The Traveling Wave Solutions and Their Bifurcations for the BBM-Like 𝐵(𝑚,𝑛) Equations

Abstract

We investigate the traveling wave solutions and their bifurcations for the BBM-like 𝐵(𝑚,𝑛) equations 𝑢_𝑡+𝛼𝑢_𝑥+𝛽(𝑢^𝑚)_𝑥−𝛾(𝑢^𝑛)_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.

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) 𝑢_𝑡+𝑢𝑢_𝑥−𝑢_𝑥𝑥𝑡=0 (1) 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 𝑢_𝑡+𝑢^𝑚_𝑥−𝑢^𝑛_𝑥𝑥𝑡=0,𝑚,𝑛>1, (2) 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).

When 𝑚=𝑛=2, (2) reduces to the BBM-like 𝐵(2,2) equation 𝑢_𝑡+𝑢²_𝑥−𝑢²_𝑥𝑥𝑡=0. (3) 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 𝑢_𝑡+𝑎𝑢_𝑥−𝑘𝑢^𝑛_𝑥+𝑏𝑢^𝑛_𝑥𝑥𝑡=0 (4) 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: 𝑢_𝑡+𝑎𝑢_𝑥−𝑘𝑢^𝑚_𝑥+𝑏𝑢^𝑛_𝑥𝑥𝑡=0. (5) 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: 𝑢_𝑡+𝑢_𝑥+𝑎𝑢^𝑚_𝑥−𝑢^𝑛_𝑥𝑥𝑡=0,𝑚,𝑛>1. (6) 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-like 𝐵(𝑚,𝑛) equations: 𝑢_𝑡+𝛼𝑢_𝑥+𝛽𝑢^𝑚_𝑥−𝛾𝑢^𝑛_𝑥𝑥𝑡=0𝑚,𝑛>1. (7) 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 Section 2, 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 sn 𝑢 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 𝑔𝜉=𝑥−𝑐𝑡,₀=4275|𝑐−𝛼|³𝛽. (8)

Proposition 1. Consider BBM-like 𝐵(3,2) equation 𝑢_𝑡+𝛼𝑢_𝑥+𝛽𝑢³_𝑥−𝛾𝑢²_𝑥𝑥𝑡=0 (9) and its traveling wave equation (𝛼−𝑐)𝜑+𝛽𝜑³+2𝛾𝑐𝜑^′2+2𝛾𝑐𝜑𝜑^′′=𝑔. (10)

For given constants 𝑐 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 𝑢₁(𝜉)=𝜑₁+𝐴₁−2𝐴₁1−cn𝜂₁𝜉,𝑘₁,𝑢₂(𝜉)=𝜑₁+𝐴₁−2𝐴₁1+cn𝜂₁𝜉,𝑘₁, (11) where 𝐴₁=𝜑₁−𝜑₂𝜑₁−𝜑₃,𝜂₁=𝛽𝐴₁,5𝛾𝑐𝑘₁=2𝐴₁+2𝜑₁−𝜑₂−𝜑₃4𝐴₁,𝜑₁=2√³100𝛽(𝛼−𝑐)−√³10Ω^2/36𝛽Ω^1/3,𝜑₂=³√52𝛽√³101−√3𝑖(𝑐−𝛼)+1+√3𝑖Ω^2/36√³4𝛽Ω^1/3,𝜑₃=³√52𝛽√³101+√3𝑖(𝑐−𝛼)+1−√3𝑖Ω^2/36√³4𝛽Ω^1/3,Ω=729𝑔²𝛽⁴−80(𝑐−𝛼)³𝛽³−27𝑔𝛽². (12)(13)(14)(15)(16)(17)(18) For the graphs of  𝑢₁(𝜉) and 𝑢₂(𝜉), see Figures 1(a) and 1(b).

(a) 𝑢=𝑢1(𝜉)

(b) 𝑢=𝑢2(𝜉).

Figure 1 

The graphs of 𝑢1(𝜉) and 𝑢2(𝜉) when 𝛼=𝛽=𝛾=1, 𝑐=2, and 𝑔=−2√3/9.

(2) When 𝑐>𝛼 and 𝑔=−𝑔₀, (9) has two trigonometric periodic blow-up solutions 𝑢₃(𝜉)=5(𝑐−𝛼)9𝛽1−3csc²𝛽𝛾𝑐⁴𝑐−𝛼𝜉,80𝛽𝑢₄(𝜉)=5(𝑐−𝛼)9𝛽1−3sec²𝛽𝛾𝑐⁴𝑐−𝛼𝜉.80𝛽 (19)(20) 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)𝑢₅(𝜉)=𝜑₃−𝜑₃−𝜑₁sn⁻²𝜂₂𝜉,𝑘₂,𝑢₆(𝜉)=𝜑₁−𝜑₂sn²𝜂₂𝜉,𝑘₂1−sn²𝜂₂𝜉,𝑘₂,𝑢₇(𝜉)=𝜑₃−𝜑₃−𝜑₂sn²𝜂₂𝜉,𝑘₂,𝑢₈(𝜉)=𝜑₂−𝜑₁𝑘²₂sn²𝜂₂𝜉,𝑘₂1−𝑘²₂sn²𝜂₂𝜉,𝑘₂, (21) where 𝜂₂=12𝛽𝜑₃−𝜑₁,𝑘5𝛾𝑐₂=𝜑₃−𝜑₂𝜑₃−𝜑₁. (22) For the graphs of 𝑢_𝑖(𝜉)(𝑖=5--8), see Figures 2–4.

(a)

(b)

(c)

(d)

Figure 2 

The varying process for graphs of 𝑢7(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)

Figure 3 

The varying process for graphs of 𝑢5(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)

Figure 4 

The varying process for graphs of 𝑢7(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.

(4) When 𝑐>𝛼 and 𝑔=𝑔₀, (9) has a hyperbolic smooth solitary wave solution 𝑢₉(𝜉)=−5(𝑐−𝛼)9𝛽1−3sech²𝛽𝛾𝑐⁴𝑐−𝛼𝜉,80𝛽 (23) a hyperbolic blow-up solution 𝑢₁₀(𝜉)=−5(𝑐−𝛼)9𝛽1+3csch²𝛽𝛾𝑐⁴𝑐−𝛼𝜉,80𝛽 (24) a hyperbolic peakon wave solution 𝑢₁₁(𝜉)=−5(𝑐−𝛼)×9𝛽1−3sech²𝛿₁−𝛽𝛾𝑐⁴𝑐−𝛼𝜉,80𝛽 (25) and a hyperbolic periodic peakon wave solution 𝑢₁₂(𝜉)=−5(𝑐−𝛼)×9𝛽1−3sech²𝛿₁+𝛽𝛾𝑐⁴𝑐−𝛼𝜉,)()𝑇),80𝛽𝜉∈[(2𝑛−1𝑇,2𝑛+1 (26) where 𝛿₁=12ln5−2√6,𝑇=5𝛾𝑐𝛽⁴𝛽5()√𝑐−𝛼ln5−26,𝑛=0,±1,±2,…. (27) 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 𝑢₁₃(𝜉)=𝜑₃+𝐴₂−2𝐴₂1−cn𝜂₃𝜉,𝑘₃,𝑢₁₄(𝜉)=𝜑₃+𝐴₂−2𝐴₂1+cn𝜂₃𝜉,𝑘₃, (28) where 𝐴₂=𝜑₃−𝜑₂𝜑₃−𝜑₁,𝜂₃=𝛽𝐴₂,𝑘5𝛾𝑐₃=2𝐴₂+2𝜑₃−𝜑₂−𝜑₁4𝐴₂. (29) For the graphs of 𝑢₁₃(𝜉) and 𝑢₁₄(𝜉), see Figures 5(a) and 6(a).

(a)

(b)

(c)

(d)

Figure 5 

The varying process for graphs of 𝑢13(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)

Figure 6 

The varying process for graphs of 𝑢14(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)  𝑢=𝑢11(𝜉)

(b) 𝑢=𝑢12(𝜉)

(c) 𝑢=𝑢15(𝜉)

(d) 𝑢=𝑢16(𝜉)

Figure 7 

The graphs of 𝑢11(𝜉), 𝑢12(𝜉) when 𝑐=2, 𝑔=4√5/27 and 𝑢15(𝜉), 𝑢16(𝜉) when 𝑐=−1, 𝑔=−10.

(6) When 𝑐<0, (9) has two elliptic periodic blow-up solutions 𝑢₁₅(𝜉)=𝜑₁−𝐴₁+2𝐴₁1−cn𝜂₄𝜉,𝑘₄,𝑢₁₆(𝜉)=𝜑₁−𝐴₁+2𝐴₁1+cn𝜂₄𝜉,𝑘₄,𝜂₄=𝛽𝐴₁,𝑘−5𝛾𝑐₄=2𝐴₁−2𝜑₁+𝜑₂+𝜑₃4𝐴₁. (30)(31) For the graphs of 𝑢₁₅(𝜉) and 𝑢₁₆(𝜉), see Figures 7(c) and 7(d).

Remark 2. When 𝑐>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. When 𝑐>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. When 𝑐>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. When 𝑐>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-like 𝐵(4,2) equation 𝑢_𝑡+𝛼𝑢_𝑥+𝛽𝑢⁴_𝑥−𝛾𝑢²_𝑥𝑥𝑡=0 (32) and its traveling wave equation (𝛼−𝑐)𝜓+𝛽𝜓⁴+2𝛾𝑐𝜓^′2+2𝛾𝑐𝜓𝜓^′′=𝑔. (33)

For given constants 𝑐 and 𝑔, there are the following results.

(1^∘) When 𝑐>𝛼 and 𝑔=0, (32) has two elliptic periodic wave solutions 𝑢₁₇(𝜉)=1−cn𝜂₅𝜉,𝑘₅₃𝛽/2(𝑐−𝛼)1+√3+√3−1cn𝜂₅𝜉,𝑘₅,𝑢₁₈(𝜉)=1+cn𝜂₅𝜉,𝑘₅₃𝛽/2(𝑐−𝛼)1+√3−√3−1cn𝜂₅𝜉,𝑘₅, (34) where 𝜂₅=√⁴3𝑐−𝛼3𝛾𝑐⁶𝛽2(),𝑘𝑐−𝛼₅=√3−12√2. (35)

(2^∘) When 𝑐<0 and 𝑔=−³(𝑐−𝛼)⁴/16𝛽, (32) has two hyperbolic blow-up solutions 𝑢₁₉(𝜉)=³√𝑐−𝛼−4−2cosh𝜂₆𝜉+√6sinh𝜂₆𝜉₃2𝛽2−2cosh𝜂₆𝜉+√6sinh𝜂₆𝜉,𝑢₂₀(𝜉)=³√𝑐−𝛼4+2cosh𝜂₆𝜉+√6sinh𝜂₆𝜉₃2𝛽−2+2cosh𝜂₆𝜉+√6sinh𝜂₆𝜉, (36) where 𝜂₆=−𝛽−𝛾𝑐³𝑐−𝛼.2𝛽 (37) For the graphs of 𝑢_𝑖(𝜉)(𝑖=17--20), see Figure 8.

(a)   𝑢=𝑢17(𝜉)

(b) 𝑢=𝑢18(𝜉)

(c) 𝑢=𝑢19(𝜉)

(d) 𝑢=𝑢20(𝜉)

Figure 8 

The graphs of 𝑢17(𝜉), 𝑢18(𝜉) when 𝑐=20, 𝑔=0 and 𝑢19(𝜉), 𝑢20(𝜉) when 𝑐=−16.

Remark 7. In order to confirm the correctness of these solutions, we have verified them by using the software Mathematica; for instance, about 𝑢₂₀(𝜉) the commands are as follows: 𝜉=𝑥−𝑐𝑡𝜂=−𝛽−𝛾𝑐³√𝑐−𝛼₃2𝛽𝑢=³√𝑐−𝛼4+2cosh𝜂𝜉+√6sinh𝜂𝜉₃2𝛽−2+2cosh𝜂𝜉+√.6sinh𝜂𝜉Simplify𝐷[][]𝑢𝑢,𝑡+𝛼𝐷𝑢,𝑥+𝛽𝐷⁴,𝑥−𝛾𝐷𝑢²,{𝑥,2},𝑡0. (38)

3. The Derivations for Proposition 1

In this section, firstly, we derive the precise expressions of the traveling wave solutions for BBM-like 𝐵(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 (𝛼−𝑐)𝜑^′+𝛽𝜑^3′+𝛾𝑐𝜑^2′′′=0. (39)

Integrating (39) once, we have (𝛼−𝑐)𝜑+𝛽𝜑³+2𝛾𝑐𝜑^′2+2𝛾𝑐𝜑𝜑^′′=𝑔, (40) where 𝑔 is an integral constant.

Letting 𝑦=𝜑^′, we obtain the following planar system 𝑑𝜑𝑑𝜉=𝑦,𝑑𝑦=()𝑑𝜉𝑔+𝑐−𝛼𝜑−𝛽𝜑³−2𝛾𝑐𝑦².2𝛾𝑐𝜑 (41) Under the transformation 𝑑𝜉=2𝛾𝑐𝜑𝑑𝜏, (42) system (41) becomes 𝑑𝜑𝑑𝜏=2𝛾𝑐𝜑𝑦,𝑑𝑦()𝑑𝜏=𝑔+𝑐−𝛼𝜑−𝛽𝜑³−2𝛾𝑐𝑦². (43)

Clearly, system (41) and system (43) have the same first integral 𝛾𝑐𝜑²𝑦²−𝑔2𝜑²−𝑐−𝛼3𝜑³+𝛽5𝜑⁵=ℎ, (44) 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 constant ℎ=0, (44) becomes 𝛾𝑐𝜑²𝑦²−𝜑²𝑔2+𝑐−𝛼3𝛽𝜑−5𝜑³=0. (45)

Solving equation 𝑔/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 ()𝑦=0,𝑔+𝑐−𝛼𝜑−𝛽𝜑³−2𝛾𝑐𝑦²=0, (46) we get three three singular points (𝜑^∗_𝑖,0)(𝑖=1,2,3), where 𝜑^∗₁=2𝛽√³18(𝛼−𝑐)−√³12Δ^2/36𝛽Δ^1/3,𝜑^∗₂=2𝛽√³2√⁶3√3−3𝑖(𝑐−𝛼)+√³4√⁶91+√3𝑖Δ^2/312𝛽Δ^1/3,𝜑^∗₃=2𝛽√³2√⁶3√3+3𝑖(𝑐−𝛼)+√³4√⁶91−√3𝑖Δ^2/312𝛽Δ^1/3,Δ=81𝑔²𝛽⁴−12(𝑐−𝛼)³𝛽³−9𝑔𝛽². (47)

According to the qualitative theory, we obtain the phase portraits of system (43) as Figures 9 and 10.

(a)  0<𝑐<𝛼,𝑔<0

(b) 0<𝑐<𝛼,𝑔=0

(c) 0<𝑐<𝛼,𝑔>0

(d) 𝑐>𝛼,𝑔≤−𝑔0

(e) 𝑐>𝛼,−𝑔0<𝑔<0

(f) 𝑐>𝛼,𝑔=0

(g) 𝑐>𝛼,0<𝑔<𝑔0

(h) 𝑐>𝛼,𝑔=𝑔0

(i) 𝑐>𝛼,𝑔>𝑔0

Figure 9 

The phase portraits of system (43) when 𝑐>0 and 𝑐≠𝛼.

(a)  𝑔<0

(b) 𝑔=0

(c) 𝑔>0

Figure 10 

The phase portraits of system (43) when 𝑐<0.

From Figures 9 and 10, one can see that there are six kinds of orbits Γ_𝑖(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 Γ₁∶𝑦=±𝛽5𝛾𝑐𝜑₁−𝜑𝜑−𝜑₂𝜑−𝜑₃,𝜑≤𝜑₁, (48) where 𝜑₂ and 𝜑₃ are complex numbers.

Substituting (48) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝜑_−∞𝑑𝑠𝜑₁−𝑠𝑠−𝜑₂𝑠−𝜑₃=𝛽𝜉,5𝛾𝑐_𝜑1𝜑𝑑𝑠𝜑₁−𝑠𝑠−𝜑₂𝑠−𝜑₃=𝛽𝜉.5𝛾𝑐 (49) 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 Γ₁∶𝑦=±𝛽𝜑5𝛾𝑐²−𝜑√𝜑₁−𝜑,𝜑≤𝜑₁, (50) where 𝜑₁=−25(𝑐−𝛼)/9𝛽, 𝜑₂=5(𝑐−𝛼)/9𝛽.

Substituting (50) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝜑_−∞𝑑𝑠𝜑₂−𝑠√𝜑₁=−𝑠𝛽𝜉,5𝛾𝑐_𝜑1𝜑𝑑𝑠𝜑₂−𝑠√𝜑₁=−𝑠𝛽𝜉.5𝛾𝑐 (51) 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 Γ₁∶𝑦=±𝛽5𝛾𝑐𝜑₁−𝜑𝜑₂−𝜑𝜑₃−𝜑,𝜑≤𝜑₁,Γ₂∶𝑦=±𝛽5𝛾𝑐𝜑−𝜑₁𝜑−𝜑₂𝜑₃−𝜑,𝜑₂≤𝜑≤𝜑₃. (52)

Substituting (52) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating them, we have ^𝜑_−∞𝑑𝑠𝜑₁−𝑠𝜑₂−𝑠𝜑₃=−𝑠𝛽𝜉,5𝛾𝑐_𝜑1𝜑𝑑𝑠𝜑₁−𝑠𝜑₂−𝑠𝜑₃=−𝑠𝛽𝜉,5𝛾𝑐_𝜑𝜑2𝑑𝑠𝑠−𝜑₁𝑠−𝜑₂𝜑₃=−𝑠𝛽𝜉,5𝛾𝑐_𝜑3𝜑𝑑𝑠𝑠−𝜑₁𝑠−𝜑₂𝜑₃=−𝑠𝛽𝜉.5𝛾𝑐 (53) Completing the integrals in the above four equations and noting that 𝑢=𝜑(𝜉), we obtain 𝑢_𝑖(𝜉)(𝑖=5--8) as (21).

(4) When 𝑐>𝛼 and 𝑔=𝑔₀, Γ₃ and Γ₄ have the expressions Γ₃∶𝑦=±𝛽𝜑5𝛾𝑐¹−𝜑√𝜑₃−𝜑,𝜑<𝜑₁,Γ₄∶𝑦=±𝛽5𝛾𝑐𝜑−𝜑₁√𝜑₃−𝜑,𝜑₁<𝜑≤𝜑₃, (54) where 𝜑₁=−5(𝑐−𝛼)/9𝛽, 𝜑₃=25(𝑐−𝛼)/9𝛽.

Substituting (54) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating them, we have ^𝜑3_𝜑𝑑𝑠𝑠−𝜑₁√𝜑₃=−𝑠𝛽𝜉,5𝛾𝑐_𝜑1−∞𝑑𝑠𝜑₁−𝑠√𝜑₃=−𝑠𝛽𝜉.5𝛾𝑐 (55) 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 Γ₅∶𝑦=±𝛽5𝛾𝑐𝜑₃−𝜑𝜑−𝜑₂𝜑−𝜑₁,𝜑≤𝜑₃, (56) where 𝜑₁ and 𝜑₂ are complex numbers.

Substituting (56) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝜑_−∞𝑑𝑠𝜑₃−𝑠𝑠−𝜑₂𝑠−𝜑₁=𝛽𝜉,5𝛾𝑐_𝜑3𝜑𝑑𝑠𝜑₃−𝑠𝑠−𝜑₂𝑠−𝜑₁=𝛽𝜉.5𝛾𝑐 (57) 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 Γ₆∶𝑦=±𝛽−5𝛾𝑐𝜑−𝜑₁𝜑−𝜑₂𝜑−𝜑₃,𝜑₁≤𝜑, (58) where 𝜑₂ and 𝜑₃ are complex numbers.

Substituting (58) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have _𝜑^+∞𝑑𝑠𝑠−𝜑₁𝑠−𝜑₂𝑠−𝜑₃=𝛽𝜉,−5𝛾𝑐_𝜑𝜑1𝑑𝑠𝑠−𝜑₁𝑠−𝜑₂𝑠−𝜑₃=𝛽𝜉.−5𝛾𝑐 (59) 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 (𝜑^∗₂,0) (see Figure 11(d)) and Γ₈ surrounding the center point (𝜑^∗₃,0) (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.

(a)

(b)

(c)

(d)

Figure 11 

The simulation of integral curves of (40) when 𝛼=𝛽=𝛾=1, 𝑐=2, 𝑔=4√5/27, (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)

Figure 12 

The simulation of integral curves of (40) when 𝛼=𝛽=𝛾=1, 𝑐=2, 𝑔=4√5/27, (b) initial value 𝜑(0)=0.5, (c) initial value 𝜑(0)=0.1, and (d) initial value 𝜑(0)=0.0001.

From Figure 11, 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 Γ₇∶𝑦=±𝛽5𝛾𝑐𝜑−𝜑₁√𝜑₃−𝜑,𝜑₁<𝜑<0, (60) where 𝜑₁=−5(𝑐−𝛼)/9𝛽, 𝜑₃=25(𝑐−𝛼)/9𝛽.

Substituting (60) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ⁰_𝜑𝑑𝑠𝑠−𝜑₁√𝜑₂=−𝑠𝛽𝜉.5𝛾𝑐 (61) Completing the integral in the above equation and noting that 𝑢=𝜑(𝜉), we obtain 𝑢₁₁(𝜉) as (25).

Similarly, from Figure 12, 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 Γ₈∶𝑦=±𝛽5𝛾𝑐𝜑−𝜑₁√𝜑₃−𝜑,0<𝜑≤𝜑₃. (62) Substituting (62) into 𝑑𝜑/𝑑𝜉=𝑦 and integrating it, we have ^𝜑₀𝑑𝑠𝑠−𝜑₁√𝜑₃=−𝑠𝛽𝜉.5𝛾𝑐 (63) Completing the integral in the above equation and noting that 𝑢=𝜑(𝜉), we obtain 𝑢₁₂(𝜉) as (26), where 𝑇=5𝛾𝑐𝛽^𝜑3₀𝑑𝑠𝑠−𝜑₁√𝜑₃=−𝑠5𝛾𝑐𝛽⁴𝛽5()√𝑐−𝛼ln5−26. (64)

Hereto, we have finished the derivations for the solutions 𝑢_𝑖(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 𝜑₁⟶−25(𝑐−𝛼),𝜑9𝛽₂⟶5(𝑐−𝛼),𝜑9𝛽₃⟶5(𝑐−𝛼),𝐴9𝛽₁⟶5(𝑐−𝛼)𝛽,𝜂₁⟶𝛽𝛾𝑐⁴𝑐−𝛼,𝑘5𝛽₁⟶0,cn𝜂₁𝜉,𝑘₁⟶cn𝛽𝛾𝑐⁴𝑐−𝛼5𝛽𝜉,0=cos𝛽𝛾𝑐⁴𝑐−𝛼𝜉,𝜂5𝛽₂⟶𝛽𝛾𝑐⁴𝑐−𝛼,𝑘80𝛽₂⟶0,sn𝜂₂𝜉,𝑘₂⟶sn𝛽𝛾𝑐⁴𝑐−𝛼80𝛽𝜉,0=sin𝛽𝛾𝑐⁴𝑐−𝛼𝜉.80𝛽 (65)

Thus, we have 𝑢₁(𝜉)⟶−25(𝑐−𝛼)+9𝛽5(𝑐−𝛼)𝛽−25(𝑐−𝛼)/𝛽1−cn𝛽/𝛾𝑐⁴()=𝑐−𝛼/5𝛽𝜉,05(𝑐−𝛼)−29𝛽5(𝑐−𝛼)/𝛽1−cos𝛽/𝛾𝑐⁴()=𝑐−𝛼/5𝛽𝜉5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽csc²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₃(𝜉)(see(,𝑢19))₂(𝜉)⟶−25(𝑐−𝛼)+9𝛽5(𝑐−𝛼)𝛽−25(𝑐−𝛼)/𝛽1+cn𝛽/𝛾𝑐⁴()=𝑐−𝛼/5𝛽𝜉,05(𝑐−𝛼)−29𝛽5(𝑐−𝛼)/𝛽1+cos𝛽/𝛾𝑐⁴()=𝑐−𝛼/5𝛽𝜉5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽sec²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₄(𝜉)(see(,𝑢20))₅(𝜉)⟶5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽sn⁻²𝛽𝛾𝑐⁴𝑐−𝛼=80𝛽𝜉,05(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽sin⁻²𝛽𝛾𝑐⁴𝑐−𝛼𝜉=80𝛽5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽csc²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₃(𝜉)(see(,𝑢19))₆(𝜉)⟶−25(𝑐−𝛼)−9𝛽5(𝑐−𝛼)×9𝛽sn²𝛽𝛾𝑐⁴𝑐−𝛼×80𝛽𝜉,01−sn²𝛽𝛾𝑐⁴𝑐−𝛼80𝛽𝜉,0⁻¹=−25(𝑐−𝛼)−9𝛽5(𝑐−𝛼)×9𝛽sin²𝛽𝛾𝑐⁴𝑐−𝛼𝜉×80𝛽1−sin²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽⁻¹=−5(𝑐−𝛼)𝛽+5(𝑐−𝛼)9𝛽×cos²𝛽𝛾𝑐⁴𝑐−𝛼𝜉×80𝛽cos²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽⁻¹=5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽×sec²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₄(𝜉)(see(,𝑢20))₇(𝜉)⟶5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)×9𝛽sn²𝛽𝛾𝑐⁴𝑐−𝛼=80𝛽𝜉,05(𝑐−𝛼),𝑢9𝛽₈(𝜉)⟶5(𝑐−𝛼)/9𝛽−0=1−05(𝑐−𝛼).9𝛽 (66)

(2^∘) When 𝑐>𝛼 and 𝑔→𝑔₀, it follows that 𝜑₁⟶−5(𝑐−𝛼),𝜑9𝛽₂⟶−5(𝑐−𝛼),𝜑9𝛽₃⟶25(𝑐−𝛼),𝜂9𝛽₂⟶𝛽𝛾𝑐⁴𝑐−𝛼,𝑘80𝛽₂⟶1,sn𝜂₂𝜉,𝑘₂⟶sn𝛽𝛾𝑐⁴𝑐−𝛼80𝛽𝜉,1=tanh𝛽𝛾𝑐⁴𝑐−𝛼𝜉,𝐴80𝛽₂⟶5(𝑐−𝛼)𝛽,𝜂₃⟶𝛽𝛾𝑐⁴𝑐−𝛼,𝑘5𝛽₃⟶1,cn𝜂₃𝜉,𝑘₃⟶cn𝛽𝛾𝑐⁴𝑐−𝛼=5𝛽𝜉,1sech𝛽𝛾𝑐⁴𝑐−𝛼𝜉.5𝛽 (67)

Thus, we have𝑢₅(𝜉)⟶25(𝑐−𝛼)−29𝛽5(𝑐−𝛼)+9𝛽5(𝑐−𝛼)×9𝛽sn⁻²𝛽𝛾𝑐⁴𝑐−𝛼80𝛽𝜉,1=25(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽×tanh⁻²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=25(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽×1+csch²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₁₀(𝜉)(see(,𝑢24))₆(𝜉)or𝑢₈(𝜉)⟶−5(𝑐−𝛼)+9𝛽5(𝑐−𝛼)×9𝛽sn²𝛽𝛾𝑐⁴𝑐−𝛼×80𝛽𝜉,11−sn²𝛽𝛾𝑐⁴𝑐−𝛼80𝛽𝜉,1⁻¹=−5(𝑐−𝛼),𝑢9𝛽₇(𝜉)⟶25(𝑐−𝛼)−29𝛽5(𝑐−𝛼)+9𝛽5(𝑐−𝛼)×9𝛽sn²𝛽𝛾𝑐⁴𝑐−𝛼80𝛽𝜉,1=25(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽×tanh²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=25(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽×1−sech²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₉(𝜉)(see(,𝑢23))₁₃(𝜉)⟶25(𝑐−𝛼)+9𝛽5(𝑐−𝛼)𝛽−25(𝑐−𝛼)/𝛽1−cn𝛽/𝛾𝑐⁴(𝑐−𝛼)/5𝛽𝜉,1=55(𝑐−𝛼)−29𝛽5(𝑐−𝛼)/𝛽1−sech𝛽/𝛾𝑐⁴(𝑐−𝛼)/5𝛽𝜉=55(𝑐−𝛼)−29𝛽5(𝑐−𝛼)𝛽×cosh𝛽𝛾𝑐⁴𝑐−𝛼𝜉×5𝛽cosh𝛽𝛾𝑐⁴𝑐−𝛼𝜉5𝛽−1⁻¹=55(𝑐−𝛼)−29𝛽5(𝑐−𝛼)𝛽+45(𝑐−𝛼)𝛽×sinh²𝛽𝛾𝑐⁴𝑐−𝛼𝜉×80𝛽2sinh²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽⁻¹=−5(𝑐−𝛼)−9𝛽5(𝑐−𝛼)𝛽×csch²𝛽𝛾𝑐⁴(𝑐−𝛼)𝜉80𝛽=𝑢₁₀(𝜉)(see(,𝑢24))₁₄(𝜉)⟶25(𝑐−𝛼)+9𝛽5(𝑐−𝛼)𝛽−25(𝑐−𝛼)/𝛽1+cn𝛽/𝛾𝑐⁴(𝑐−𝛼)/5𝛽𝜉,1=55(𝑐−𝛼)−29𝛽5(𝑐−𝛼)/𝛽1+sech𝛽/𝛾𝑐⁴(𝑐−𝛼)/5𝛽𝜉=55(𝑐−𝛼)−29𝛽5(𝑐−𝛼)𝛽×cosh𝛽𝛾𝑐⁴𝑐−𝛼𝜉×5𝛽cosh𝛽𝛾𝑐⁴𝑐−𝛼𝜉5𝛽+1⁻¹=55(𝑐−𝛼)−9𝛽−25(𝑐−𝛼)𝛽+45(𝑐−𝛼)𝛽×cosh²𝛽𝛾𝑐⁴𝑐−𝛼𝜉×80𝛽2cosh²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽⁻¹=−5(𝑐−𝛼)+9𝛽5(𝑐−𝛼)𝛽×sech²𝛽𝛾𝑐⁴𝑐−𝛼𝜉80𝛽=𝑢₉(𝜉)(see(23)). (68)

Hereto, we have completed the derivations for Proposition 1.

4. The Derivations for Proposition 6

In this section, we derive the precise expressions of the traveling wave solutions for BBM-like 𝐵(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: 𝑑𝜓𝑑𝜉=𝑦,𝑑𝑦=()𝑑𝜉𝑔+𝑐−𝛼𝜓−𝛽𝜓⁴−2𝛾𝑐𝑦²,2𝛾𝑐𝜓 (69) with the first integral 𝛾𝑐𝜓²𝑦²−𝑔2𝜓²−𝑐−𝛼3𝜓³+𝛽6𝜓⁶=ℎ, (70) where 𝑔 and ℎ are the integral constants.

When 𝑐>𝛼, ℎ=0, and 𝑔=0, (70) becomes 𝑦=±𝑐−𝛼3𝛾𝑐𝜓𝛽1−2()𝜓𝑐−𝛼³,0≤𝜓≤³2(𝑐−𝛼)𝛽. (71)

Substituting (71) into the first equation of (69) and integrating it from 0 to 𝜓 or 𝜓 to ³2(𝑐−𝛼)/𝛽, respectively, it follows that ^𝜓₀𝑑𝑠𝑠1−𝛽/2(𝑐−𝛼)𝑠³=𝑐−𝛼𝜉,3𝛾𝑐_{3𝜓2(𝑐−𝛼)/𝛽}𝑑𝑠𝑠1−𝛽/2(𝑐−𝛼)𝑠³=𝑐−𝛼𝜉.3𝛾𝑐 (72) 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).

When 𝑐<0, ℎ=0, and 𝑔=−³(𝑐−𝛼)⁴/16𝛽, (70) becomes 𝑦=±𝛽−6𝛾𝑐𝜓−𝑝𝜓²+𝑞𝜓+𝑟, (73) where 𝑝=³𝑐−𝛼2𝛽,𝑞=³4(𝑐−𝛼)𝛽,𝑟=³27(𝑐−𝛼)²4𝛽². (74)

Substituting (73) into the first equation of (69) and integrating it from −∞ to 𝜓 or 𝜓 to +∞, respectively, it follows that ^𝜓_−∞𝑑𝑠𝑝−𝑠𝑠²=𝛽+𝑞𝑠+𝑟𝜉,,−6𝛾𝑐𝜓<𝑝_𝜓+∞𝑑𝑠𝑠−𝑝𝑠²=𝛽+𝑞𝑠+𝑟𝜉,.−6𝛾𝑐𝑝<𝜓 (75) 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 Proposition 6.

5. Conclusion

In this paper, we have investigated the nonlinear wave solutions and their bifurcations for BBM-like 𝐵(𝑚,2)(𝑚=3,4) equations. For BBM-like 𝐵(3,2) equation, we obtain some precise expressions of traveling wave solutions (see 𝑢_𝑖(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 𝑢_𝑖(𝜉)(𝑖=17--20)). We would like to study the BBM-like 𝐵(𝑚,𝑛) equations further.

Conflict of Interests

The authors declare that they do not have any commercial or associative interest that represents a conflict of interests in connection with the work submitted.

Acknowledgment

This work is supported by the National Natural Science Foundation (no. 11171115).