Some Explicit Expressions and Interesting Bifurcation Phenomena for Nonlinear Waves in Generalized Zakharov Equations

Abstract

Using bifurcation method of dynamical systems, we investigate the nonlinear waves for the generalized Zakharov equations 𝑢_𝑡𝑡−𝑐²_𝑠𝑢_𝑥𝑥=𝛽(|𝐸|²)_𝑥𝑥,𝑖𝐸_𝑡+𝛼𝐸_𝑥𝑥−𝛿₁𝑢𝐸+𝛿₂|𝐸|²𝐸+𝛿₃|𝐸|⁴𝐸=0, where 𝛼,𝛽,𝛿₁,𝛿₂,𝛿₃, and 𝑐_𝑠 are real parameters, 𝐸=𝐸(𝑥,𝑡) is a complex function, and 𝑢=𝑢(𝑥,𝑡) is a real function. We obtain the following results. (i) Three types of explicit expressions of nonlinear waves are obtained, that is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. (ii) Under different parameter conditions, these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves. We point out that there are two sets of kink waves which are called tall-kink waves and low-kink waves, respectively. (iii) Five kinds of interesting bifurcation phenomena are revealed. The first kind is that the 1-blow-up waves can be bifurcated from the periodic-blow-up and 2-blow-up waves. The second kind is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves. The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves. The fourth kind is that the low-kink waves can be bifurcated from four types of nonlinear waves, the symmetric solitary waves, the 1-blow-up waves, the tall-kink waves, and the antisymmetric solitary waves. The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves. We also show that the exp-function expressions include some results given by pioneers.

1. Introduction

Since the exact solutions to nonlinear wave equations help to understand the characteristics of nonlinear equations, seeking exact solutions of nonlinear equations is an important subject. For this purpose, there have been many methods such as the Jacobi elliptic function method [1, 2], F-expansion method [3, 4], and (𝐺^′/𝐺)-expansion method [5, 6].

Recently, the bifurcation method of dynamical systems [7–9] has been introduced to study the nonlinear partial differential equations. Up to now, the method is widely used in literatures such as [10–16].

In this paper, we consider the generalized Zakharov equations [17], which read as 𝑢_𝑡𝑡−𝑐²_𝑠𝑢_𝑥𝑥=𝛽|𝐸|²_𝑥𝑥,𝑖𝐸_𝑡+𝛼𝐸_𝑥𝑥−𝛿₁𝑢𝐸+𝛿₂|𝐸|²𝐸+𝛿₃|𝐸|⁴𝐸=0, (1) where 𝛼,𝛽,𝛿₁,𝛿₂,𝛿₃, and 𝑐_𝑠 are real parameters. 𝐸=𝐸(𝑥,𝑡) is a complex function which represents the envelop of the electric field, and 𝑢=𝑢(𝑥,𝑡) is a real function which represents the plasma density measured from its equilibrium value. Huang and Zhang [17] used Fan's direct algebraic method to obtain some exact travelling wave solutions of (1) as follows: 𝑢^±_𝑎0=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑎0²,𝐸^±_𝑎0=𝜑^±_𝑎0𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑏0=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑏0²,𝐸^±_𝑏0=𝜑^±_𝑏0𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑐0=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑐0²,𝐸^±_𝑐0=𝜑^±_𝑐0𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑑0=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑑0²,𝐸^±_𝑑0=𝜑^±_𝑑0𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑒0=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑒0²,𝐸^±_𝑒0=𝜑^±_𝑒0𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (2) where 𝛾and𝑤 are two constants and 𝜑^±_𝑎0=±−12𝑝3𝑞−9𝑞²−48𝑝𝑟cos2√,𝜑−𝑝𝜉_±𝑏0=±2𝑝𝑞−1+tanh√,𝜑𝑝𝜉_±𝑐0=±2𝑝𝑞−1−tanh√,𝜑𝑝𝜉_±𝑑0=±12𝑝−3𝑞+9𝑞²−48𝑝𝑟cosh2√,𝜑𝑝𝜉_±𝑒0=±2𝑝𝑞−1−coth√𝑝𝜉,𝜉=𝑥−𝑐𝑡,𝑝=𝛼𝛾²−𝜔𝛼,𝑞=𝛽𝛿₁𝛼𝑐²−𝑐²_𝑠−𝛿₂𝛼,𝛿𝑟=−³𝛼,𝑐=2𝛼𝛾. (3)(4)(5)(6)(7)(8)(9)(10)(11)(12) When 𝛼=1,𝛽=−1,𝛿₁=−2,𝛿₂=2𝜆,𝛿₃=0,and𝑐_𝑠=1, (1) reduce to the equations 𝑢_𝑡𝑡−𝑢_𝑥𝑥+|𝐸|²_𝑥𝑥=0,𝑖𝐸_𝑡+𝐸_𝑥𝑥+2𝑢𝐸+2𝜆|𝐸|²𝐸=0. (13) El-Wakil et al. [18] used the extended Jacobi elliptic function expansion method to obtain some Jacobi elliptic function expression solutions of (13).

When 𝜆=0, (13) reduce to the equations 𝑢_𝑡𝑡−𝑢_𝑥𝑥+|𝐸|²_𝑥𝑥=0,𝑖𝐸_𝑡+𝐸_𝑥𝑥+2𝑢𝐸=0. (14) By multisymplectic numerical method, Wang [19] proved the preservation of discrete normal conservation law of (14) theoretically and investigated the propagation and collision behaviors of the solitary waves numerically. There are also many other researchers studying (1) or its special case; for more information, one can see [20–24].

In this paper, we investigate the nonlinear waves and the bifurcation phenomena of (1). Coincidentally, under some transformations, (1) reduce to a planar system (54) which is similar to the planar system obtained by Feng and Li [16]. Many exact explicit parametric representations of solitary waves, kink and antikink waves, and periodic waves were obtained in [16], and their work is very important for the 𝜙⁶ model. In order to find the travelling wave solutions of (1), here we consider (1) by using the bifurcation method mentioned above; firstly, we obtain three types of explicit nonlinear wave solutions, this is, the fractional expressions, the trigonometric expressions, and the exp-function expressions. Secondly, we point out that these expressions represent symmetric and antisymmetric solitary waves, kink and antikink waves, symmetric periodic and periodic-blow-up waves, and 1-blow-up and 2-blow-up waves under different parameter conditions. Thirdly, we reveal five kinds of interesting bifurcation phenomena mentioned in the abstract above.

The remainder of this paper is organized as follows. In Section 2, we give some notations and state our main results. In Section 3, we give derivations for our results. A brief conclusion is given in Section 4.

2. Main Results

In this section, we state our main results. To relate conveniently, let us give some notations which will be used in the latter statement and the derivations.

Let 𝑙_𝑖(𝑖=1,2,…,7) represent the following seven curves: 𝑙₁𝑞∶𝑝=²,𝑙4𝑟𝑟>0,𝑞<0₂∶𝑝=3𝑞²,𝑙16𝑟𝑟>0,𝑞<0₃,𝑙∶𝑝=0𝑟>0,𝑞<0₄,𝑙∶𝑝=0𝑟>0,𝑞>0₅,𝑙∶𝑝=0𝑟<0,𝑞>0₆∶𝑝=3𝑞²,𝑙16𝑟𝑟<0,𝑞>0₇𝑞∶𝑝=².4𝑟𝑟<0,𝑞<0 (15)(16)(17)(18)(19)(20)(21)

Let 𝐴_𝑖(𝑖=1,2,…,12) represent the regions surrounded by the curves 𝑙_𝑖(𝑖=1,2,…,7) and the coordinate axes (see Figure 1).

(a) 𝑟>0

(b) 𝑟<0

Figure 1 

The locations of the regions 𝐴𝑖(𝑖=1,2,…,12) and curves 𝑙𝑖(𝑖=1,2,…,7).

Let 𝑢()=𝛽𝑥,𝑡𝑐²−𝑐²_𝑠𝜑²(𝜉),𝐸(𝑥,𝑡)=𝜑(𝜉)𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},Δ=𝑞²𝜑−4𝑝𝑟,₁=−𝑞+Δ,𝜑2𝑟₂=−𝑞−Δ,𝐻2𝑟𝜑,𝑦=ℎ, (22)(23)(24)(25)(26)(27) where 𝐻(𝜑,𝑦) is the first integral which will be given later and ℎ is the integral constant.

In order to search for the solutions of (1) and studing the bifurcation phenomena, we only need to get the solution 𝜑(𝜉) according to (22) and (23). For convenience, throughout the following work we only discuss the solution 𝜑(𝜉). Now let us state our main results in the following Propositions 1, 2, and 3.

2.1. When the Orbit Γ Is Defined by 𝐻(𝜑,𝑦)=𝐻(0,0)

Proposition 1. (i) For 𝑝=0, (1) have two fractional nonlinear wave solutions 𝑢^±_𝑎=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑎²,𝐸^±_𝑎=𝜑^±_𝑎𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (28) where 𝜑^±_𝑎=±6𝑞3𝑞²𝜉².−4𝑟 (29) If (𝑞,𝑝)∈𝑙₄, then 𝜑^±_𝑎 are 1-blow-up waves (refer to Figure 2(d)). If (𝑞,𝑝)∈𝑙₃, then 𝜑^±_𝑎 are 2-blow-up waves (refer to Figure 3(d)). If (𝑞,𝑝)∈𝑙₅, then 𝜑^±_𝑎 are symmetric solitary waves (refer to Figure 4(d)).
(ii) For 𝑝<0, (1) have two nonlinear wave solutions 𝑢^±_𝑏=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑏²,𝐸^±_𝑏=𝜑^±_𝑏𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (30) where 𝜑^±_𝑏=±−12𝑝3𝑞+9𝑞²−48𝑝𝑟cos2√.−𝑝𝜉 (31) These solutions have the following properties and wave shapes.
(1) When (𝑞,𝑝)∈𝐴₁ or 𝐴₂, then 𝜑^±_𝑎0 and 𝜑^±_𝑏 are periodic-blow-up waves (refer to Figure 2(a) or Figure 3(a)). Specially, in region 𝐴₁ when 𝑝→0−0, the periodic-blow-up waves 𝜑^±_𝑎0 become 1-blow-up waves 𝜑^±_𝑎, and for the varying process, see Figure 2, while the periodic-blow-up waves 𝜑^±_𝑏 become a trivial wave 𝜑=0. In region 𝐴₂ when 𝑝→0−0, the periodic-blow-up waves 𝜑^±_𝑏 become 2-blow-up waves 𝜑^±_𝑎, and for the varying process, see Figure 3.
 (2) When (𝑞,𝑝)∈𝐴₁₂, then 𝜑^±_𝑎0 and 𝜑^±_𝑏 are symmetric periodic waves (refer to Figure 5(a)). If 𝑝→0−0, the symmetric periodic waves 𝜑^±_𝑎0 become symmetric solitary waves 𝜑^±_𝑎, and for the varying process, see Figure 4. If 𝑝→3𝑞²/16𝑟+0, the symmetric periodic waves 𝜑^±_𝑎0 and 𝜑^±_𝑏 become two trivial waves 𝜑=±−(3𝑞/4𝑟), and for the varying process, see Figure 5.
(iii) For 𝑝>0, (1) have four nonlinear wave solutions 𝑢^±_𝑐=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑐²,𝐸^±_𝑐=𝜑^±_𝑐𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑑=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑑²,𝐸^±_𝑑=𝜑^±_𝑑𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (32) where 𝜑^±_𝑐=±4𝜆𝑝𝜆²𝑒^2√𝑝𝜉+𝜆₀/12𝑒^{√−2𝑝𝜉},𝜑−𝜆𝑞_±𝑑=±4𝜆𝑝𝜆²𝑒^{√−2𝑝𝜉}+𝜆₀/12𝑒^2√𝑝𝜉,−𝜆𝑞 (33)𝜆 is a nonzero arbitrary real constant and 𝜆₀=3𝑞²−16𝑝𝑟. Corresponding to 𝜆>0 or 𝜆<0, these solutions have the following properties and wave shapes.
(1) For the case of 𝜆>0, there are four properties as follows.(1)_aWhen (𝑞,𝑝)∈𝑙₂, that is, 𝑟>0,𝑞<0, and 𝑟=3𝑞²/16𝑝, then 𝜑^±_𝑐 and 𝜑^±_𝑑 become 𝜑^±_𝑐1=±4𝑝𝜆𝑒^2√𝑝𝜉,𝜑−𝑞_±𝑑1=±4𝑝𝜆𝑒^{√−2𝑝𝜉},−𝑞 (34) which represent four low-kink waves (refer to Figure 6(d) or Figure 7(d)). Specially, let 𝜆=−𝑞>0, then 𝜑^±_𝑐1 and 𝜑^±_𝑑1 become 𝜑^±_𝑏0 and 𝜑^±_𝑐0.(1)_b If (𝑞,𝑝) belongs to one of 𝐴₃,𝐴₇,𝐴₈, and 𝜆≠𝜆₀/12, then 𝜑^±_𝑐≠𝜑^±_𝑑, and they represent four symmetric solitary waves (refer to Figure 6(a)). Specially, when (𝑞,𝑝)∈𝐴₃ and 𝑝→3𝑞²/16𝑟−0, then the four symmetric solitary waves 𝜑^±_𝑐 and 𝜑^±_𝑑 become the four low-kink waves 𝜑^±_𝑐1 and 𝜑^±_𝑑1, and for the varying process, see Figure 6.(1)_c If (𝑞,𝑝) belongs to one of 𝐴₄,𝐴₅,𝐴₆, 𝑙₁, and 𝜆≠𝜆₀/12, then 𝜑^±_𝑐≠𝜑^±_𝑑, and they represent four 1-blow-up waves (refer to Figure 7(a)). Specially, when (𝑞,𝑝)∈𝐴₄ and 𝑝→3𝑞²/16𝑟+0, then the four 1-blow-up waves 𝜑^±_𝑐 and 𝜑^±_𝑑 become the four low-kink waves 𝜑^±_𝑐1 and 𝜑^±_𝑑1, and for the varying process, see Figure 7.(1)_d If (𝑞,𝑝) belongs to one of 𝐴₃,𝐴₇,𝐴₈, and 𝜆=𝜆₀/12, then 𝜑^±_𝑐=𝜑^±_𝑑=𝜑^±_𝑑0. Specially, when (𝑞,𝑝)∈𝐴₃ and 𝑝→3𝑞²/16𝑟−0, then 𝜑^±_𝑑0 tend to two trivial solutions 𝜑=±−(3𝑞/4𝑟).

(a)

(b)

(c)

(d)

Figure 2 

(The 1-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of 𝜑±𝑎0 when (𝑞,𝑝)∈𝐴1 and 𝑝→0−0, where 𝑟=1, 𝑞=4 and (a) 𝑝=0−10−1, (b) 𝑝=0−10−2, (c) 𝑝=0−10−3, and (d) 𝑝=0−10−4.

(a)

(b)

(c)

(d)

Figure 3 

(The 2-blow-up waves are bifurcated from the periodic-blow-up waves.) The varying process for the figures of 𝜑±𝑏 when (𝑞,𝑝)∈𝐴2 and 𝑝→0−0, where 𝑟=1,  𝑞=−4 and (a) 𝑝=0−1, (b) 𝑝=0−0.5, (c) 𝑝=0−0.1, and (d) 𝑝=0−0.01.

(a)

(b)

(c)

(d)

Figure 4 

(The symmetric solitary waves are bifurcated from the symmetric periodic waves.) The varying process for the figures of 𝜑±𝑎0 when (𝑞,𝑝)∈𝐴12 and 𝑝→0−0, where 𝑟=−1, 𝑞=4 and (a) 𝑝=0−0.5, (b) 𝑝=0−0.1, (c) 𝑝=0−0.01, and (d) 𝑝=0−0.001.

(a)

(b)

(c)

(d)

Figure 5 

(The symmetric periodic waves become two trivial waves.) The varying process for the figures of 𝜑±𝑎0 and 𝜑±𝑏 when (𝑞,𝑝)∈𝐴12 and 𝑝→3𝑞2/16𝑟+0, where 𝑟=−1,  𝑞=4,and𝑙6∶𝑝=3𝑞2/16𝑟=−3 and (a) 𝑝=−3+10−1, (b) 𝑝=−3+10−2, (c) 𝑝=−3+10−3, and (d) 𝑝=−3+10−6.

(a)

(b)

(c)

(d)

Figure 6 

(The four low-kink waves are bifurcated from four symmetric solitary waves.) The varying process for the figures of 𝜑±𝑐 and 𝜑±𝑑 when (𝑞,𝑝)∈𝐴3,𝜆>0,𝜆≠𝜆0/12 and 𝑝→3𝑞2/16𝑟−0, where 𝑟=1,𝑞=−4,𝜆=4,and𝑙2∶𝑝=3𝑞2/16𝑟=3 and (a) 𝑝=3−10−1, (b) 𝑝=3−10−3, (c) 𝑝=3−10−7, and (d) 𝑝=3−10−9.

(a)

(b)

(c)

(d)

Figure 7 

(The four low-kink waves are bifurcated from four 1-blow-up waves.) The varying process for the figures of 𝜑±𝑐 and 𝜑±𝑑 when (𝑞,𝑝)∈𝐴4,𝜆>0,𝜆≠𝜆0/12, and 𝑝→3𝑞2/16𝑟+0, where 𝑟=1,𝑞=−4,𝜆=4,and𝑙2∶𝑝=3𝑞2/16𝑟=3 and (a) 𝑝=3+10−1, (b) 𝑝=3+10−3, (c) 𝑝=3+10−6, and (d) 𝑝=3+10−9.

(2) For the case of 𝜆<0, there are three properties as follows. (2)_a If (𝑞,𝑝)∈𝑙₂, that is, 𝑟>0,𝑞<0, and 𝑟=3𝑞²/16𝑝, then 𝜑^±_𝑐 and 𝜑^±_𝑑 become 𝜑^±_𝑐1 and 𝜑^±_𝑑1 which represent four 1-blow-up waves (refer to Figure 8(d)). Specially, let 𝜆=𝑞<0, then 𝜑^±_𝑐1 and 𝜑^±_𝑑1 become the hyperbolic 1-blow-up wave solutions 𝜑^±_𝑒0 and𝜑^±_𝑑2=±2𝑝𝑞−1+coth√𝑝𝜉. (35)(2)_b If (𝑞,𝑝)∈𝐴₃ and 𝜆≠−𝜆₀/12, then 𝜑^±_𝑐≠𝜑^±_𝑑, and they represent four 2-blow-up waves (refer to Figure 8(a)). Specially, when 𝑝→3𝑞²/16𝑟−0, then the four 2-blow-up waves become four 1-blow-up waves 𝜑^±_𝑐1 and 𝜑^±_𝑑1, and for the varying process, see Figure 8.(2)_c If (𝑞,𝑝)∈𝐴₃ and 𝜆=−𝜆₀/12, then 𝜑^±_𝑐=𝜑^±_𝑑=𝜑^±_𝑐𝑑 of forms 𝜑^±_𝑐𝑑=±12𝑝−3𝑞−9𝑞²−48𝑝𝑟cosh2√,𝑝𝜉 (36) which represent hyperbolic blow-up waves. When 𝑝→3𝑞²/16𝑟−0, then 𝜑^±_𝑐𝑑 tend to two trivial solutions 𝜑=±−(3𝑞/4𝑟).

(a)

(b)

(c)

(d)

Figure 8 

(The four 2-blow-up waves become four 1-blow-up waves.) The varying process for the figures of 𝜑±𝑐 and 𝜑±𝑑 when (𝑞,𝑝)∈𝐴3,𝜆<0,𝜆≠−𝜆0/12, and 𝑝→3𝑞2/16𝑟−0, where 𝑟=1, 𝑞=−4,𝜆=−4,and𝑙2∶𝑝=3𝑞2/16𝑟=3 and (a) 𝑝=3−10−1, (b) 𝑝=3−10−3, (c) 𝑝=3−10−6, and (d) 𝑝=3−10−9.

2.2. When the Orbit Γ Is Defined by 𝐻(𝜑,𝑦)=𝐻(𝜑₁,0)

Proposition 2. If (𝑞,𝑝) belongs to one of the regions 𝐴₁,𝐴₂,𝐴₃,𝐴₄,𝐴₁₁,and𝐴₁₂ or curves 𝑙₁,𝑙₂,𝑙₃,𝑙₆, and 𝑙₇, then (1) have four nonlinear wave solutions 𝑢^±_𝑒=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑒²,𝐸^±_𝑒=𝜑^±_𝑒𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑓=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑓²,𝐸^±_𝑓=𝜑^±_𝑓𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (37) where 𝜑^±_𝑒=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^𝜃𝜉+4𝜇²𝑟²𝑒^2𝜃𝜉,𝜑^±_𝑓=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^−𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^−𝜃𝜉+4𝜇²𝑟²𝑒^−2𝜃𝜉, (38)𝜇 is a nonzero arbitrary real constant, Δ is given in (24), and 𝜃=1𝑟ΔΔ−𝑞. (39) Let 𝜇₀=𝑞+2Δ.2𝑟 (40) About 𝜇₀, one has the following fact: 𝜇₀>0for𝑞,𝑝∈𝐴₁,𝐴₂,𝐴₃,𝑙₃,=0for𝑞,𝑝∈𝑙₂,<0for𝑞,𝑝∈𝐴₄,𝐴₁₁,𝐴₁₂,𝑙₁,𝑙₆,𝑙₇. (41) Corresponding to 𝜇>0 and 𝜇<0, these solutions have the following properties and wave shapes.
(1^o)   For the case of 𝜇>0, there are six properties as follows.(1^o)_a If (𝑞,𝑝)∈𝑙₂, that is, 𝑟>0,𝑞<0, and 𝑟=3𝑞²/16𝑝, then 𝜑^±_𝑒 and 𝜑^±_𝑓 become 𝜑^∓_𝑒1=∓4𝜇𝑝16𝑝𝑒^{√−2𝑝𝜉},𝜑−𝜇𝑞_∓𝑓1=∓4𝜇𝑝16𝑝𝑒^2√𝑝𝜉,−𝜇𝑞 (42)(43) which represent four low-kink waves (refer to Figure 9(d)). Specially, let 𝜇=−(16𝑝/𝑞)>0, then 𝜑^∓_𝑒1=𝜑^∓_𝑐0 and 𝜑^∓_𝑓1=𝜑^∓_𝑏0.(1^o)_b  If (𝑞,𝑝)∈𝐴₁,𝐴₂,𝐴₃, and 𝜇≠|𝜇₀|, then 𝜑^±_𝑒≠𝜑^±_𝑓, and they represent four tall-kink waves (refer to Figure 9(a)). Specially, when (𝑞,𝑝)∈𝐴₃ and 𝑝→3𝑞²/16𝑟−0, then 𝜑^±_𝑒 and 𝜑^±_𝑓 become 𝜑^∓_𝑒1 and 𝜑^∓_𝑓1, and for the varying process, see Figure 9.(1^o)_c  If (𝑞,𝑝)∈𝐴₄,𝐴₁₁,𝐴₁₂,𝑙₆, and 𝜇≠|𝜇₀|, then 𝜑^±_𝑒≠𝜑^±_𝑓. When (𝑞,𝑝)∈𝐴₄, they represent four antisymmetric solitary waves (refer to Figure 10(a)). When (𝑞,𝑝)∈𝐴₁₁,𝐴₁₂,and𝑙₆, they represent four symmetric solitary waves (refer to Figure 11(a)). Specially, when (𝑞,𝑝)∈𝐴₄, if 𝑝→3𝑞²/16𝑟+0, then 𝜑^±_𝑒 and 𝜑^±_𝑓 become 𝜑^∓_𝑒1 and 𝜑^∓_𝑓1, and for the varying process, see Figure 10. When (𝑞,𝑝)∈𝐴₁₁ and 𝑝→𝑞²/4𝑟+0, then 𝜑^±_𝑒 and 𝜑^±_𝑓 tend to two trivial solutions 𝜑=±−(𝑞/2𝑟), and for the varying process, see Figure 11.(1^o)_d If (𝑞,𝑝)∈𝐴₃ and 𝜇=|𝜇₀|, then 𝜑^±_𝑒=𝜑^∓_𝑓=𝜑^∓_𝑔 of forms 𝜑^∓_𝑔=∓(1/𝑟)Δ−𝑞2Δ+𝑞sinh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉), (44) which represent two tall-kink waves and tend to a trivial wave 𝜑=0 when 𝑝→3𝑞²/16𝑟−0.(1^o)_e If (𝑞,𝑝)∈𝐴₄ and 𝜇=|𝜇₀|, then 𝜑^±_𝑒=𝜑^±_𝑓=𝜑^∓_ℎ of forms 𝜑^∓_ℎ=∓(1/𝑟)Δ−𝑞2Δ+𝑞cosh((𝜃/2)𝜉)𝑞−4Δ+𝑞+2Δcosh(𝜃𝜉), (45) which represent two antisymmetric solitary waves, tend to a trivial wave 𝜑=0 when 𝑝→3𝑞²/16𝑟+0, and tend to two trivial waves 𝜑=±−(𝑞/2𝑟) when 𝑝→𝑞²/4𝑟−0.(1^o)_f If (𝑞,𝑝)∈𝐴₁₁,𝐴₁₂,𝑙₆ and 𝜇=|𝜇₀|, then 𝜑^±_𝑒=𝜑^±_𝑓=𝜑^±_ℎ which represent two symmetric solitary waves. Specially, when (𝑞,𝑝)∈𝐴₁₁ and 𝑝→𝑞²/4𝑟−0, then 𝜑^±_ℎ tend to two trivial waves 𝜑=±−(𝑞/2𝑟).
(2^o)   For the case of 𝜇<0, there are five properties as follows.(2^o)_a If (𝑞,𝑝)∈𝑙₂, then 𝜑^±_𝑒1 and 𝜑^±_𝑓1 represent four 1-blow-up waves (refer to Figure 12(d)). Specially, let 𝜇=16𝑝/𝑞<0, then 𝜑^±_𝑒1=𝜑^±_𝑒0 and 𝜑^±_𝑓=𝜑^∓_𝑑2. (2^o)_b If (𝑞,𝑝)∈𝐴₁,𝐴₂,𝐴₃,𝐴₄ and 𝜇≠−|𝜇₀|, then 𝜑^±_𝑒≠𝜑^±_𝑓, and they represent four pairs of 1-blow-up waves (refer to Figure 12(a)). In particular, when (𝑞,𝑝)∈𝐴₄, if 𝑝→3𝑞²/16𝑟+0, then the four pairs of 1-blow-up waves become two pairs of 1-blow-up waves, and the varying process is displayed in Figure 12. If 𝑝→𝑞²/4𝑟−0, then the four pairs of 1-blow-up waves become two trivial waves 𝜑=±−(𝑞/2𝑟). (2^o)_c If (𝑞,𝑝)∈𝐴₁₁,𝐴₁₂,𝑙₆ and 𝜇≠−|𝜇₀|, then 𝜑^±_𝑒≠𝜑^±_𝑓, and they represent four tall-kink waves.(2^o)_d If 𝜇=−|𝜇₀|, when (𝑞,𝑝)∈𝐴₃, then 𝜑^±_𝑒=𝜑^±_𝑓=𝜑^∓_ℎ which represent two pairs of 1-blow-up waves. When (𝑞,𝑝)∈𝐴₄, then 𝜑^±_𝑒=𝜑^∓_𝑓=𝜑^∓_𝑔 which represent two pairs of 1-blow-up waves.(2^o)_e If (𝑞,𝑝)∈𝐴₁₁,𝐴₁₂,𝑙₆ and 𝜇=−|𝜇₀|, then 𝜑^±_𝑒=𝜑^∓_𝑓=𝜑^∓_𝑔 which represent two tall-kink waves.

(a)

(b)

(c)

(d)

Figure 9 

(The four low-kink waves are bifurcated from the four tall-kink waves.) The varying process for the figures of 𝜑±𝑒 and 𝜑±𝑓 when 𝜇>0,𝜇≠|𝜇0|,(𝑞,𝑝)∈𝐴3, and 𝑝→3𝑞2/16𝑟−0, where 𝑟=1,𝑞=−4,𝑙2∶𝑝=3𝑞2/16𝑟=3,and𝜇=4 and (a) 𝑝=3−0.5, (b) 𝑝=3−10−2, (c) 𝑝=3−10−4, and (d) 𝑝=3−10−6.

(a)

(b)

(c)

(d)

Figure 10 

(The four low-kink waves are bifurcated from the four antisymmetric solitary waves.) The varying process for the figures of 𝜑±𝑒 and 𝜑±𝑓 when 𝜇>0,𝜇≠|𝜇0|,(𝑞,𝑝)∈𝐴4, and 𝑝→3𝑞2/16𝑟+0, where 𝑟=1,𝑞=−4,𝑙2∶𝑝=3𝑞2/16𝑟=3,and𝜇=4 and (a) 𝑝=3+10−1, (b) 𝑝=3+10−3, (c) 𝑝=3+10−5, and (d) 𝑝=3+10−7.

(a)

(b)

(c)

(d)

Figure 11 

(The four symmetric solitary waves become two trivial waves.) The varying process for the figures of 𝜑±𝑒 and 𝜑±𝑓 when 𝜇>0,𝜇≠|𝜇0|,(𝑞,𝑝)∈𝐴11, and 𝑝→𝑞2/4𝑟+0, where 𝑟=−1, 𝑞=4,𝑙7∶𝑝=𝑞2/4𝑟=−4,and𝜇=1 and (a) 𝑝=−3.5, (b) 𝑝=−4+10−1, (c) 𝑝=−4+10−2, and (d) 𝑝=−4+10−5.

(a)

(b)

(c)

(d)

Figure 12 

(The two pairs of 1-blow-up waves are bifurcated from the four pairs of 1-blow-up waves.) The varying process for the figures of 𝜑±𝑒 and 𝜑±𝑓 when 𝜇<0,𝜇≠−|𝜇0|,(𝑞,𝑝)∈𝐴4, and 𝑝→3𝑞2/16𝑟+0, where 𝑟=1,𝑞=−4,𝑙2∶𝑝=3𝑞2/16𝑟=3,and𝜇=−4 and (a) 𝑝=3+10−2, (b) 𝑝=3+10−4, (c) 𝑝=3+10−6, and (d) 𝑝=3+10−9.

2.3. When the Orbit Γ Is Defined by 𝐻(𝜑,𝑦)=𝐻(𝜑₂,0)

Proposition 3. (i) If (𝑞,𝑝) belongs to one of 𝐴₃,𝐴₄,𝐴₁₁, and 𝑙₂, then (1) have four nonlinear wave solutions 𝑢^±_𝑖=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑖²,𝐸^±_𝑖=𝜑^±_𝑖𝑒^{𝑖(𝛾𝑥−𝜔𝑡)},𝑢^±_𝑗=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑗²,𝐸^±_𝑗=𝜑^±_𝑗𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (46) where 𝜑^±_𝑖=±𝑞+Δ2Δ−𝑞𝑟𝜂𝑞+4Δ+𝑞−2Δcos𝜂𝜉cos2𝜉,𝜑^±_𝑗=±𝑞+Δ2Δ−𝑞𝑟𝜂𝑞+4Δ−𝑞−2Δcos𝜂𝜉sin2𝜉,𝜂=1𝑟4𝑝𝑟−𝑞𝑞+Δ. (47)(48)(49)
When (𝑞,𝑝)∈𝐴₃,𝐴₄, or 𝑙₂, then 𝜑^±_𝑖 and 𝜑^±_𝑗 represent periodic blow-up wave solutions (refer to Figure 13(a)). When (𝑞,𝑝)∈𝐴₁₁, then 𝜑^±_𝑖 and 𝜑^±_𝑗 represent periodic wave solutions (refer to Figure 14(a)).
(ii) If (𝑞,𝑝)∈𝑙₁ or 𝑙₇, then (1) have two fractional wave solutions 𝑢^±_𝑘=𝛽𝑐²−𝑐²_𝑠𝜑^±_𝑘²,𝐸^±_𝑘=𝜑^±_𝑘𝑒^{𝑖(𝛾𝑥−𝜔𝑡)}, (50) where 𝜑^±_𝑘=±−𝑞2𝑟𝑞𝜉𝑞²𝜉².−12𝑟 (51)
When (𝑞,𝑝)∈𝑙₁,  𝜑^±_𝑘 represent two 1-blow-up waves (refer to Figure 13(d)). When (𝑞,𝑝)∈𝑙₇,  𝜑^±_𝑘 represent two tall-kink waves (refer to Figure 14(d)).
(iii) If (𝑞,𝑝)∈𝐴₄ and 𝑝→𝑞²/4𝑟−0, then the periodic blow-up waves 𝜑^±_𝑖 tend to two trivial solutions 𝜑=±−(𝑞/2𝑟), while 𝜑^±_𝑗 tend to the 1-blow-up waves 𝜑^±_𝑘, and the varying process is displayed in Figure 13. If (𝑞,𝑝)∈𝐴₁₁ and 𝑝→𝑞²/4𝑟+0, then the periodic waves 𝜑^±_𝑖 tend to two trivial solutions 𝜑=±−(𝑞/2𝑟), while 𝜑^±_𝑗 tend to the tall-kink waves 𝜑^±_𝑘, and for the varying process, see Figure 14.

(a)

(b)

(c)

(d)

Figure 13 

(The 1-blow-up waves are bifurcated from the periodic blow-up waves.) The varying process for the figures of 𝜑±𝑗 when (𝑞,𝑝)∈𝐴4 and 𝑝→𝑞2/4𝑟−0, where 𝑟=1,𝑞=−4,and𝑙1∶𝑝=𝑞2/4𝑟=4 and (a) 𝑝=4−10−1, (b) 𝑝=4−10−2, (c) 𝑝=4−10−4, and (d) 𝑝=4−10−6.

(a)

(b)

(c)

(d)

Figure 14 

(The two tall-kink waves are bifurcated form two periodic waves.) The varying process for the figures of 𝜑±𝑗 when (𝑞,𝑝)∈𝐴11 and 𝑝→𝑞2/4𝑟+0, where 𝑟=−1,𝑞=4,and𝑙7∶𝑝=𝑞2/4𝑟=−4 and (a) 𝑝=−3.5, (b) 𝑝=−4+10−1, (c) 𝑝=−4+10−3, and (d) 𝑝=−4+10−5.

3. The Derivations of Main Results

To derive our results, substituting (23) and 𝑢(𝑥,𝑡)=𝑢(𝜉) with 𝜉=𝑥−𝑐𝑡 into (1), it follows that 𝑐²−𝑐²_𝑠𝑢^′′=𝛽𝜑^2′′,𝛼𝜑^′′+𝑖2𝛼𝛾−𝑐𝜑^′+𝑤−𝛼𝛾²𝜑−𝛿₁𝑢𝜑+𝛿₂𝜑³+𝛿₃𝜑⁵=0. (52)

Integrating the first equation of (52) twice with respect to 𝜉 and taking the integral constants to zero, we get (22). Substituting (22) into the second equation of (52) and letting 𝑐=2𝛼𝛾, it follows that 𝜑^′′=𝑝𝜑+𝑞𝜑³+𝑟𝜑⁵. (53)

Form (53) we obtain the planar system 𝑑𝜑𝑑𝜉=𝑦,𝑑𝑦𝑑𝜉=𝑝𝜑+𝑞𝜑³+𝑟𝜑⁵, (54) with the first integral 𝐻𝜑,𝑦=𝑦²−𝑝𝜑²+𝑞2𝜑⁴+𝑟3𝜑⁶=ℎ. (55)

According to the qualitative theory, we obtain the bifurcation phase portraits of system (54) as Figures 15 and 16. Using the information given by Figures 15 and 16, we give derivations to Propositions 1, 2, and 3, respectively.

Figure 15 

The bifurcation phase portraits of system (54) for 𝑟>0.

Figure 16 

The bifurcation phase portraits of system (54) for 𝑟<0.

3.1. The Derivations to Proposition 1

When the orbit Γ is defined by 𝐻(𝜑,𝑦)=𝐻(0,0), from (55) we obtain 𝑦=±𝑝𝜑²+𝑞2𝜑⁴+𝑟3𝜑⁶. (56)

Substituting (56) into the first equation of (54) and integrating it, we have ^𝜑_𝑙𝑑𝑠𝑝𝑠²+𝑞/2𝑠⁴+(𝑟/3)𝑠⁶=𝜉, (57) where 𝑙 is an arbitrary constant or ±∞.

For 𝑝=0, letting 𝑙=−(3𝑞/2𝑟) or ±∞ and completing the above integral, it follows that 𝜑=±6𝑞3𝑞²𝜉²,−4𝑟 (58) which yields 𝜑^±_𝑎 as (29).

For 𝑝<0, completing (57) and solving for 𝜑, it follows that 𝜑=±−12𝑝3𝑞−9𝑞²−48𝑝𝑟sin𝜆+2√,−𝑝𝜉 (59) where 𝜆=𝜆(𝑙) is an arbitrary real constant. Let 𝜆=±(𝜋/2), respectively, and we obtain the solutions 𝜑^±_𝑎0 and 𝜑^±_𝑏 as (3) and (31).

When (𝑞,𝑝)∈𝐴₁, that is, 𝑟>0,𝑞>0,and𝑝<0, let 𝑓𝑝=3𝑞−9𝑞²−48𝑝𝑟cos2√,𝑔𝑝−𝑝𝜉=3𝑞+9𝑞²−48𝑝𝑟cos2√−𝑝𝜉. (60)

Thus, we have lim_𝑝→0−0𝜑^±_𝑎0=lim_𝑝→0−0±−12𝑝𝑓𝑝=±lim_𝑝→0−0−12𝑓^′𝑝=±lim_𝑝→0−029𝑞²×2√−48𝑝𝑟sin−𝑝𝜉/2√2√−𝑝𝜉−4𝑟cos−𝑝𝜉^−11/2=±6𝑞3𝑞²𝜉²−4𝑟=𝜑^±_𝑎(see(29)),lim_𝑝→0−0𝜑^±_𝑏=lim_𝑝→0−0±−12𝑝𝑔𝑝=±lim_𝑝→0−0−12𝑝𝑔𝑝=0. (61)

When (𝑞,𝑝)∈𝐴₂, that is, 𝑟>0,𝑞<0,and𝑝<0, similarly we get lim_𝑝→0−0𝜑^±_𝑏=𝜑^±_𝑎. (62)

When (𝑞,𝑝)∈𝐴₁₂, that is, 𝑟<0,𝑞>0,and𝑝<0, if 𝑝→3𝑞²/16𝑟+0, then 𝑓(𝑝) and 𝑔(𝑝)→0.

Thus, lim_𝑝→0−0𝜑^±_𝑎0or𝜑^±_𝑏=±−3𝑞.4𝑟 (63)

For 𝑝>0, completing (57) and solving for 𝜑, it follows that 𝜑=±4𝜆𝑝𝑒^2√𝑝𝜉𝜆²𝑒^4√𝑝𝜉+𝑞²/4−4𝑝𝑟/3−𝜆𝑞𝑒^2√𝑝𝜉, (64) where 𝜆=𝜆(𝑙) is an arbitrary real constant.

Note that if 𝜑(𝜉) is a solution of (53), so is 𝜑(−𝜉). Thus, from (64) we obtain the solutions 𝜑^±_𝑐 and 𝜑^±_𝑑 as in (33).

For the case of 𝜆>0. When (𝑞,𝑝)∈𝑙₂, that is, 𝑟>0,𝑞<0 and 𝑟=3𝑞²/16𝑝. In (33) letting 𝑟=3𝑞²/16𝑝, we get (34). Furthermore, in (34) letting 𝜆=−𝑞>0, it follows that 𝜑^±_𝑐1=±4𝑝−𝑞𝑒^2√𝑝𝜉+1=±4𝑝𝑒^{−√𝑝𝜉}−𝑞𝑒^√𝑝𝜉+𝑒^{−√𝑝𝜉}=±2𝑝𝑒−𝑞1−^√𝑝𝜉−𝑒^{−√𝑝𝜉}𝑒^√𝑝𝜉+𝑒^{−√𝑝𝜉}=±2𝑝√−𝑞1−tanh𝑝𝜉=±2𝑝𝑞−1+tanh√𝑝𝜉=𝜑^±_𝑏0((4,𝜑see))_±𝑑1=±4𝑝−𝑞𝑒^{√−2𝑝𝜉}+1=±4𝑝𝑒^√𝑝𝜉−𝑞𝑒^√𝑝𝜉+𝑒^{−√𝑝𝜉}=±2𝑝𝑒−𝑞1+^√𝑝𝜉−𝑒^{−√𝑝𝜉}𝑒^√𝑝𝜉+𝑒^{−√𝑝𝜉}=±2𝑝√−𝑞1+tanh𝑝𝜉=±2𝑝𝑞−1−tanh√𝑝𝜉=𝜑^±_𝑐0(see(5)). (65)

When 𝜆=𝜆₀/12, by (33) it follows that 𝜑^±_𝑐=𝜑^±_𝑑=±24𝑝−6𝑞+9𝑞²−48𝑝𝑟𝑒^√𝑝𝜉+𝑒^{−√𝑝𝜉}=±12𝑝−3𝑞+9𝑞²−48𝑝𝑟cosh2√𝑝𝜉=𝜑^±_𝑑0(see(6)). (66)

For the case of 𝜆<0, similarly we can obtain (7), (35), and (36), and here we omit the process. Hereto, we have completed the derivations for Proposition 1.

3.2. The Derivations to Proposition 2

When the orbit Γ is defined by 𝐻(𝜑,𝑦)=𝐻(𝜑₁,0), from (55) we obtain 𝑦=±𝑝𝜑²+𝑞2𝜑⁴+𝑟3𝜑⁶+𝐻𝜑₁,0=±𝑟3𝜑²₁−𝜑²²𝜑²+𝜇₀, (67) where 𝜑₁ and 𝜇₀ are given in (25) and (40). Substituting (67) into the first equation of (54) and integrating it, we have ^𝜑_𝑚𝑑𝑠(𝑟/3)𝜑²₁−𝑠²²𝑠²+𝜇₀=𝜉, (68) where 𝑚 is an arbitrary constant or ±∞.

Completing the above integral and solving for 𝜑, it follows that 𝜑=±𝜑²₁−4𝑎₁𝜇𝑒^𝜃𝜉𝜇²𝑒^2𝜃𝜉−2𝜇𝑏₁𝑒^𝜃𝜉+𝑏²₁−4𝑎₁, (69) where 𝜃 is given in (39), 𝜇=𝜇(𝑚) is an arbitrary constant, and 𝑎₁=3ΔΔ−𝑞4𝑟²,𝑏₁=𝑞−4Δ.2𝑟 (70)

Similar to the derivations for 𝜑^±_𝑐 and 𝜑^±_𝑑, we get 𝜑^±_𝑒 and 𝜑^±_𝑓 (see (38)) from (69).

For the case of 𝜇>0, when (𝑞,𝑝)∈𝑙₂, that is, 𝑟>0,𝑞<0,and𝑟=3𝑞²/16𝑝, then 𝜃=2√𝑝 and 𝑞+2Δ=0. From (38) and (41), it is easy to check that 𝜑^±_𝑒 and 𝜑^±_𝑓 become 𝜑^∓_𝑒1 and 𝜑^∓_𝑓1 (see (42)), respectively. Furthermore, in (42) letting 𝜇=−(16𝑝/𝑞), it follows that 𝜑^∓_𝑒1=∓−4𝑝𝑞1+𝑒^{√−2𝑝𝜉}=∓2𝑝𝑞−1−tanh√𝑝𝜉=𝜑^∓_𝑐0((5,𝜑see))_∓𝑓1=∓−4𝑝𝑞1+𝑒^2√𝑝𝜉=∓2𝑝𝑞−1+tanh√𝑝𝜉=𝜑^∓_𝑏0(see(4)). (71)

If (𝑞,𝑝)∈𝐴₃ and 𝜇=|𝜇₀|, that is, 𝑟>0,𝜇=𝜇₀>0, and 2Δ+𝑞>0, we have𝜑^±_𝑒=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^𝜃𝜉+4𝜇²𝑟²𝑒^2𝜃𝜉=±Δ−𝑞/2𝑟2Δ+𝑞1−𝑒^𝜃𝜉𝑞+2Δ²+2𝑞+2Δ4Δ−𝑞𝑒^𝜃𝜉+𝑞+2Δ²𝑒^2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞1−𝑒^𝜃𝜉𝑞+2Δ+24Δ−𝑞𝑒^𝜃𝜉+𝑞+2Δ𝑒^2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞𝑒^{−𝜃𝜉/2}−𝑒^𝜃𝜉/224Δ−𝑞+𝑞+2Δ𝑒^−𝜃𝜉+𝑒^𝜃𝜉=∓(1/𝑟)Δ−𝑞2Δ+𝑞sinh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉)=𝜑^∓_𝑔((,𝜑see44))_±𝑓=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^−𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^−𝜃𝜉+4𝜇²𝑟²𝑒^−2𝜃𝜉=±Δ−𝑞/2𝑟2Δ+𝑞1−𝑒^−𝜃𝜉𝑞+2Δ²+2𝑞+2Δ4Δ−𝑞𝑒^−𝜃𝜉+𝑞+2Δ²𝑒^−2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞1−𝑒^−𝜃𝜉𝑞+2Δ+24Δ−𝑞𝑒^−𝜃𝜉+𝑞+2Δ𝑒^−2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞𝑒^𝜃𝜉/2−𝑒^{−𝜃𝜉/2}24Δ−𝑞+𝑞+2Δ𝑒^−𝜃𝜉+𝑒^𝜃𝜉=±(1/𝑟)Δ−𝑞2Δ+𝑞sinh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉)=𝜑^±_𝑔(see(44)). (72)

If (𝑞,𝑝)∈𝐴₄ and 𝜇=|𝜇₀|, that is, 𝑟>0,𝜇=−𝜇₀>0, and 2Δ+𝑞<0, we have 𝜑^±_𝑒=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^𝜃𝜉+4𝜇²𝑟²𝑒^2𝜃𝜉=±Δ−𝑞/2𝑟2Δ+𝑞1+𝑒^𝜃𝜉𝑞+2Δ²+2𝑞+2Δ4Δ−𝑞𝑒^𝜃𝜉+𝑞+2Δ²𝑒^2𝜃𝜉=∓(1/2𝑟)Δ−𝑞2Δ+𝑞1+𝑒^𝜃𝜉𝑞+2Δ+24Δ−𝑞𝑒^𝜃𝜉+𝑞+2Δ𝑒^2𝜃𝜉=∓(1/2𝑟)Δ−𝑞2Δ+𝑞𝑒^{−𝜃𝜉/2}+𝑒^𝜃𝜉/224Δ−𝑞+𝑞+2Δ𝑒^−𝜃𝜉+𝑒^𝜃𝜉=∓(1/𝑟)Δ−𝑞2Δ+𝑞cosh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉)=𝜑^∓_ℎ((,𝜑see45))_±𝑓=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^−𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^−𝜃𝜉+4𝜇²𝑟²𝑒^−2𝜃𝜉=±Δ−𝑞/2𝑟2Δ+𝑞1+𝑒^−𝜃𝜉𝑞+2Δ²+2𝑞+2Δ4Δ−𝑞𝑒^−𝜃𝜉+𝑞+2Δ²𝑒^−2𝜃𝜉=∓(1/2𝑟)Δ−𝑞2Δ+𝑞1+𝑒^−𝜃𝜉𝑞+2Δ+24Δ−𝑞𝑒^−𝜃𝜉+𝑞+2Δ𝑒^−2𝜃𝜉=∓(1/2𝑟)Δ−𝑞2Δ+𝑞𝑒^{−𝜃𝜉/2}+𝑒^𝜃𝜉/224Δ−𝑞+𝑞+2Δ𝑒^−𝜃𝜉+𝑒^𝜃𝜉=∓(1/𝑟)Δ−𝑞2Δ+𝑞cosh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉)=𝜑^∓_ℎ(see(45)). (73)

If (𝑞,𝑝)∈𝐴₁₁,𝐴₁₂,𝑙₆ and 𝜇=|𝜇₀|, that is, 𝑟<0,𝜇=−𝜇₀>0, and 2Δ+𝑞>0, we have𝜑^±_𝑒=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^𝜃𝜉+4𝜇²𝑟²𝑒^2𝜃𝜉=±Δ−𝑞/2𝑟2Δ+𝑞1+𝑒^𝜃𝜉𝑞+2Δ²+2𝑞+2Δ4Δ−𝑞𝑒^𝜃𝜉+𝑞+2Δ²𝑒^2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞1+𝑒^𝜃𝜉𝑞+2Δ+24Δ−𝑞𝑒^𝜃𝜉+𝑞+2Δ𝑒^2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞𝑒^{−𝜃𝜉/2}+𝑒^𝜃𝜉/224Δ−𝑞+𝑞+2Δ𝑒^−𝜃𝜉+𝑒^𝜃𝜉=±(1/𝑟)Δ−𝑞2Δ+𝑞cosh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉)=𝜑^±_ℎ((,𝜑see45))_±𝑓=±Δ−𝑞/2𝑟2Δ+𝑞−2𝜇𝑟𝑒^−𝜃𝜉2Δ+𝑞²+4𝜇𝑟4Δ−𝑞𝑒^−𝜃𝜉+4𝜇²𝑟²𝑒^−2𝜃𝜉=±Δ−𝑞/2𝑟2Δ+𝑞1+𝑒^−𝜃𝜉𝑞+2Δ²+2𝑞+2Δ4Δ−𝑞𝑒^−𝜃𝜉+𝑞+2Δ²𝑒^−2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞1+𝑒^−𝜃𝜉𝑞+2Δ+24Δ−𝑞𝑒^−𝜃𝜉+𝑞+2Δ𝑒^−2𝜃𝜉=±(1/2𝑟)Δ−𝑞2Δ+𝑞𝑒^{−𝜃𝜉/2}+𝑒^𝜃𝜉/224Δ−𝑞+𝑞+2Δ𝑒^−𝜃𝜉+𝑒^𝜃𝜉=±(1/𝑟)Δ−𝑞2Δ+𝑞cosh((𝜃/2)𝜉)−𝑞+4Δ+𝑞+2Δcosh(𝜃𝜉)=𝜑^±_ℎ(see(45)). (74)

For the case of 𝜇<0, similarly we can obtain the relations of the solutions 𝜑^±_𝑒,𝜑^±_𝑓,𝜑^±_𝑔, and 𝜑^±_ℎ, and here we omit the process. Hereto, we have completed the derivations for Proposition 2.

3.3. The Derivations to Proposition 3

When the orbit Γ is defined by 𝐻(𝜑,𝑦)=𝐻(𝜑₂,0), firstly, if (𝑞,𝑝) belongs to one of 𝐴₃,𝐴₄,𝐴₁₁, and 𝑙₂, then from (55) we obtain 𝑦=±𝑝𝜑²+𝑞2𝜑⁴+𝑟3𝜑⁶+𝐻𝜑₂,0=±𝑟3𝜑²−𝜑²₂²𝜑²+𝛿₀, (75) where 𝜑₂ is given in (16) and 𝛿₀=(𝑞−2Δ)/2𝑟. Substituting (75) into the first equation of (54) and integrating it, we have ^𝜑_𝑛𝑑𝑠(𝑟/3)𝑠²−𝜑²₂²𝑠²+𝛿₀=𝜉, (76) where 𝑛 is an arbitrary constant or ±∞.

Completing the above integral and solving for 𝜑, it follows that 𝜑=±𝜑²₂−2𝑎₂𝑏₂+𝛿₀,sin𝛿+𝜂𝜉 (77) where 𝜂 is given in (49),  𝛿=𝛿(𝑛) is an arbitrary constant, and 𝑎₂=3ΔΔ+𝑞4𝑟²,𝑏₂=−𝑞+4Δ.2𝑟 (78)

In (77) letting 𝛿=±(𝜋/2), respectively, we obtain the solutions 𝜑^±_𝑖 and 𝜑^±_𝑗 as (47) and (48).

Secondly, if (𝑞,𝑝)∈𝑙₁ or 𝑙₇, that is, Δ=0 and 𝜑²₁=−𝛿₀=−(𝑞/2𝑟), thus from (75), we obtain 𝑦=±𝑟3𝜑²+𝑞2𝑟³. (79)

Similarly, we have 𝜑=±−𝑞2𝑟𝑞𝜉𝑞²𝜉²,−12𝑟 (80) which yields 𝜑^±_𝑘 as in (51).

When 𝑝→𝑞²/4𝑟, it follows that Δ=𝑞²−4𝑝𝑟→0,𝜂=1𝑟4𝑝𝑟−𝑞𝑞+Δ=−ΔΔ+𝑞𝑟𝜂→0,cos𝜂𝜉=1−²𝜉²2+𝜂⁴𝜉⁴Δ4!+⋯=1+Δ+𝑞𝜉2𝑟²+𝑜Δ². (81)

Thus, we have lim_{𝑝→(𝑞2/4𝑟)}𝜑^±_𝑖=±lim_{𝑝→𝑞2/4𝑟}𝑞+Δ2Δ−𝑞𝑟𝜂𝑞+4Δ+𝑞−2Δcos𝜂𝜉cos2𝜉=±−𝑞²2𝑟𝑞=±−𝑞,2𝑟lim_{𝑝→𝑞2/4𝑟}𝜑^±_𝑗=lim_{𝑝→𝑞2/4𝑟}±𝑞+Δ2Δ−𝑞𝑟𝜂𝑞+4Δ−𝑞−2Δcos𝜂𝜉sin2𝜉=lim_{𝑝→𝑞2/4𝑟}±𝑞+Δ2Δ−𝑞1−cos𝜂𝜉2𝑟𝑞+4Δ−𝑞−2Δcos𝜂𝜉=lim_{𝑝→𝑞2/4𝑟}±−Δ𝑞+Δ2Δ−𝑞Δ+𝑞𝜉2𝑟²+𝑜(Δ)²××Δ2𝑟𝑞+4Δ−𝑞−2Δ1+Δ+𝑞𝜉2𝑟²+𝑜Δ^2−11/2=lim_{𝑝→𝑞2/4𝑟}±−Δ+𝑞²2Δ−𝑞𝜉2𝑟²(Δ×Δ+𝑜)2𝑟6−𝑞−2Δ1+Δ+𝑞𝜉2𝑟²+𝑜(Δ)^−11/2=±𝑞³/2𝑟𝜉²2𝑟6−𝑞²/2𝑟𝜉²=±−𝑞2𝑟𝑞𝜉𝑞²𝜉²−12𝑟=𝜑^±_𝑘(see(51)). (82)

Hereto, we have completed the derivations for our main results.

4. Conclusions

In this paper, we have investigated the explicit expressions of the nonlinear waves and bifurcation phenomena in (1). Firstly, we obtained three types of explicit nonlinear wave solutions. The first type is the fractional expressions 𝜑^±_𝑎 and 𝜑^±_𝑘 (see (29) and (51)). The second type is the trigonometric expressions 𝜑^±_𝑖 and 𝜑^±_𝑗 (see (47) and (48)). The third type is the exp-function expressions 𝜑^±_𝑐, 𝜑^±_𝑑, 𝜑^±_𝑒, and 𝜑^±_𝑓 (see (33) and (38)).

Secondly, we revealed five kinds of interesting bifurcation phenomena in (1). The first phenomena is that the 1-blow-up waves can be bifurcated from the periodic-blow-up (see Figure 2 or Figure 13) and 2-blow-up waves (see Figure 8). The second phenomena is that the 2-blow-up waves can be bifurcated from the periodic-blow-up waves (see Figure 3). The third kind is that the symmetric solitary waves can be bifurcated from the symmetric periodic waves (see Figure 4). The fourth kind is that the low-kink waves can be bifurcated from the symmetric solitary waves (see Figure 6), the 1-blow-up waves (see Figure 7), the tall-kink waves (see Figure 9), and the antisymmetric solitary waves (see Figure 10). The fifth kind is that the tall-kink waves can be bifurcated from the symmetric periodic waves (see Figure 14).

Thirdly, we showed that some previous results are our special cases. For instants, 𝜑^±_𝑏0, 𝜑^±_𝑐0, and 𝜑^±_𝑒0 are included in 𝜑^±_𝑐, 𝜑^±_𝑑, 𝜑^±_𝑒, and 𝜑^±_𝑓 (see (4), (5), (7), (33), and (38)). 𝜑^±_𝑑0 are included in 𝜑^±_𝑐 and 𝜑^±_𝑑 (see (6) and (33)).

Finally, we employed the software Mathematica to check the correctness of these solutions. For example, the commands for 𝑢_𝑎 and 𝐸_𝑎 are as follows:  𝑤=𝛼𝛾²,  𝑝=(𝛼𝛾²−𝑤)/𝛼,  𝑞=𝛽𝛿₁/𝛼(𝑐²−𝑐²_𝑠)−𝛿₂/𝛼,  𝑟=−𝛿₃/𝛼,  𝑐=2𝛼𝛾,  𝜉=𝑥−𝑐𝑡,  𝜑_𝑎=6𝑞/(3𝑞²𝜉²−4𝑟),  𝑢_𝑎=(𝛽/(𝑐²−𝑐²_𝑠))𝜑²_𝑎,  𝐸_𝑎=𝜑_𝑎Exp[𝑖(𝛾𝑥−𝑤𝑡)],  Simplify[D[𝑢_𝑎,{𝑡,2}]−𝑐²_𝑠D[𝑢_𝑎,{𝑥,2}]−𝛽D[𝜑²_𝑎,{𝑥,2}]]  Simplify[𝑖D[𝐸_𝑎,𝑡]+𝛼D[𝐸_𝑎,{𝑥,2}]−𝛿₁𝑢_𝑎𝐸_𝑎+𝛿₂𝜑²_𝑎𝐸_𝑎+𝛿₃𝜑⁴_𝑎𝐸_𝑎] 0 0

Acknowledgment

This paper is supported by the National Natural Science Foundation (no. 11171115), Natural Science Foundation of Guangdong Province (no. S2012040007959) and the Fundamental Research Funds for the Central Universities (no. 2012ZM0057).