New Types of Nonlinear Waves and Bifurcation Phenomena in Schamel-Korteweg-de Vries Equation

Abstract

We study the nonlinear waves described by Schamel-Korteweg-de Vries equation 𝑢_𝑡+(𝑎𝑢^1/2+𝑏𝑢)𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0. Two new types of nonlinear waves called compacton-like waves and kink-like waves are displayed. Furthermore, two kinds of new bifurcation phenomena are revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which are the bell-shape solitary waves, the blow-up waves, the valley-shape solitary waves, the kink-like waves, and the compacton-like waves. The second phenomenon is that the periodic-blow-up wave can be bifurcated from the smooth periodic wave.

1. Introduction and Preliminary

Consider the following Schamel-Korteweg-de Vries (S-KdV) equation [1, 2]: 𝑢_𝑡+𝑎𝑢^1/2+𝑏𝑢𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0, (1) where 𝑎, 𝑏, and 𝛿 are constants.

Equation (1) arises in plasma physics in the study of ion acoustic solitons when electron trapping is present and also it governs the electrostatic potential for a certain electron distribution in velocity space. Tagare and Chakraborti [1] showed that (1) has solitary wave solution by applying direct integral method. Lee and Sakthivel [3] gave some exact traveling wave solutions of (1) by using exp-function method.

When 𝑏=0, (1) becomes the Schamel equation [4]: 𝑢_𝑡+𝑎𝑢^1/2𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0. (2)

When 𝑎=0, (1) becomes a well-known KdV equation 𝑢_𝑡+𝑏𝑢𝑢_𝑥+𝛿𝑢_𝑥𝑥𝑥=0, (3) which has been studied successively by many authors (e.g., [5–8]).

The concept of compacton, soliton with compact support or strict localization of solitary waves, appeared in the work of Rosenau and Hyman [9] where a genuinely nonlinear dispersive equation 𝐾(𝑛;𝑛) is defined by 𝑢_𝑡+𝑎𝑢^𝑛_𝑥+𝑢^𝑛_𝑥𝑥𝑥=0. (4) They found certain solitary wave solutions which vanish identically outside a finite core region. These solutions are called compactons.

Several studies have been conducted in [10–20]. The aim of these studies was to examine if other nonlinear dispersive equations may generate compactons structures.

In order to investigate the nonlinear waves of (1), letting 𝑐>0 be wave speed and substituting 𝑢=𝜑(𝜉) with 𝜉=𝑥−𝑐𝑡 into (1), it follows that −𝑐𝜑^′+𝑎𝜑^1/2𝜑^′+𝑏𝜑𝜑^′+𝛿𝜑^′′′=0. (5) Integrating (5), we get 2−𝑐𝜑+3𝑎𝜑^3/2+𝑏2𝜑²+𝛿𝜑^′′=0. (6) Setting 𝜑^′=𝑦 yields the following planar system: 𝜑^′=𝑦,𝑦^′=𝑐𝛿𝜑−2𝑎𝜑3𝛿^3/2−𝑏𝜑2𝛿². (7) Obviously, system (7) is a Hamiltonian system with Hamiltonian function 𝐻𝜑,𝑦=𝑦²−𝑐𝛿𝜑²+8𝑎𝜑15𝛿^5/2+𝑏𝜑3𝛿³. (8) If one puts 𝑓𝜑=𝑐𝛿𝜑−2𝑎𝜑3𝛿^3/2−𝑏𝜑2𝛿²,Δ=4𝑎²+18𝑏𝑐, (9) then one can see the following facts.

When Δ>0, 𝑓(𝜑) has three zero points 𝜑₀, 𝜑₁, and 𝜑₂ which possess expressions 𝜑₀=0,𝜑₁=−2𝑎−√Δ3𝑏²,𝜑₂=−2𝑎+√Δ3𝑏². (10) When Δ=0, 𝑓(𝜑) has two zero points 𝜑₀ and 𝜑^∗ which are denoted by 𝜑₀=0,𝜑^∗=4𝑎²9𝑏². (11) When Δ<0, 𝑓(𝜑) has one zero point 𝜑₀=0.

Letting (𝜑,0) be one of the singular points of system (7), then the characteristic values at (𝜑,0) are 𝜆=±𝑓^′𝜑. (12)

From the qualitative theory of dynamical systems, we get the following conclusions:(1)if 𝑓^′(𝜑)>0, then (𝜑,0) is a saddle point,(2)if 𝑓^′(𝜑)<0, then (𝜑,0) is a center point,(3)if 𝑓^′(𝜑)=0, then (𝜑,0) is a degenerate saddle point.

On 𝑎−𝑏 parametric plane, let 𝑙₁, 𝑙₂, and 𝑙₃, respectively, represent the following three curves: 𝑙₁𝑙∶𝑏=0,₂∶𝑏=−16𝑎²,𝑙75𝑐₃∶𝑏=−2𝑎².9𝑐 (13) Let 𝐴_𝑖  (𝑖=1,2,…,6) represent the regions surrounded by 𝑙₁, 𝑙₂, 𝑙₃, and the coordinate axes (see Figures 1 and 2).

Figure 1 

The bifurcation phase portraits of system (7) when 𝛿>0.

Figure 2 

The bifurcation phase portraits of system (7) when 𝛿<0.

According to the previous analysis, we obtain the bifurcation phase portraits of system (7) as in Figures 1 and 2.

In this paper, we study the nonlinear waves and their bifurcations in (1) by using the bifurcation method of dynamical systems [21–23]. We point out that there are two new types of nonlinear waves, kink-like waves and compacton-like waves [24–33]. Furthermore, we reveal two kinds of new bifurcation phenomena which are introduced in the abstract.

This paper is organized as follows. In Section 2, we display the two new types of nonlinear waves. We show the two kinds of new bifurcation phenomena in Sections 3 and 4. A brief conclusion is given in Section 5.

2. Two New Types of Nonlinear Waves

In this section, we display two new types of nonlinear waves defined by (1).

2.1. Kink-Like Waves

Proposition 1. (1) When the parameters satisfy 𝛿>0, 𝑎>0, and −16𝑎²/75𝑐<𝑏<0, (1) has a kink-like wave solution 𝑢⁺_𝑎=𝜑(𝜉) and an antikink-like wave solution 𝑢⁻_𝑎=𝜑(𝜉), respectively, which are hidden in the following equations: ^𝜑_𝜑1/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₁=𝜉,𝜉∈−𝜉⁰₁,∞_𝜑𝜑1/2,𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₁=−𝜉,𝜉∈−∞,𝜉⁰₁, (14)(15) where 𝜉⁰₁=^𝜑1₀^/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₁ (16) and ℎ₁=𝐻(𝜑₁,0).
(2)   When the parameters satisfy one of the following Cases.
Case  1.  𝛿<0, 𝑎>0, and −2𝑎²/9𝑐<𝑏<0,
Case  2.  𝛿<0, and 𝑏>0,
Case  3.  𝛿<0, 𝑎>0, and 𝑏=0,
Case  4.  𝛿<0, 𝑎>0, and 𝑏=−2𝑎²/9𝑐.
Equation (1) has a kink-like wave solution 𝑢⁺_𝑏=𝜑(𝜉) and an antikink-like wave solution 𝑢⁻_𝑏=𝜑(𝜉), respectively, which are hidden in the following equations: ^𝜑_𝜑2/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₂=𝜉,𝜉∈−𝜉⁰₂,∞_𝜑𝜑2/2,𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₂=−𝜉,𝜉∈−∞,𝜉⁰₂, (17) where 𝜉⁰₂=^𝜑2₀^/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₂ (18) and ℎ₂=𝐻(𝜑₂,0).

Proof. (1) Under the condition 𝛿>0, 𝑎>0, and −16𝑎²/75𝑐<𝑏<0, (𝜑₁,0) is a saddle point and on its left side there are two orbits Γ^±₁ connecting with it (see Figure 3(a₁)).
In (8), letting ℎ₁=𝐻(𝜑₁,0), it follows that Γ^±₁∶𝑦=±𝑐𝛿𝜑²−8𝑎𝜑15𝛿^5/2−𝑏𝜑3𝛿³+ℎ₁0<𝜑<𝜑₁. (19)
On Γ^±₁ suppose 𝜑(0)=𝜑₁/2. Substituting (19) into (7) and integrating them along Γ⁺₁ and Γ⁻₁, respectively, we get (14)–(16).
(2) Under one of Cases  1–4, (𝜑₂,0) is a saddle point and on its left side there are two orbits Γ^±₂ connecting with it (see Figures 3(a₂)–3(a₄)).
In (8), letting ℎ₂=𝐻(𝜑₂,0), it follows that Γ^±₂∶𝑦=±𝑐𝛿𝜑²−8𝑎𝜑15𝛿^5/2−𝑏𝜑3𝛿³+ℎ₂0<𝜑<𝜑₂. (20)
Similar to the proof of (1), we get the results of (2).

Figure 3 

The graphs of orbits Γ±1 and Γ±2 when (a1) 𝛿>0 and (𝑎,𝑏)∈𝐴2, (a2) 𝛿<0 and (𝑎,𝑏)∈𝐴2 or 𝐴3 or 𝑙2, (a3) 𝛿<0 and (𝑎,𝑏)∈𝐴1 or 𝐴6 or 𝑙1(𝑎>0), and (a4) 𝛿<0 and (𝑎,𝑏)∈𝑙3.

Next, we simulate the planar graphs of the kink-like waves for those data given in Example 2.

Example 2 (Corresponding to Proposition 1  (1)). Giving 𝑎=1/2, 𝑏=−2/45, 𝑐=1, and 𝛿=1, we get 𝜑₁=117.81152949 and 𝜑(0)=𝜑₁/2=58.90576474. Note that orbits Γ^±₁ have expressions (19). From (19) we get 𝑦⁺₁(0)=38.27939786 and 𝑦⁻₁(0)=−38.27939786. These imply that Γ⁺₁ passes point (𝜑(0),𝑦⁺₁(0)) and Γ⁻₁ passes point (𝜑(0),𝑦⁻₁(0)). Thus letting 𝜑(0)=𝜑₁/2 and 𝜑^′(0)=𝑦⁺₁(0) as the initial conditions of (6), we get the simulation of the integral curve which corresponds to Γ⁺₁ as Figure 4(a₁). Meanwhile, choosing 𝜑(0)=𝜑₁/2 and 𝜑^′(0)=𝑦⁻₁(0) as the initial conditions of (6), we get the simulation of the integral curve which corresponds to Γ⁻₁ as in Figure 4(a₂).

Figure 4 

(Corresponding to Example 2). The simulation of integral curve of (6) when 𝑎=1/2, 𝑏=−2/45, 𝑐=1, and 𝛿=1. (a1) Initial conditions 𝜑(0)=58.90576474 and 𝜑′(0)=38.27939786. (a2) Initial conditions 𝜑(0)=58.90576474 and 𝜑′(0)=−38.27939786.

2.2. Compacton-Like Waves

Proposition 3. Let 𝜑_𝑘 be an initial value when parameters and initial value satisfy one of the following Cases.
Case  1. 𝛿>0, 𝑏>0, and 𝜑_𝑘>𝛾₁,
Case  2. 𝛿>0, 𝑎>0, 𝑏=0, and 𝜑_𝑘>𝛾₁,
Case  3. 𝛿>0, 𝑎>0, −16𝑎²/75𝑐<𝑏<0, and 𝛾₁<𝜑_𝑘<𝜑₁,
Case  4. 𝛿<0, 𝑏>0, and 0<𝜑_𝑘<𝜑₂,
Case  5. 𝛿<0, 𝑎>0, 𝑏=0, and 0<𝜑_𝑘<𝜑₂,
Case  6. 𝛿<0, 𝑎<0, 𝑏≤0, and 𝜑_𝑘>0,
Case  7. 𝛿<0, 𝑎>0, 𝑏<−2𝑎²/9𝑐, and 𝜑_𝑘>0,
Case  8. 𝛿<0, 𝑎>0, 𝑏=−2𝑎²/9𝑐, and 𝜑_𝑘≠𝛾₂(=𝜑₂),
Case  9. 𝛿<0, 𝑎>0, −2𝑎²/9𝑐<𝑏<0, and 𝜑_𝑘>𝛾₂ or 0<𝜑_𝑘<𝜑₂.
Equation (1) has compacton-like wave solutions 𝑢⁺_𝑐=𝜑(𝜉) and 𝑢⁻_𝑐=𝜑(𝜉), respectively, which are hidden in the following equations: ^𝜑_𝜑𝑘/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ_𝜑𝑘=𝜉,𝜉∈−𝜉⁰₃,𝜉⁰₄,_{𝜑𝜑𝑘/2}𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ_𝜑𝑘=−𝜉,𝜉∈−𝜉⁰₄,𝜉⁰₃, (21) where 𝜉⁰₃=^𝜑𝑘₀^/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ_𝜑𝑘,𝜉⁰₄=2^𝜑𝑘_𝜑𝑘/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ_𝜑𝑘+𝜉⁰₃, (22)(23) and ℎ_𝜑𝑘=𝐻(𝜑_𝑘,0).

Proof. Under one of Cases  1–9, there is an orbit Γ_𝜑𝑘 passing point (𝜑_𝑘,0) (see Figure 5 (a₁)–5(a₆)).
In (8), letting ℎ_𝜑𝑘=𝐻(𝜑_𝑘,0), it follows that Γ_𝜑𝑘∶𝑦²=𝑐𝛿𝜑²−8𝑎𝜑15𝛿^5/2−𝑏𝜑3𝛿³+ℎ_𝜑𝑘0<𝜑<𝜑_𝑘. (24)
On Γ_𝜑𝑘 suppose 𝜑(0)=𝜑_𝑘/2. Substituting (24) into (7) and integrating it along Γ_𝜑𝑘, respectively, we obtain (21)–(23).

Figure 5 

The graphs of orbit Γ𝜑𝑘 when (a1) 𝛿>0 and (𝑎,𝑏)∈𝐴1 or 𝐴6 or 𝑙1(𝑎>0), (a2) 𝛿>0 and (𝑎,𝑏)∈𝐴2, (a3) 𝛿<0 and (𝑎,𝑏)∈𝐴1 or 𝐴6 or 𝑙1(𝑎>0), (a4) 𝛿<0 and (𝑎,𝑏)∈𝐴4 or 𝐴5 or 𝑙1(𝑎<0), (a5) 𝛿<0 and (𝑎,𝑏)∈𝑙3, and (a6) 𝛿<0 and (𝑎,𝑏)∈𝐴2 or 𝐴3 or 𝑙2.

Next, we simulate the planar graphs of the compacton-like waves for those data given in Example 4.

Example 4 (Corresponding to Proposition 3 Case  (3)). Giving 𝑎=1/2, 𝑏=−157/3000, 𝑐=1, and 𝛿=1, we get 𝜑₁=62.46353941 and 𝛾₁=43.51656288. Note that orbit Γ_𝜑𝑘 has expression (24). Letting 𝜑_𝑘=(𝜑₁+𝛾₁)/2, it follows that 𝜑(0)=26.49502557. From (24) we get 𝑦(0)=±10.49025102. Thus letting 𝜑(0)=26.49502557 and 𝜑^′(0)=10.49025102 as the initial conditions of (6), we get the simulation of the integral curve as in Figure 6(a₁). Meanwhile, choosing 𝜑(0)=26.49502557 and 𝜑^′(0)=−10.49025102 as the initial conditions of (6), we get the simulation of the integral curve as Figure 6(a₂).

Figure 6 

(Corresponding to Example 4). The simulation of integral curve of (6) when 𝑎=1, 𝑏=−157/3000, 𝑐=1, and 𝛿=1. (a1) Initial conditions 𝜑(0)=26.49502557 and 𝜑′(0)=10.49025102. (a2) Initial conditions 𝜑(0)=26.49502557 and 𝜑′(0)=−10.49025102.

3. Bifurcation of the Kink Waves

In this section, we show that the kink waves can be bifurcated from five other waves.

3.1. Bifurcation from Bell-Shape Solitary Waves

Proposition 5. For 𝑎𝑏≠0 and 𝛿>0, (1) has two nonlinear wave solutions 𝑢_𝑑=4𝛼𝜆𝜆²𝑒^−𝜏1𝜉−2𝜆𝛽+𝛽²−4𝛼𝑒^𝜏1𝜉2,𝑢_𝑒=4𝛼𝜆𝜆²𝑒^𝜏1𝜉−2𝜆𝛽+𝛽²−4𝛼𝑒^{−𝜏1𝜉2}, (25) where 𝛼=𝑝𝑞,𝛽=−𝑝+𝑞,𝜏₁=12𝑐𝛿,𝑝=√𝛾₁=−4𝑎+√16𝑎²+75𝑏𝑐,√5𝑏𝑞=−4𝑎−16𝑎²+75𝑏𝑐,5𝑏 (26)(27)(28) and 𝜆≠0 is an arbitrary real number. These solutions possess the following properties.(1)If 𝜆>0, 𝑎>0, and 𝑏=−16𝑎²/75𝑐, then 𝑢_𝑑 and 𝑢_𝑒 become𝑢^∗_𝑑=4𝛼𝜆𝑒^−𝜏1𝜉−2𝛽²,𝑢^∗_𝑒=4𝛼𝜆𝑒^𝜏1𝜉−2𝛽², (29)which represent a kink wave and an antikink wave.
In particular, when 𝛼=𝑝𝑞=−3𝑐/𝑏, 𝛽=−(𝑝+𝑞)=8𝑎/5𝑏, 𝑐=−16𝑎²/75𝑏, 𝜆=𝑎₀, and 𝜉=𝑥−𝑐𝑡, 𝑢^∗_𝑑 and 𝑢^∗_𝑒 become 𝑢₁=16𝑎²₀𝑎²4𝑎exp±(2/5)−1/3𝑏𝛿𝑎𝑥+16𝑎²/75𝑏𝑡−5𝑎₀𝑏², (30) which was given by Lee and Sakthivel [3]. This implies that 𝑢₁ is the special case of 𝑢_𝑑 or 𝑢_𝑒.(2)If 𝜆=𝛽²−4𝛼, then 𝑢_𝑑=𝑢_𝑒 and become𝑢^∗_𝑑𝑒=2𝛼𝛽²−4𝛼cosh𝜏₁𝜉−𝛽². (31)
When (𝑎,𝑏) belongs to one of the regions 𝐴₂, 𝐴₆, 𝑢^∗_𝑑𝑒 represents a hyperbolic solitary wave.
In particular, when 𝛼=𝑝𝑞=−3𝑐/𝑏, and 𝛽=−(𝑝+𝑞)=8𝑎/5𝑏, 𝑢^∗_𝑑𝑒 becomes 𝑢₂=15𝑐4𝑎+√16𝑎²+75𝑏𝑐cosh(1/2)𝑐/𝛿𝜉², (32) which was obtained by Tagare and Chakraborti [3]. This implies that 𝑢₂ is the special case of 𝑢_𝑑 or 𝑢_𝑒.(3)Under one of the following Cases.
Case  1. 𝜆>0, 𝜆≠𝛽²−4𝛼, and (𝑎,𝑏) belongs to one of the regions 𝐴₂, 𝐴₆,
Case  2. 𝜆<0, 𝜆≠−𝛽²−4𝛼, and (𝑎,𝑏)∈𝐴₁, 𝑢_𝑑≠𝑢_𝑒 and they represent two bell-shape solitary waves.
In particular, when (𝑎,𝑏)∈𝐴₂ in Case  1 and 𝑏→0−0, 𝑢_𝑑 and 𝑢_𝑒 become 𝑢^∘_𝑑=225𝑐²4𝑎²1+𝑒^{−𝜏1𝜉4},𝑢^∘_𝑒=225𝑐²4𝑎²1+𝑒^𝜏1𝜉4, (33)(34) which are the solutions of the Schamel equation.
When (𝑎,𝑏)∈𝐴₂ in Case  1 and 𝑏→−16𝑎²/75𝑐+0, the two bell-shape solitary waves 𝑢_𝑑 and 𝑢_𝑒 become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 7 and 8.

(a)

(b)

(c)

(d)

Figure 7 

(The kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph of 𝑢𝑑 when 𝑐>0, 𝛿>0, 𝑎>0, 𝜆>0 and 𝑏→−16𝑎2/75𝑐+0. Where 𝑐=1, 𝛿=1, 𝑎=1, 𝜆=50 and (a) 𝑏=−16𝑎2/75𝑐+10−1. (b) 𝑏=−16𝑎2/75𝑐+10−3. (c) 𝑏=−16𝑎2/75𝑐+10−4. (d) 𝑏=−16𝑎2/75𝑐+10−7.

(a)

(b)

(c)

(d)

Figure 8 

(The anti-kink wave is bifurcated from the bell-shape solitary wave). The varying process for the graph of 𝑢𝑒 when 𝑐>0, 𝛿>0, 𝑎>0, 𝜆>0 and 𝑏→−16𝑎2/75𝑐+0. Where 𝑐=1, 𝛿=1, 𝑎=1, 𝜆=50 and (a) 𝑏=−16𝑎2/75𝑐+10−1. (b) 𝑏=−16𝑎2/75𝑐+10−3. (c) 𝑏=−16𝑎2/75𝑐+10−4. (d) 𝑏=−16𝑎2/75𝑐+10−7.

Proof. In (8), letting ℎ₀=𝐻(0,0), it follows that 𝑦=±𝑐𝛿𝜑²−8𝑎𝜑15𝛿^5/2−𝑏𝜑3𝛿³. (35) Substituting (35) into d𝜑/d𝜉=𝑦, we have 𝑑𝜑𝑑𝜉=±𝑐𝛿𝜑²−8𝑎𝜑15𝛿^5/2−𝑏𝜑3𝛿³. (36) Let 𝜑=𝑤², (36) becomes 2𝑤d𝑤𝑑𝜉=±𝑐𝛿𝑤⁴−8𝑎𝑤15𝛿⁵−𝑏𝑤3𝛿⁶. (37) Integrating (37), we have ^𝑤_𝜈𝑑𝑠𝑠𝑐/𝛿−(8𝑎/15𝛿)𝑠−(𝑏/3𝛿)𝑠²1=±2𝜉, (38) where 𝜈 is an arbitrary constant.
Completing the previous integral and solving the equation for 𝜑, it follows that 𝜑=4𝛼𝜆𝑒^∓𝜏1𝜉𝜆²𝑒^{∓2𝜏1𝜉}−2𝜆𝛽𝑒^∓𝜏1𝜉+𝛽²−4𝛼², (39) where 𝜆=𝜆(𝜈) is an arbitrary real number. From (39) we obtain the solutions 𝑢_𝑑 and 𝑢_𝑒 as (25).
In (25) letting 𝑏=−16𝑎²/75𝑐, we get (29). From (25) and (29), we get the result (1) of Proposition 5.
When 𝜆=𝛽²−4𝛼, via (25) it follows that 𝑢_𝑑=𝑢_𝑒=4𝛼𝜆𝑒^−𝜏1𝜉+𝑒^𝜏1𝜉−2𝛽²=2𝛼𝜆cosh𝜏₁𝜉−𝛽²=𝑢^∗_𝑑𝑒 (40) (see (31)).
Thus, we get the result (2) of Proposition 5.
In (38), letting 𝜈=𝑝 (see (27)), it follows that 𝜆=𝛽𝑝+2𝛼𝑝2√=−16𝑎²+75𝑏𝑐.5𝑏 (41)
Letting 𝑏→0−0, then lim_𝑏→0−0𝛼𝜆=lim_𝑏→0−03𝑐𝑏⋅5𝑏2√16𝑎²+75𝑏𝑐=lim_𝑏→015𝑐2√16𝑎²=+75𝑏𝑐15𝑐,8𝑎lim_𝑏→0−0𝛽𝜆=lim_𝑏→0−0−8𝑎⋅5𝑏5𝑏2√16𝑎²+75𝑏𝑐=−1,lim_𝑏→0−0𝛽²−4𝛼𝜆²=lim_𝑏→0−0𝛽𝜆²𝛼−4𝜆⋅1𝜆=1. (42)(43)(44) We have lim_𝑏→0−0𝑢_𝑑=lim_𝑏→0−04𝛼𝜆𝜆²𝑒^−𝜏1𝜉−2𝜆𝛽+𝛽²−4𝛼𝑒^𝜏1𝜉2=lim_𝑏→0−04𝛼/𝜆𝑒^−𝜏1𝜉−2𝛽/𝜆+𝛽²−4𝛼/𝜆²⋅𝑒^𝜏1𝜉2=𝑢^∘_𝑑 (45) (see (33)).
Similarly, we have lim_𝑏→0−0𝑢_𝑒=𝑢^∘_𝑒 (46) (see (34)).
From (41)–(46), we get result (3) of Proposition 5.

3.2. Bifurcation from Blow-Up Waves

Proposition 6. For 𝑎𝑏≠0 and 𝛿>0, (1) has two nonlinear wave solutions as 𝑢_𝑑 and 𝑢_𝑒. These solutions possess the following properties.
(1) Under one of the following Cases.
Case  1. 𝜆>0, 𝜆≠𝛽²−4𝛼, and (𝑎,𝑏) belongs to one of the regions 𝐴₁, 𝐴₃,
Case  2. 𝜆<0, 𝜆≠−𝛽²−4𝛼, and (𝑎,𝑏) belongs to any one of the regions 𝐴₂, 𝐴₃, and 𝐴₆, 𝑢_𝑑≠𝑢_𝑒 and they represent two blow-up waves.
In particular, when (𝑎,𝑏)∈𝐴₃ in Case  1 and 𝑏→−16𝑎²/75𝑐−0, the two blow-up waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 9 and 10.

(a)

(b)

(c)

(d)

Figure 9 

(The kink wave is bifurcated from the blow-up wave). The varying process for the graph of 𝑢𝑑 when 𝑐>0, 𝛿>0, 𝑎>0, 𝜆>0, and 𝑏→−16𝑎2/75𝑐−0, where 𝑐=1, 𝛿=1, 𝑎=1, 𝜆=1, and (a) 𝑏=−16𝑎2/75𝑐−10−2. (b) 𝑏=−16𝑎2/75𝑐−10−4. (c) 𝑏=−16𝑎2/75𝑐−10−5. (d) 𝑏=−16𝑎2/75𝑐−10−9.

(a)

(b)

(c)

(d)

Figure 10 

(The antikink wave is bifurcated from the blow-up wave). The varying process for the graph of 𝑢𝑒 when 𝑐>0, 𝛿>0, 𝑎>0, 𝜆>0, and 𝑏→−16𝑎2/75𝑐−0, where 𝑐=1, 𝛿=1, 𝑎=1, 𝜆=1, and (a) 𝑏=−16𝑎2/75𝑐−10−2. (b) 𝑏=−16𝑎2/75𝑐−10−4. (c) 𝑏=−16𝑎2/75𝑐−10−5. (d) 𝑏=−16𝑎2/75𝑐−10−9.

Similar to the proof of Proposition 5, we get the results of Proposition 6.

3.3. Bifurcation from Valley-Shape Solitary Waves

Proposition 7. When the parameters satisfy 𝛿>0 and (𝑎,𝑏)∈𝐴₃, (1) has two valley-shape solitary wave solutions 𝑢_𝑓=𝜑(𝜉) and 𝑢_𝑔=𝜑(𝜉), respectively, which are hidden in the following equations: ^𝜑_𝜑2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₁=𝜉0<𝜑<𝜑₁,_𝜑𝜑2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ₁=−𝜉0<𝜑<𝜑₁. (47)(48) In particular, when 𝑏→−16𝑎²/75𝑐−0, the two valley-shape solitary waves become a kink wave and an antikink wave with the expressions (29). For the varying process, see Figures 11 and 12.

(a)

(b)

(c)

(d)

Figure 11 

(The kink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph of 𝑢𝑓 when 𝑐>0, 𝛿>0, 𝑎>0, and 𝑏→−16𝑎2/75𝑐−0, where 𝑐=1, 𝛿=1, 𝑎=1/2, and (a) 𝑏=−16𝑎2/75𝑐−10−3. (b) 𝑏=−16𝑎2/75𝑐−10−7. (c) 𝑏=−16𝑎2/75𝑐−10−8. (d) 𝑏=−16𝑎2/75𝑐−10−10.

(a)

(b)

(c)

(d)

Figure 12 

(The antikink wave is bifurcated from the valley-shape solitary wave). The varying process for the graph of 𝑢𝑔 when 𝑐>0, 𝛿>0, 𝑎>0, and 𝑏→−16𝑎2/75𝑐−0, where 𝑐=1, 𝛿=1, 𝑎=1/2, and (a) 𝑏=−16𝑎2/75𝑐−10−3. (b) 𝑏=−16𝑎2/75𝑐−10−7. (c) 𝑏=−16𝑎2/75𝑐−10−8. (d) 𝑏=−16𝑎2/75𝑐−10−10.

Proof. When 𝛿>0 and (𝑎,𝑏)∈𝐴₃, (𝜑₁,0) is a saddle point and on its left side there is an orbit Γ₃ connecting with it (see Figure 13).
In (8), letting ℎ₁=𝐻(𝜑₁,0), it follows that Γ₃∶𝑦²=𝑐𝛿𝜑²−8𝑎𝜑15𝛿^5/2−𝑏𝜑3𝛿³+ℎ₁0<𝜑<𝜑₁. (49) Substituting (49) into (7) and integrating it along the orbit Γ₃, we get (47) and (48).
Letting 𝑏→−16𝑎²/75𝑐−0, it follows that lim_{𝑏→−16𝑎2/75𝑐−0}ℎ₁=lim_{𝑏→−16𝑎2/75𝑐−0}𝐻𝜑₁,0=lim_{𝑏→−16𝑎2/75𝑐−0}−𝑐𝛿𝜑²₁+8𝑎𝜑15𝛿^15/2+𝑏𝜑3𝛿³¹=0. (50)
When ℎ₁→0, completing the integrals in (47) and (48), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition 7.

Figure 13 

The graph of orbit Γ3 when 𝛿>0 and (𝑎,𝑏)∈𝐴3.

3.4. Bifurcation from Kink-Like Waves

Proposition 8. When the parameters satisfy 𝛿>0, (𝑎,𝑏)∈𝐴₂, and 𝑏→−16𝑎²/75𝑐+0, the kink-like wave and the antikink-like wave, respectively, become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures 14 and 15.

(a)

(b)

(c)

(d)

Figure 14 

(The kink wave is bifurcated from the kink-like wave). The varying process for the graph of 𝑢+𝑎 when 𝑐>0, 𝛿>0, 𝑎>0, and 𝑏→−16𝑎2/75𝑐+0, where 𝑐=1, 𝛿=1, 𝑎=1/2, and (a) 𝑏=−16𝑎2/75𝑐+10−3. (b) 𝑏=−16𝑎2/75𝑐+10−4. (c) 𝑏=−16𝑎2/75𝑐+10−6. (d) 𝑏=−16𝑎2/75𝑐+10−8.

(a)

(b)

(c)

(d)

Figure 15 

(The antikink wave is bifurcated from the antikink-like wave). The varying process for the graph of 𝑢−𝑎 when 𝑐>0, 𝛿>0, 𝑎>0, and 𝑏→−16𝑎2/75𝑐+0, where 𝑐=1, 𝛿=1, 𝑎=1/2, and (a) 𝑏=−16𝑎2/75𝑐+10−3. (b) 𝑏=−16𝑎2/75𝑐+10−4. (c) 𝑏=−16𝑎2/75𝑐+10−6. (d) 𝑏=−16𝑎2/75𝑐+10−8.

Proof. Letting 𝑏→−16𝑎²/75𝑐+0, it follows that ℎ₁→0 (see (50)) and lim_{𝑏→−16𝑎2/75𝑐+0}𝜉⁰₁=^𝜑1₀^/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2+16𝑎²/225𝑐𝛿𝑠³=lim_𝜈→0^𝜑1_𝜈^/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2+16𝑎²/225𝑐𝛿𝑠³=∞. (51)
When ℎ₁→0, and 𝜉⁰₁→∞, completing the integrals in (17), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition 8.

3.5. Bifurcation from Compacton-Like Waves

Proposition 9. When the parameters satisfy 𝛿>0, (𝑎,𝑏)∈𝐴₂, and 𝑏→−16𝑎²/75𝑐+0, the two compacton-like waves become a kink wave and an anti-kink wave with the expressions (29).
For the varying process, see Figures 16 and 17.

(a)

(b)

(c)

(d)

Figure 16 

(The kink wave is bifurcated from the compacton-like wave). The varying process for the graph of 𝑢+𝑐 when 𝑐>0, 𝛿>0, 𝑎>0, and 𝑏→−16𝑎2/75𝑐+0, where 𝑐=1, 𝛿=1, 𝑎=1/2, and (a) 𝑏=−16𝑎2/75𝑐+10−3. (b) 𝑏=−16𝑎2/75𝑐+10−4. (c) 𝑏=−16𝑎2/75𝑐+10−6. (d) 𝑏=−16𝑎2/75𝑐+10−8.

(a)

(b)

(c)

(d)

Figure 17 

(The antikink wave is bifurcated from the compacton-like wave). The varying process for the graph of 𝑢−𝑐 when 𝑐>0, 𝛿>0, 𝑎>0, and 𝑏→−16𝑎2/75𝑐+0, where 𝑐=1, 𝛿=1, 𝑎=1/2, and (a) 𝑏=−16𝑎2/75𝑐+10−3. (b) 𝑏=−16𝑎2/75𝑐+10−4. (c) 𝑏=−16𝑎2/75𝑐+10−6. (d) 𝑏=−16𝑎2/75𝑐+10−8.

Proof. Letting 𝑏→−16𝑎²/75𝑐+0, it follows that lim_{𝑏→−16𝑎2/75𝑐+0}ℎ_𝜑𝑘=lim_{𝑏→−16𝑎2/75𝑐+0}𝐻𝜑_𝑘,0=lim_{𝑏→−16𝑎2/75𝑐+0}−𝑐𝛿𝜑²_𝑘+8𝑎𝜑15𝛿^𝑘5/2+𝑏𝜑3𝛿^3𝑘=0, (52)lim_{𝑏→−16𝑎2/75𝑐+0}𝜉⁰₃=^𝜑𝑘₀^/2𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2+16𝑎²/225𝑐𝛿𝑠³=lim_𝜈→0^𝜑𝑘_𝜈𝑑𝑠(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2+16𝑎²/225𝑐𝛿𝑠³=∞, (53)lim_{𝑏→−16𝑎2/75𝑐+0}𝜉⁰₄=lim_{𝑏→−16𝑎2/75𝑐+0}2^𝜑𝑘_𝜑𝑘/2(𝑑𝑠)(𝑐/𝛿)𝑠²−(8𝑎/15𝛿)𝑠^5/2−(𝑏/3𝛿)𝑠³+ℎ_𝜑𝑘^−1/2+𝜉⁰₃=∞. (54)
When ℎ_𝜑𝑘→0, 𝜉⁰₃→∞, and 𝜉⁰₄→∞, completing the integrals in (21), we get the kink wave solution and the antikink wave solution as (29).
Hereto, we have completed the proof for the Proposition 9.

4. Bifurcation of Smooth Periodic Wave

Proposition 10. For 𝑎𝑏≠0 and 𝛿<0, (1) has a nonlinear wave solution 𝑢_ℎ=−2𝛼𝛽²−4𝛼sin𝜏₂𝜉+𝜂+𝛽², (55) where 𝜏₂=12−𝑐𝛿, (56) and 𝜂 is an arbitrary real number. The solution possesses the following properties.(1)if (𝑎,𝑏) belongs to any one of the regions 𝐴₁, 𝐴₆, then 𝑢_ℎ represents periodic blow-up wave solution,(2)if (𝑎,𝑏) belongs to 𝐴₂, then 𝑢_ℎ represents periodic wave solution.
In particular, when 𝑏→0−0, the periodic wave becomes a periodic blow-up wave. For the varying process, see Figure 18. When 𝑏→−16𝑎²/75𝑐+0, the periodic wave tends to a trivial wave 𝑢=225𝑐²/16𝑎². For the varying process, see Figure 19.

(a)

(b)

(c)

(d)

Figure 18 

(The periodic blow-up wave is bifurcated from the periodic wave). The varying process for the graph of 𝑢ℎ when 𝑐>0, 𝛿<0, 𝑎>0, 𝜂>0, and 𝑏→0−0, where 𝑐=1, 𝛿=−1, 𝑎=1, 𝜂=1.57, and (a) 𝑏=0−0.2. (b) 𝑏=0−0.1. (c) 𝑏=0−0.05. (d) 𝑏=0−0.01.

(a)

(b)

(c)

(d)

Figure 19 

(The periodic wave become the trivial wave). The varying process for the graph of 𝑢ℎ when 𝑐>0, 𝛿<0, 𝑎>0, 𝜂<0, and 𝑏→−16𝑎2/75𝑐+0, where 𝑐=1, 𝛿=−1, 𝑎=1, 𝜂=−4.7, and (a) 𝑏=−16𝑎2/75𝑐+10−2. (b) 𝑏=−16𝑎2/75𝑐+10−3. (c) 𝑏=−16𝑎2/75𝑐+10−4. (d) 𝑏=−16𝑎2/75𝑐+10−7.

Proof. Completing the integral in (38), we get 𝑢_ℎ as (55). 𝜂=𝜂(𝜈) is an arbitrary real number.
When (𝑎,𝑏)∈𝐴₂, in (38) letting 𝜈=𝑞 (see (28)), we have 𝜂=arcsin−𝛽𝑞−2𝛼𝑞𝛽².−4𝛼 (57) Letting 𝑏→0−0, then lim_𝑏→0−0𝛽²−4𝛼𝛽²=lim_𝑏→0−075𝑏𝑐16𝑎²+1=1,lim_𝑏→0−0𝜂=lim_𝑏→0−0arcsin−𝛽𝑞−2𝛼𝑞𝛽²−4𝛼=lim_𝑏→0−01arcsin𝛽²−4𝛼/𝛽²+2𝑞𝛽²−4𝛼/𝛼²=𝜋=arcsin12. (58) We have lim_𝑏→0−0𝑢_ℎ=lim_𝑏→0−0−2𝛼𝛽²−4𝛼sin𝜏₂𝜉+𝜂+𝛽²=lim_𝑏→0−02𝛼/𝛽𝛽²−4𝛼/𝛽²sin𝜏₂𝜉+𝜂−1²=225𝑐²16𝑎²cos𝜏₂𝜉−1². (59)
Obviously, 𝑢_ℎ will blow up when 𝜉=2𝑘𝜋/𝜏₂(𝑘∈𝑧).
Hereto, we have completed the proofs for all propositions.

5. Conclusion

In this paper, we have studied the bifurcation behavior of S-KdV equation. Two new types of nonlinear waves called kink-like waves and compacton-like waves have been displayed in Propositions 1–3. Furthermore, two kinds of new bifurcation phenomena have been revealed. The first phenomenon is that the kink waves can be bifurcated from five types of nonlinear waves which have been stated in Propositions 5–9. The second phenomenon is that the periodic blow-up wave can be bifurcated from the periodic wave which has been explained in Proposition 10. At the same time, we have got three new explicit expressions for traveling waves which were given in (25) and (55). Two previous results are our some special cases (see (30) and (32)).

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 11171115) and the Science and Technology Foundation of Guizhou (no. LKS[2012]14).