) equations [16–18]. He presented a method called the extend sine-cosine method to seek exact solitary-wave solutions with compact support and exact special solutions with solitary patterns of (2).Whenm=n=2, (2) reduces to the BBM-like B(2,2) equation
(3)ut+(u2)x-(u2)xxt=0.
Jiang et al. [19] employed the bifurcation method of dynamical systems to investigate (3). Under different parametric conditions, they gave various sufficient conditions to guarantee the existence of smooth and nonsmooth traveling wave solutions. Furthermore, through some special phase orbits, they obtained some solitary wave solutions expressed by implicit functions, periodic cusp wave solution, compacton solution, and peakon solution.Wazwaz [20] introduced a system of nonlinear variant RLW equations
(4)ut+aux-k(un)x+b(un)xxt=0
and derived some compact and noncompact exact solutions by using the sine-cosine method and tanh method.Feng et al. [21] studied the following generalized variant RLW equations:
(5)ut+aux-k(um)x+b(un)xxt=0.
By using four different ansatzs, they obtained some exact solutions such as compactons, solitary pattern solutions, solitons, and periodic solutions.Kuru [22–24] considered the following BBM-like equations with a fully nonlinear dispersive term:
(6)ut+ux+a(um)x-(un)xxt=0,m,n>1.
By means of the factorization technique, he obtained the traveling wave solutions of (6) in terms of the Weierstrass functions.In the present paper, we use the bifurcation method and numerical simulation approach of dynamical systems to study the following BBM-likeB(m,n) equations:
(7)ut+αux+β(um)x-γ(un)xxt=0m,n>1.
For BBM-like B(3,2) equation, we obtain some precise expressions of traveling wave solutions, which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. We also reveal the relationships among these solutions theoretically. For BBM-like B(4,2) equation, we construct two elliptic periodic wave solutions and two hyperbolic blow-up solutions.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. Firstly, 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. Secondly, for the sake of simplification, we only consider the case α>0, β>0, and γ>0 (the other cases can be considered similarly). To relate conveniently, for given constant wave speed c, let
(8)ξ=x-ct,g0=4275|c-α|3β.Proposition 1.
Consider BBM-likeB(3,2) equation
(9)ut+αux+β(u3)x-γ(u2)xxt=0
and its traveling wave equation
(10)(α-c)φ+βφ3+2γc(φ′)2+2γcφφ′′=g.For given constantsc and g, there are the following results.(
1
) When 0<c<α or c>α, g<-g0, (9) has two elliptic periodic blow-up solutions
(11)u1(ξ)=φ1+A1-2A11-cn(η1ξ,k1),u2(ξ)=φ1+A1-2A11+cn(η1ξ,k1),
where
(12)A1=(φ1-φ2)(φ1-φ3),(13)η1=βA15γc,(14)k1=2A1+2φ1-φ2-φ34A1,(15)φ1=21003β(α-c)-103Ω2/36βΩ1/3,(16)φ2=53(2β103(1-3i)(c-α)+(1+3i)Ω2/3)643βΩ1/3,(17)φ3=53(2β103(1+3i)(c-α)+(1-3i)Ω2/3)643βΩ1/3,(18)Ω=729g2β4-80(c-α)3β3-27gβ2.
For the graphs of  u1(ξ) and u2(ξ), see Figures 1(a) and 1(b).The graphs ofu1(ξ) and u2(ξ) when α=β=γ=1, c=2, and g=-23/9.
(a)
u
=
u
1
(
ξ
)
(b)
u
=
u
2
(
ξ
).(
2
) When c>α and g=-g0, (9) has two trigonometric periodic blow-up solutions
(19)u3(ξ)=5(c-α)9β(1-3csc2(βγcc-α80β4ξ)),(20)u4(ξ)=5(c-α)9β(1-3sec2(βγcc-α80β4ξ)).
The graphs of u3(ξ) and u4(ξ) are similar to Figure 1.(
3
) When c>α and -g0<g<g0, (9) has two elliptic periodic blow-up solutions u5(ξ), u6(ξ) and two symmetric elliptic periodic wave solutions u7(ξ), u8(ξ)(21)u5(ξ)=φ3-(φ3-φ1)sn-2(η2ξ,k2),u6(ξ)=φ1-φ2sn2(η2ξ,k2)1-sn2(η2ξ,k2),u7(ξ)=φ3-(φ3-φ2)sn2(η2ξ,k2),u8(ξ)=φ2-φ1k22sn2(η2ξ,k2)1-k22sn2(η2ξ,k2),
where
(22)η2=12β(φ3-φ1)5γc,k2=φ3-φ2φ3-φ1.
For the graphs of ui(ξ)(i=5–8), see Figures 2–4.The varying process for graphs ofu7(ξ) and u8(ξ) when c>α and g→-g0+0 where α=β=γ=1, c=2, and (a) g=-g0+10-1. (b) g=-g0+10-2. (c) g=-g0+10-4. (d) g=-g0+10-6.
(a)
(b)
(c)
(d)The varying process for graphs ofu5(ξ) when c>α and g→g0-0 where α=β=γ=1, c=2, and (a) g=g0-10-1. (b) g=g0-10-2. (c) g=g0-10-4. (d) g=g0-10-6.
(a)
(b)
(c)
(d)The varying process for graphs ofu7(ξ) when c>α and g→g0-0 where α=β=γ=1, c=2, and (a) g=g0-10-1. (b) g=g0-10-2. (c) g=g0-10-4. (d) g=g0-10-6.
(a)
(b)
(c)
(d)(
4
) When c>α and g=g0, (9) has a hyperbolic smooth solitary wave solution
(23)u9(ξ)=-5(c-α)9β(1-3sech2(βγcc-α80β4ξ)),
a hyperbolic blow-up solution
(24)u10(ξ)=-5(c-α)9β(1+3csch2(βγcc-α80β4ξ)),
a hyperbolic peakon wave solution
(25)u11(ξ)=-5(c-α)9β×(1-3sech2(δ1-βγcc-α80β4|ξ|)),
and a hyperbolic periodic peakon wave solution
(26)u12(ξ)=-5(c-α)9β×(1-3sech2(δ1+βγcc-α80β4|ξ|)),000000000000ξ∈[(2n-1)T,(2n+1)T),
where
(27)δ1=12ln(5-26),T=5γcββ5(c-α)4|ln(5-26)|,n=0,±1,±2,….
For the graphs of u9(ξ) and u10(ξ), see Figures 4(d) and 3(d). For the graphs of u11(ξ) and u12(ξ), see Figures 7(a) and 7(b).(
5
) When c>α and g>g0, (9) has two elliptic periodic blow-up solutions
(28)u13(ξ)=φ3+A2-2A21-cn(η3ξ,k3),u14(ξ)=φ3+A2-2A21+cn(η3ξ,k3),
where
(29)A2=(φ3-φ2)(φ3-φ1),η3=βA25γc,k3=2A2+2φ3-φ2-φ14A2.
For the graphs of u13(ξ) and u14(ξ), see Figures 5(a) and 6(a).The varying process for graphs ofu13(ξ) when c>α and g→g0+0 where α=β=γ=1, c=2, and (a) g=g0+1/5. (b) g=g0+10-2. (c) g=g0+10-6. (d) g=g0+10-8.
(a)
(b)
(c)
(d)The varying process for graphs ofu14(ξ) when c>α and g→g0+0 where α=β=γ=1, c=2, and (a) g=g0+1/5. (b) g=g0+10-2. (c) g=g0+10-6. (d) g=g0+10-8.
(a)
(b)
(c)
(d)The graphs ofu11(ξ), u12(ξ) when c=2, g=45/27 and u15(ξ), u16(ξ) when c=-1, g=-10.
(a)
u=u11(ξ)
(b)
u
=
u
12
(
ξ
)
(c)
u
=
u
15
(
ξ
)
(d)
u
=
u
16
(
ξ
)(
6
) When c<0, (9) has two elliptic periodic blow-up solutions
(30)u15(ξ)=φ1-A1+2A11-cn(η4ξ,k4),u16(ξ)=φ1-A1+2A11+cn(η4ξ,k4),(31)η4=βA1-5γc,k4=2A1-2φ1+φ2+φ34A1.
For the graphs of u15(ξ) and u16(ξ), see Figures 7(c) and 7(d).Remark 2.
Whenc>α and g→-g0-0, the elliptic periodic blow-up solutions u1(ξ) and u2(ξ) become the trigonometric periodic blow-up solutions u3(ξ) and u4(ξ), respectively.Remark 3.
Whenc>α and g→-g0+0, the elliptic periodic blow-up solutions u5(ξ) and u6(ξ) become the trigonometric periodic blow-up solutions u3(ξ) and u4(ξ), respectively. The symmetric elliptic periodic wave solutions u7(ξ) and u8(ξ) become a trivial solution u(ξ)=5(c-α)/9β, and for the varying process, see Figure 2.Remark 4.
Whenc>α and g→g0-0, the elliptic periodic blow-up solution u5(ξ) becomes the hyperbolic blow-up solution u10(ξ), for the varying process, see Figure 3. The elliptic periodic wave solutions u7(ξ) become the hyperbolic smooth solitary wave solution u9(ξ), and for the varying process, see Figure 4. The elliptic solutions u6(ξ) and u8(ξ) become a trivial solution u(ξ)=-5(c-α)/9β.Remark 5.
Whenc>α and g→g0+0, the elliptic periodic blow-up solution u13(ξ) becomes the hyperbolic blow-up solution u10(ξ), and for the varying process, see Figure 5. The elliptic periodic blow-up solution u14(ξ) becomes the hyperbolic smooth solitary wave solution u9(ξ), and for the varying process, see Figure 6.Proposition 6.
Consider BBM-likeB(4,2) equation
(32)ut+αux+β(u4)x-γ(u2)xxt=0
and its traveling wave equation
(33)(α-c)ψ+βψ4+2γc(ψ′)2+2γcψψ′′=g.For given constantsc and g, there are the following results.(1∘) When c>α and g=0, (32) has two elliptic periodic wave solutions
(34)u17(ξ)=1-cn(η5ξ,k5)β/2(c-α)3(1+3+(3-1)cn(η5ξ,k5)),u18(ξ)=1+cn(η5ξ,k5)β/2(c-α)3(1+3-(3-1)cn(η5ξ,k5)),
where
(35)η5=34c-α3γcβ2(c-α)6,k5=3-122.(2∘) When c<0 and g=-(c-α)4/16β3, (32) has two hyperbolic blow-up solutions
(36)u19(ξ)=c-α3(-4-2cosh(η6ξ)+6sinh(η6|ξ|))2β3(2-2cosh(η6ξ)+6sinh(η6|ξ|)),u20(ξ)=c-α3(4+2cosh(η6ξ)+6sinh(η6|ξ|))2β3(-2+2cosh(η6ξ)+6sinh(η6|ξ|)),
where
(37)η6=-β-γcc-α2β3.
For the graphs of ui(ξ)(i=17–20), see Figure 8.The graphs ofu17(ξ), u18(ξ) when c=20, g=0 and u19(ξ), u20(ξ) when c=-16.
(a)
u=u17(ξ)
(b)
u
=
u
18
(
ξ
)
(c)
u
=
u
19
(
ξ
)
(d)
u
=
u
20
(
ξ
)Remark 7.
In order to confirm the correctness of these solutions, we have verified them by using the software Mathematica; for instance, aboutu20(ξ) the commands are as follows:
(38)ξ=x-ctη=-β-γcc-α32β3u=c-α3(4+2cosh[ηξ]+6sinh[ηξ])2β3(-2+2cosh[ηξ]+6sinh[ηξ]).Simplify[[u2,{x,2},t]D[u,t]+αD[u,x]+βD[u4,x]-γD[u2,{x,2},t]]0.


## 3. The Derivations for Proposition1

In this section, firstly, we derive the precise expressions of the traveling wave solutions for BBM-likeB(3,2) equation. Secondly we show the relationships among these solutions theoretically. Substituting u(x,t)=φ(ξ) with ξ=x-ct into (9), it follows that
(39)(α-c)φ′+β(φ3)′+γc(φ2)′′′=0.Integrating (39) once, we have
(40)(α-c)φ+βφ3+2γc(φ′)2+2γcφφ′′=g,
where g is an integral constant.Lettingy=φ′, we obtain the following planar system
(41)dφdξ=y,dydξ=g+(c-α)φ-βφ3-2γcy22γcφ.
Under the transformation
(42)dξ=2γcφdτ,
system (41) becomes
(43)dφdτ=2γcφy,dydτ=g+(c-α)φ-βφ3-2γcy2.Clearly, system (41) and system (43) have the same first integral
(44)γcφ2y2-g2φ2-c-α3φ3+β5φ5=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 (41) from that of system (43).When the integral constanth=0, (44) becomes
(45)γcφ2y2-φ2(g2+c-α3φ-β5φ3)=0.Solving equationg/2+((c-α)/3)φ-(β/5)φ3=0, we get three roots φ1, φ2, and φ3 as (15), (16), and (17).On the other hand, solving equation(46)y=0,g+(c-α)φ-βφ3-2γcy2=0,
we get three three singular points (φi*,0)(i=1,2,3), where
(47)φ1*=2β183(α-c)-123Δ2/36βΔ1/3,φ2*=2β2336(3-3i)(c-α)+4396(1+3i)Δ2/312βΔ1/3,φ3*=2β2336(3+3i)(c-α)+4396(1-3i)Δ2/312βΔ1/3,Δ=81g2β4-12(c-α)3β3-9gβ2.According to the qualitative theory, we obtain the phase portraits of system (43) as Figures 9 and 10.The phase portraits of system (43) when c>0 and c≠α.
(a)
0<c<α,g<0
(b)
0
<
c
<
α
,
g
=
0
(c)
0
<
c
<
α
,
g
>
0
(d)
c
>
α
,
g
≤
-
g
0
(e)
c
>
α
,
-
g
0
<
g
<
0
(f)
c
>
α
,
g
=
0
(g)
c
>
α
,
0
<
g
<
g
0
(h)
c
>
α
,
g
=
g
0
(i)
c
>
α
,
g
>
g
0The phase portraits of system (43) when c<0.
(a)
g<0
(b)
g
=
0
(c)
g
>
0From Figures9 and 10, one can see that there are six kinds of orbits Γi(i=1–6), where Γ1(Γ6) passes (φ1,0), Γ2 passes (φ2,0) and (φ3,0), Γ3 and Γ4 pass (φ1,0) and (φ3,0), and Γ5 passes (φ3,0). Now, we will derive the explicit expressions of solutions for the BBM-like B(3,2) equation, respectively.(
1
) When 0<c<α or c>α, g<-g0, Γ1 has the expression
(48)Γ1:y=±β5γc(φ1-φ)(φ-φ2)(φ-φ3),φ≤φ1,
where φ2 and φ3 are complex numbers.Substituting (48) into dφ/dξ=y and integrating it, we have
(49)∫-∞φ‍ds(φ1-s)(s-φ2)(s-φ3)=β5γc|ξ|,∫φφ1‍ds(φ1-s)(s-φ2)(s-φ3)=β5γc|ξ|.
Completing the integrals in the above two equations and noting that u=φ(ξ), we obtain u1(ξ) and u2(ξ) as (11).(
2
) When c>α and g=-g0, Γ1 has the expression
(50)Γ1:y=±β5γc(φ2-φ)φ1-φ,φ≤φ1,
where φ1=-25(c-α)/9β, φ2=5(c-α)/9β.Substituting (50) into dφ/dξ=y and integrating it, we have
(51)∫-∞φ‍ds(φ2-s)φ1-s=β5γc|ξ|,∫φφ1‍ds(φ2-s)φ1-s=β5γc|ξ|.
Completing the integrals in the two above equations and noting that u=φ(ξ), we obtain u3(ξ) and u4(ξ) as (19) and (20).(
3
) When c>α and -g0<g<g0, Γ1 and Γ2 have the expressions
(52)Γ1:y=±β5γc(φ1-φ)(φ2-φ)(φ3-φ),φ≤φ1,Γ2:y=±β5γc(φ-φ1)(φ-φ2)(φ3-φ),φ2≤φ≤φ3.Substituting (52) into dφ/dξ=y and integrating them, we have
(53)∫-∞φ‍ds(φ1-s)(φ2-s)(φ3-s)=β5γc|ξ|,∫φφ1‍ds(φ1-s)(φ2-s)(φ3-s)=β5γc|ξ|,∫φ2φ‍ds(s-φ1)(s-φ2)(φ3-s)=β5γc|ξ|,∫φφ3‍ds(s-φ1)(s-φ2)(φ3-s)=β5γc|ξ|.
Completing the integrals in the above four equations and noting that u=φ(ξ), we obtain ui(ξ)(i=5–8) as (21).(
4
) When c>α and g=g0, Γ3 and Γ4 have the expressions
(54)Γ3:y=±β5γc(φ1-φ)φ3-φ,φ<φ1,Γ4:y=±β5γc(φ-φ1)φ3-φ,φ1<φ≤φ3,
where φ1=-5(c-α)/9β, φ3=25(c-α)/9β.Substituting (54) into dφ/dξ=y and integrating them, we have
(55)∫φφ3‍ds(s-φ1)φ3-s=β5γc|ξ|,∫-∞φ1‍ds(φ1-s)φ3-s=β5γc|ξ|.
Completing the integrals in the two equations above and noting that u=φ(ξ), we obtain u9(ξ) and u10(ξ) as (23) and (24).(
5
) When c>α and g0<g, Γ5 has the expression
(56)Γ5:y=±β5γc(φ3-φ)(φ-φ2)(φ-φ1),φ≤φ3,
where φ1 and φ2 are complex numbers.Substituting (56) into dφ/dξ=y and integrating it, we have
(57)∫-∞φ‍ds(φ3-s)(s-φ2)(s-φ1)=β5γc|ξ|,∫φφ3‍ds(φ3-s)(s-φ2)(s-φ1)=β5γc|ξ|.
Completing the integrals in the two above equations and noting that u=φ(ξ), we obtain u13(ξ) and u14(ξ) as (28).(
6
) When c<0, Γ6 has the expression
(58)Γ6:y=±β-5γc(φ-φ1)(φ-φ2)(φ-φ3),φ1≤φ,
where φ2 and φ3 are complex numbers.Substituting (58) into dφ/dξ=y and integrating it, we have
(59)∫φ+∞‍ds(s-φ1)(s-φ2)(s-φ3)=β-5γc|ξ|,∫φ1φ‍ds(s-φ1)(s-φ2)(s-φ3)=β-5γc|ξ|.
Completing the integrals in the two above equations and noting that u=φ(ξ), we obtain u15(ξ) and u16(ξ) as (30).(
7
) When c>α and g=g0, there are two special kinds of orbits Γ7 surrounding the center point (φ2*,0) (see Figure 11(d)) and Γ8 surrounding the center point (φ3*,0) (see Figure 12(a)), which are the boundaries of two families of closed orbits. Note that the periodic waves of (9) correspond to the periodic integral curves of (40), and the periodic integral curves correspond to the closed orbits of system (41). For given constants c, g and the corresponding initial value φ(0), we simulate the integral curves of (40) as shown in Figures 11 and 12.The simulation of integral curves of (40) when α=β=γ=1, c=2, g=45/27, (a) initial value φ(0)=-0.2, (b) initial value φ(0)=-0.01, and (c) initial value φ(0)=-0.00001.
(a)
(b)
(c)
(d)The simulation of integral curves of (40) when α=β=γ=1, c=2, g=45/27, (b) initial value φ(0)=0.5, (c) initial value φ(0)=0.1, and (d) initial value φ(0)=0.0001.
(a)
(b)
(c)
(d)From Figure11, we see that when the initial value φ(0) tends to 0-0, the periodic integral curve tends to peakon. This implies that the orbit Γ7 corresponds to peakon. On φ-y plane, Γ7 has the expression
(60)Γ7:y=±β5γc(φ-φ1)φ3-φ,φ1<φ<0,
where φ1=-5(c-α)/9β, φ3=25(c-α)/9β.Substituting (60) into dφ/dξ=y and integrating it, we have
(61)∫φ0‍ds(s-φ1)φ2-s=β5γc|ξ|.
Completing the integral in the above equation and noting that u=φ(ξ), we obtain u11(ξ) as (25).Similarly, from Figure12, we see that when the initial value φ(0) tends to 0+0, the periodic integral curve tends to a periodic peakon. This implies that the orbit Γ8 corresponds to a periodic peakon. On φ-y plane, Γ8 has the expression
(62)Γ8:y=±β5γc(φ-φ1)φ3-φ,0<φ≤φ3.
Substituting (62) into dφ/dξ=y and integrating it, we have
(63)∫0φ‍ds(s-φ1)φ3-s=β5γc|ξ|.
Completing the integral in the above equation and noting that u=φ(ξ), we obtain u12(ξ) as (26), where
(64)T=5γcβ|∫0φ3‍ds(s-φ1)φ3-s|=5γcββ5(c-α)4|ln(5-26)|.Hereto, we have finished the derivations for the solutionsui(ξ)(i=1–16). In what follows, we shall derive the relationships among these solutions.(1∘) When c>α and g→-g0, it follows that
(65)φ1⟶-25(c-α)9β,φ2⟶5(c-α)9β,φ3⟶5(c-α)9β,A1⟶5(c-α)β,η1⟶βγcc-α5β4,k1⟶0,cn(η1ξ,k1)⟶cn(βγcc-α5β4ξ,0)=cos(βγcc-α5β4ξ),η2⟶βγcc-α80β4,k2⟶0,sn(η2ξ,k2)⟶sn(βγcc-α80β4ξ,0)=sin(βγcc-α80β4ξ).Thus, we have(66)u1(ξ)⟶-25(c-α)9β+5(c-α)βmmii-25(c-α)/β1-cn(β/γc(c-α)/5β4ξ,0)=5(c-α)9β-25(c-α)/β1-cos(β/γc(c-α)/5β4ξ)=5(c-α)9β-5(c-α)βcsc2(βγcc-α80β4ξ)=u3(ξ)(see(19)),u2(ξ)⟶-25(c-α)9β+5(c-α)βmmii-25(c-α)/β1+cn(β/γc(c-α)/5β4ξ,0)=5(c-α)9βmmii-25(c-α)/β1+cos(β/γc(c-α)/5β4ξ)=5(c-α)9β-5(c-α)βsec2(βγcc-α80β4ξ)=u4(ξ)(see(20)),u5(ξ)⟶5(c-α)9β-5(c-α)βsn-2(βγcc-α80β4ξ,0)=5(c-α)9β-5(c-α)βsin-2(βγcc-α80β4ξ)=5(c-α)9β-5(c-α)βcsc2(βγcc-α80β4ξ)=u3(ξ)(see(19)),u6(ξ)⟶(-25(c-α)9β-5(c-α)9β×sn2(βγcc-α80β4ξ,0))×(1-sn2(βγcc-α80β4ξ,0))-1=(-25(c-α)9β-5(c-α)9β×sin2(βγcc-α80β4ξ))×(1-sin2(βγcc-α80β4ξ))-1=(-5(c-α)β+5(c-α)9β×cos2(βγcc-α80β4ξ))×(cos2(βγcc-α80β4ξ))-1=5(c-α)9β-5(c-α)β×sec2(βγcc-α80β4ξ)=u4(ξ)(see(20)),u7(ξ)⟶5(c-α)9β-(5(c-α)9β-5(c-α)9β)mii×sn2(βγcc-α80β4ξ,0)=5(c-α)9β,u8(ξ)⟶5(c-α)/9β-01-0=5(c-α)9β.(2∘) When c>α and g→g0, it follows that
(67)φ1⟶-5(c-α)9β,φ2⟶-5(c-α)9β,φ3⟶25(c-α)9β,η2⟶βγcc-α80β4,k2⟶1,sn(η2ξ,k2)⟶sn(βγcc-α80β4ξ,1)=tanh(βγcc-α80β4ξ),A2⟶5(c-α)β,η3⟶βγcc-α5β4,k3⟶1,cn(η3ξ,k3)⟶cn(βγcc-α5β4ξ,1)=sech(βγcc-α5β4ξ).Thus, we have(68)u5(ξ)⟶25(c-α)9β-(25(c-α)9β+5(c-α)9β)×sn-2(βγcc-α80β4ξ,1)=25(c-α)9β-5(c-α)β×tanh-2(βγcc-α80β4ξ)=25(c-α)9β-5(c-α)β×(1+csch2(βγcc-α80β4ξ))=u10(ξ)(see(24)),u6(ξ)(oru8(ξ))⟶(-5(c-α)9β+5(c-α)9β×sn2(βγcc-α80β4ξ,1))u6(ξ)(oru8(ξ))⟶×(1-sn2(βγcc-α80β4ξ,1))-1u6(ξ)(oru8(ξ))=-5(c-α)9β,u7(ξ)⟶25(c-α)9β-(25(c-α)9β+5(c-α)9β)×sn2(βγcc-α80β4ξ,1)=25(c-α)9β-5(c-α)β×tanh2(βγcc-α80β4ξ)=25(c-α)9β-5(c-α)β×(1-sech2(βγcc-α80β4ξ))=u9(ξ)(see(23)),u13(ξ)⟶25(c-α)9β+5(c-α)β-25(c-α)/β1-cn(β/γc(c-α)/5β4ξ,1)=55(c-α)9β-25(c-α)/β1-sech(β/γc(c-α)/5β4ξ)=55(c-α)9β-((β/γcc-α/5β4ξ)(βγcc-α5β4ξ)25(c-α)β×cosh(βγcc-α5β4ξ))×(cosh(βγcc-α5β4ξ)-1)-1=55(c-α)9β-(25(c-α)β+45(c-α)β×sinh2(βγcc-α80β4ξ))×(2sinh2(βγcc-α80β4ξ))-1=-5(c-α)9β-5(c-α)β×csch2(βγc(c-α)80β4ξ)=u10(ξ)(see(24)),u14(ξ)⟶25(c-α)9β+5(c-α)β-25(c-α)/β1+cn(β/γc(c-α)/5β4ξ,1)=55(c-α)9β-25(c-α)/β1+sech(β/γc(c-α)/5β4ξ)=55(c-α)9β-((β/γcc-α/5β4ξ)(β/γcc-α5β4ξ)25(c-α)β×cosh(βγcc-α5β4ξ))×(cosh(βγcc-α5β4ξ)+1)-1=55(c-α)9β-(-25(c-α)β+45(c-α)β×cosh2(βγcc-α80β4ξ))×(2cosh2(βγcc-α80β4ξ))-1=-5(c-α)9β+5(c-α)β×sech2(βγcc-α80β4ξ)=u9(ξ)(see(23)).Hereto, we have completed the derivations for Proposition1.


## 4. The Derivations for Proposition6

In this section, we derive the precise expressions of the traveling wave solutions for BBM-likeB(4,2) equation. Similar to the derivations in Section 3, substituting u(x,t)=ψ(ξ) with ξ=x-ct into (32) and integrating it, we have the following planar system:
(69)dψdξ=y,dydξ=g+(c-α)ψ-βψ4-2γcy22γcψ,
with the first integral
(70)γcψ2y2-g2ψ2-c-α3ψ3+β6ψ6=h,
where g and h are the integral constants.Whenc>α, h=0, and g=0, (70) becomes
(71)y=±c-α3γcψ(1-β2(c-α)ψ3),0≤ψ≤2(c-α)β3.Substituting (71) into the first equation of (69) and integrating it from 0 to ψ or ψ to 2(c-α)/β3, respectively, it follows that
(72)∫0ψ‍dss(1-(β/2(c-α))s3)=c-α3γc|ξ|,∫ψ2(c-α)/β3‍dss(1-(β/2(c-α))s3)=c-α3γc|ξ|.
Completing the integrals in the two above equations and noting that u=ψ(ξ), we obtain the two elliptic periodic wave solutions u17(ξ) and u18(ξ) as (34).Whenc<0, h=0, and g=-(c-α)4/16β3, (70) becomes
(73)y=±β-6γc|ψ-p|ψ2+qψ+r,
where
(74)p=c-α2β3,q=4(c-α)β3,r=27(c-α)24β23.Substituting (73) into the first equation of (69) and integrating it from -∞ to ψ or ψ to +∞, respectively, it follows that
(75)∫-∞ψ‍ds(p-s)s2+qs+r=β-6γc|ξ|,(ψ<p),∫ψ+∞‍ds(s-p)s2+qs+r=β-6γc|ξ|,(p<ψ).
Completing the integrals in the two above equations and noting that u=ψ(ξ), we obtain the two hyperbolic blow-up solutions u19(ξ) and u20(ξ) as (36).Hereto, we have completed the derivations for Proposition6.


## 5. Conclusion

In this paper, we have investigated the nonlinear wave solutions and their bifurcations for BBM-likeB(m,2)(m=3,4) equations. For BBM-like B(3,2) equation, we obtain some precise expressions of traveling wave solutions (see ui(ξ)(i=1–16)), which include periodic blow-up and periodic wave solution, peakon and periodic peakon wave solution, and solitary wave and blow-up solution. We also reveal the relationships among these solutions theoretically (see Remarks 2–5 and the corresponding derivations). For BBM-like B(4,2) equation, we construct two elliptic periodic wave solutions and two hyperbolic blow-up solutions (see ui(ξ)(i=17–20)). We would like to study the BBM-like B(m,n) equations further.



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*Source: 490341-2013-08-20.xml*