Exact Traveling Wave Solutions of Explicit Type, Implicit Type, and Parametric Type for K(m,n) Equation

Abstract

By using the integral bifurcation method, we study the nonlinear 𝐾(𝑚,𝑛) equation for all possible values of 𝑚 and 𝑛. Some new exact traveling wave solutions of explicit type, implicit type, and parametric type are obtained. These exact solutions include peculiar compacton solutions, singular periodic wave solutions, compacton-like periodic wave solutions, periodic blowup solutions, smooth soliton solutions, and kink and antikink wave solutions. The great parts of them are different from the results in existing references. In order to show their dynamic profiles intuitively, the solutions of 𝐾(𝑛,𝑛), 𝐾(2𝑛−1,𝑛), 𝐾(3𝑛−2,𝑛), 𝐾(4𝑛−3,𝑛), and 𝐾(𝑚,1) equations are chosen to illustrate with the concrete features.

1. Introduction

In this paper, we will investigate some new traveling-wave phenomena of the following nonlinear dispersive 𝐾(𝑚,𝑛) equation [1]:𝑢_𝑡+𝜎𝑢^𝑚_𝑥+𝑢^𝑛_𝑥𝑥𝑥=0,𝑚>1,𝑛≥1, (1.1) where 𝑚 and𝑛 are integers and 𝜎 is a real parameter. This is a family of fully KdV equations. When 𝜎=1, (1.1) as a role of nonlinear dispersion in the formation of patterns in liquid drops was studied by Rosenau and Hyman [1]. In [2–6], the studies show that the model equation (1.1) supports compact solitary structure. In [3], especially Rosenau’s study shows that the branch + (i.e., 𝜎=1) supports compact solitary waves and the branch − (i.e., 𝜎=−1) supports motion of kinks, solitons with spikes, cusps or peaks. In [7, 8], Wazwaz developed new solitary wave solutions of (1.1) with compact support and solitary patterns with cusps or infinite slopes under 𝜎=±1, respectively. In [9], by using the extend decomposition method, Zhu and Lü obtained exact special solutions with solitary patterns for (1.1). In [10], by using homotopy perturbation method (HPM), Domairry et al. studied the (1.1); under particular cases, they obtained some numerical and exact compacton solutions of the nonlinear dispersive 𝐾(2,2) and 𝐾(3,3) equations with initial conditions. In [11], by variational iteration method, Tian and Yin obtained new solitary solutions for nonlinear dispersive equations 𝐾(𝑚,𝑛); under particular values of 𝑚 and 𝑛, they obtained shock-peakon solutions for 𝐾(2,2) equation and shock-compacton solutions for 𝐾(3,3) equation. In [12], the nonlinear equation 𝐾(𝑚,𝑛) is studied by Wazwaz for all possible values of 𝑚 and 𝑛. In [13], by using Adomian decomposition method, Zhu and Gao obtained new solitary-wave special solutions with compact support for (1.1). In [14], by using a new method which is different from the Adomian decomposition method, Shang studied (1.1) and obtained new exact solitary-wave solutions with compact. In [15, 16], 1-soliton solutions of the 𝐾(𝑚,𝑛) equation with generalized evolution are obtained by Biswas. In [17], the bright and dark soliton solutions for 𝐾(𝑚,𝑛) equation with 𝑡-dependent coefficients are obtained by Triki and Wazwaz, especially, when 𝑚=𝑛, the 𝐾(𝑛,𝑛) equation was studied by many authors; see [18–24] and references cited therein. Defocusing branch, Deng et al. [25] obtained exact solitary and periodic traveling wave solutions of 𝐾(2,2) equation. Also, under some particular values of 𝑚 and 𝑛, many authors considered some particular cases of 𝐾(𝑚,𝑛) equation. Ismail and Taha [26] implemented a finite difference method and a finite element method to study two types of equations 𝐾(2,2) and 𝐾(3,3). A single compacton as well as the interaction of compactons has been numerically studied. Then, Ismail [27] made an extension to the work in [26], applied a finite difference method on 𝐾(2,3) equation, and obtained numerical solutions of 𝐾(2,3) equation [28]. Frutos and Lopez-Marcos [29] presented a finite difference method for the numerical integration of 𝐾(2,2) equation. Zhou and Tian [30] studied soliton solution of 𝐾(2,2) equation. Xu and Tian [31] investigated the peaked wave solutions of 𝐾(2,2) equation. Zhou et al. [32] obtained kink-like wave solutions and antikink-like wave solutions of 𝐾(2,2) equation. He and Meng [33] obtain some new exact explicit peakon and smooth periodic wave solutions of the 𝐾(3,2) equation by the bifurcation method of planar systems and qualitative theory of polynomial differential system.

From the aforementioned references, and references cited therein, it has been shown that (1.1) is a very important physical and engineering model. This is a main reason for us to study it again. In this paper, by using the integral bifurcation method [34–36], we mainly investigate some new exact solutions such as explicit solutions of Jacobian elliptic function type with low-power, implicit solutions of Jacobian elliptic function type, periodic solutions of parametric type, and so forth. We also investigate some new traveling wave phenomena and their dynamic properties.

The rest of this paper is organized as follows. In Section 2, we will derive the equivalent two-dimensional planar system of (1.1) and its first integral. In Section 3, by using the integral bifurcation method, we will obtain some new traveling wave solutions and discuss their dynamic properties; some phenomena of new traveling waves are illustrated with the concrete features.

2. The Equivalent Two-Dimensional Planar System to (1.1) and Its First Integral Equations

We make a transformation 𝑢(𝑡,𝑥)=𝜙(𝜉) with 𝜉=𝑥−𝑣𝑡, where the 𝑣 is a nonzero constant as wave velocity. Thus, (1.1) can be reduced to the following ODE:−𝑣𝜙^′+𝜎𝜙^𝑚′+𝜙^{𝑛′′′}=0. (2.1) Integrating (2.1) once and setting the integral constant as zero yields−𝑣𝜙+𝜎𝜙^𝑚+𝑛(𝑛−1)𝜙^𝑛−2𝜙^′2+𝑛𝜙^𝑛−1𝜙^′′=0. (2.2) Let 𝜙^′=(𝑑𝜙/𝑑𝜉)=𝑦. The equation (2.2) can be reduced to a 2D planar system:𝑑𝜙𝑑𝜉=𝑦,𝑑𝑦=𝑑𝜉𝑣𝜙−𝜎𝜙^𝑚−𝑛(𝑛−1)𝜙^𝑛−2𝑦²𝑛𝜙^𝑛−1, (2.3) where 𝜙≠0. Obviously, the solutions of (2.2) include the solutions of (2.3) and constant solution 𝜙=0. We notice that the second equation in (2.3) is not continuous when 𝜙=0; that is, the function 𝜙^′′(𝜉) is not defined by the singular line 𝜙=0. Therefore, we make the following transformation:𝑑𝜉=𝑛𝜙^𝑛−1𝑑𝜏, (2.4) where 𝜏 is a free parameter. Under the transformation (2.4), (2.3), and 𝜙=0 combine to make one 2D system as follows:𝑑𝜙𝑑𝜏=𝑛𝜙^𝑛−1𝑦,𝑑𝑦𝑑𝜏=𝑣𝜙−𝜎𝜙^𝑚−𝑛(𝑛−1)𝜙^𝑛−2𝑦². (2.5) Clearly, (2.5) is equivalent to (2.2). It is easy to know that (2.3) and (2.5) have the same first integral as follows:𝑦²=ℎ+(2𝑣/(𝑛+1))𝜙^𝑛+1−(2𝜎/(𝑛+𝑚))𝜙^𝑛+𝑚𝑛𝜙^2𝑛−2, (2.6) where ℎ is an integral constant. From (2.6), we define a function as follows:𝐻𝜙,𝑦=𝑛𝜙^2𝑛−2𝑦²+2𝜎𝜙𝑛+𝑚^𝑚+𝑛−2𝑣𝜙𝑛+1^𝑛+1=ℎ. (2.7) It is easy to verify that (2.5) satisfies𝑑𝜙=1𝑑𝜏2𝜙^𝑛−1𝜕𝐻,𝜕𝑦𝑑𝑦1𝑑𝜏=−2𝜙^𝑛−1𝜕𝐻.𝜕𝜙 (2.8) Therefore, (2.5) is a Hamiltonian system and 1/2𝜙ⁿ⁻¹ is an integral factor. In fact, (2.7) can be rewritten as the form 𝐻=𝐸+𝑇, where 𝐸=(1/2)𝑀𝑦²=(1/2)𝑀(𝜙^′)² and 𝑇=(2𝜎/(𝑛+𝑚))𝜙^𝑚+𝑛−(2𝑣/(𝑛+1))𝜙^𝑛+1 with 𝑀=2𝑛𝜙^2𝑛−2.𝐸 denotes kinetic energy, and 𝑇 denotes potential energy. Especially, when 𝑛=1,𝑀 becomes a constant 2. In this case, the kinetic energy 𝐸 only depends on movement velocity 𝜙^′ of particle; it does not depend on potential function 𝜙. So, according to Theorem 3.2 in [37], it is easy to know that (2.5) is a stable and nonsingular system when 𝑛=1; in this case its solutions have not singular characters. When 𝑛>1, (2.5) becomes a singular system; in this case some solutions of (2.5) have singular characters.

For the equilibrium points of the system (2.5), we have the following conclusion.

Case 1. When 𝑚 is even number, (2.5) has two equilibrium points 𝑂(0,0) and 𝐴₀((𝑣/𝜎)^1/(𝑚−1),0). From (2.7), we obtain ℎ_𝑂=𝐻(0,0)=0,ℎ_𝐴0()=−2𝑣𝑚−1()𝑣𝑚+𝑛)(𝑛+1𝜎^{(𝑛+1)/(𝑚−1)}. (2.9)

Case 2. When 𝑚 is odd number and 𝜎𝑣>0, (2.5) has three equilibrium points 𝑂(0,0) and 𝐴_1,2(±(𝑣/𝜎)^1/(𝑚−1),0). From (2.7), we also obtain ℎ_𝑂=𝐻(0,0)=0 and ℎ_𝐴12()𝑣=−𝑚−1()𝑣𝑚+𝑛)(𝑛+1𝜎^{(𝑛+1)/(𝑚−1)},ℎ_𝐴2=(−1)^𝑛+22(𝑚−1)𝑣()𝑣𝑚+𝑛)(𝑛+1𝜎^{(𝑛+1)/(𝑚−1)}. (2.10) Obviously, if 𝑛 is odd, then ℎ_𝐴1=ℎ_𝐴2. If 𝑛 is even, then ℎ_𝐴1≠ℎ_𝐴2. Then ℎ_𝑂=𝐻(0,0)=0 whether 𝑚 is odd number or even number.

3. Exact Solutions of Explicit Type, Implicit Type, and Parametric Type and Their Properties

3.1. Exact Solutions and Their Properties of (1.1) under ℎ=ℎ_𝑂

Taking ℎ=ℎ_𝑂=0, (2.6) can be reduced to𝑦²=(2𝑣/(𝑛+1))𝜙^𝑛+1−(2𝜎/(𝑛+𝑚))𝜙^𝑛+𝑚𝑛𝜙^2𝑛−2. (3.1)(i) When 𝑚=𝑛>1, (3.1) can be rewritten as𝑦=±(2𝑛𝑣/(𝑛+1))𝜙^𝑛+1−𝜎𝜙^2𝑛𝑛𝜙^𝑛−1. (3.2) Substituting (3.2) into the first expression in (2.5) yields𝑑𝜙𝑑𝜏=±𝜙2𝑛𝑣𝜙𝑛+1^𝑛−1−𝜎𝜙^𝑛−12. (3.3) Noticing that equation (2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−𝜎(𝜙^𝑛−1)²=0 has two roots 𝜙=0and𝜙=[2𝑛𝑣/(𝑛+1)𝜎]^1/(𝑛−1), we take ([2𝑛𝑣/(𝑛+1)𝜎]^1/(𝑛−1),0) as the initial value. Using this initial value, integrating (3.2) yields^𝜙_{[()𝜎]2𝑛𝑣/𝑛+11/(𝑛+1)}𝑑𝜙𝜙(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−𝜎𝜙^𝑛−12=±^𝜏₀𝑑𝜏. (3.4) After completing the aforementioned integral, we solve this equation; thus we obtain()𝑣𝜙=2𝑛𝑛+1𝑛²(𝑛−1)²𝑣²𝜏²+(𝑛+1)²𝜎^1/(n−1). (3.5) Substituting (3.5) into (2.4), then integrating it yields𝜉=2𝑛(𝑛−1)√𝜎𝑛()𝑣arctan𝑛−1(𝑛+1)√𝜎𝜏,𝜎>0,𝜉=−2𝑛(𝑛−1)√−𝜎tanh⁻¹𝑛(𝑛−1)𝑣(𝑛+1)√𝜏−𝜎,𝜎<0. (3.6) Thus, we respectively obtain a periodic wave solution and solitary wave solution of parametric type for the equation 𝐾(𝑛,𝑛) as follows:(𝜏)=()𝑣𝑢=𝜙2𝑛𝑛+1𝑛²(𝑛−1)²𝑣²𝜏²+(𝑛+1)²𝜎^1/(𝑛−1),𝜉=2𝑛(𝑛−1)√𝜎𝑛()𝑣arctan𝑛−1(𝑛+1)√𝜎𝜏,𝜎>0,(𝜏)=()𝑣𝑢=𝜙2𝑛𝑛+1𝑛²(𝑛−1)²𝑣²𝜏²+(𝑛+1)²𝜎^1/(𝑛−1),𝜉=−2𝑛(𝑛−1)√−𝜎tanh⁻¹𝑛(𝑛−1)𝑣(𝑛+1)√𝜏−𝜎,𝜎<0. (3.7)(3.8) On the other hand, (3.1) can be rewritten as𝑦=±(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−𝜎𝜙^2(𝑛−1)𝑛𝜙^𝑛−2. (3.9) Using ([2𝑛𝑣/(𝑛+1)𝜎]^1/(𝑛−1),0) as the initial value, substituting (3.9) into the first expression in (2.3) directly, we obtain an integral equation as follows:^𝜙_{[]2𝑛𝑣/(𝑛+1)𝜎1/(𝑛−1)}𝑛𝜙^𝑛−2𝑑𝜙(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−𝜎𝜙^2(𝑛−1)=±^𝜉₀𝑑𝜉. (3.10) Completing the aforementioned integral equation, then solving it, we obtain a periodic solution and a hyperbolic function solution as follows:𝑢(𝑥,𝑡)=𝜙(𝜉)=2𝑛𝑣(𝑛+1)𝜎cos²(𝑛−1)√𝜎𝜉2𝑛^1/(𝑛−1),𝜎>0, (3.11)𝑢(𝑥,𝑡)=𝜙(𝜉)=2𝑛𝑣(𝑛+1)𝜎cosh²(𝑛−1)√−𝜎𝜉2𝑛^1/(𝑛−1),𝜎<0. (3.12) Obviously, the solution (3.7) is equal to the solution (3.11); also the solution (3.8) is equal to the solution (3.12). Similarly, taking the (0,0) as initial value, substituting (3.9) into the first expression in (2.3), then integrating them, we obtain another periodic solution and another hyperbolic function solution of 𝐾(𝑛,𝑛) equation as follows.𝑢(𝑥,𝑡)=𝜙(𝜉)=2𝑛𝑣(𝑛+1)𝜎sin²(𝑛−1)√𝜎𝜉2𝑛^1/(𝑛−1),𝜎>0, (3.13)𝑢(𝑥,𝑡)=𝜙(𝜉)=2𝑛𝑣(𝑛+1)𝜎sinh²(𝑛−1)√−𝜎𝜉2𝑛^1/(𝑛−1),𝜎<0. (3.14) In fact, the solutions (3.11) and (3.13) have been appeared in [35], so we do not list similar solutions anymore at here. Next, we discuss a interesting problem as follows.

When 𝜎>0, from (3.11) and (3.13), we can construct two compacton solutions as follows:𝑢(𝑥,𝑡)=𝜙(𝜉)=2𝑛𝑣(𝑛+1)𝜎cos²(𝑛−1)√𝜎𝜉2𝑛^1/(𝑛−1),𝜎>0,−𝑛𝜋𝑛−1≤𝜉≤𝑛𝜋,𝑛−10,otherwise, (3.15)𝑢(𝑥,𝑡)=𝜙(𝜉)=2𝑛𝑣(𝑛+1)𝜎sin²(𝑛−1)√𝜎𝜉2𝑛^1/(𝑛−1),𝜎>0,0≤𝜉≤2𝑛𝜋,𝑛−10,otherwise. (3.16) The shape of compacton solutions (3.15) and (3.16) changes gradually as the value of parameter 𝑛 increases. For example, when 𝑛=2,15,400, respectively, the shapes of compacton solution (3.15) are shown in Figure 1.(ii) When 𝑛=1,𝑚>1, (3.1) can be directly reduced to𝑦=±𝜙𝑣−2𝜎𝜙𝑚+1^𝑚−1. (3.17) Equation (3.17) is a nonsingular equation. Using ([2𝜎/(𝑚+1)𝑣]^𝑛−1,0) as initial value and then substituting (3.17) into the first expression in (2.3) directly, we obtain a smooth solitary wave solution and a periodic wave solution of 𝐾(𝑚,1) equation as follows:𝑢()(𝜉)=()𝑣𝑥,𝑡=𝜙𝑚+12𝜎sech²(𝑚−1)√𝑣2𝜉^1/(𝑚−1),𝑣>0, (3.18)𝑢()(𝜉)=()𝑣𝑥,𝑡=𝜙𝑚+12𝜎sec²(𝑚−1)√−𝑣2𝜉^1/(𝑚−1),𝑣<0. (3.19) Also, the shape of solitary wave solution (3.18) changes gradually as the value of parameter 𝑚 increases. When 𝑚=2,20,200, respectively, its shapes of compacton solution (3.18) are shown in Figure 2.(iii)When 𝑛 is even number and 𝑚=2𝑛−1, (3.1) can be reduced to𝑦=±(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(3𝑛−1))𝜙^3(𝑛−1)𝑛𝜙^𝑛−2. (3.20) It is easy to know that (2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(3𝑛−1))𝜙^3(𝑛−1)=0 has three roots 𝜙=0 and 𝜙=𝛼,𝛾 with 𝛼,𝛾=±[(3𝑛−1)𝑣/(𝑛+1)𝜎]^1/(𝑛−1) when 𝜎𝑣>0. In fact, 𝛾=−𝛼. Using these three roots as initial value, respectively, then substituting (3.20) into the first expression in (2.3), we obtain three integral equations as follows:^𝜙_𝛼𝑛𝜙^𝑛−2𝑑𝜙(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(3𝑛−1))𝜙^3(𝑛−1)=±^𝜉₀𝑑𝜉,_0𝜙𝑛𝜙^𝑛−2𝑑𝜙(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(3𝑛−1))𝜙^3(𝑛−1)=±^𝜉₀𝑑𝜉._𝛾𝜙𝑛𝜙^𝑛−2𝑑𝜙(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(3𝑛−1))𝜙^3(𝑛−1)=±^𝜉₀𝑑𝜉, (3.21) Completing the previous three integral equations, then solving them, we obtain three periodic solutions of Jacobian elliptic function for 𝐾(2𝑛−1,𝑛) equation as follows:𝑢(𝑥,𝑡)=𝜙(𝜉)=𝛼nc²(𝑛−1)√2𝛼12𝑛𝜉,√2^1/(𝑛−1),𝑛=evennumber, (3.22)𝑢()(𝜉)=−𝛼𝑥,𝑡=𝜙sn²(𝑛−1)√2𝛼/2𝑛𝜉,1/√22dn²(𝑛−1)√2𝛼/2𝑛𝜉,1/√2^1/(𝑛−1),𝑛=evennumber, (3.23)𝑢(𝑥,𝑡)=𝜙(𝜉)=𝛾nc²(𝑛−1)√2𝛼/2𝑛𝜉,1/√2^1/(𝑛−1),𝑛=evennumber. (3.24)

(a) 𝑛=2

(b) 𝑛=15

(c) 𝑛=400

Figure 1 

The solution 𝑢 in (3.15) shows a shape of compacton for parameters 𝑣=2,and𝜎=1.

(a) 𝑚=2

(b) 𝑚=20

(c) 𝑛=200

Figure 2 

The solution 𝑢 in (3.18) shows a shape of compacton for parameters 𝑣=2,and𝜎=1.

The solutions (3.22) and (3.24) show two shapes of periodic wave with blowup form, which are shown in Figures 3(a) and 3(c). The solution (3.23) shows a shape of periodic cusp wave, which is shown in Figure 3(b).(iv) When 𝑚=3𝑛−2,𝑛>1, (3.1) can be directly reduced to𝑦=±(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(4𝑛−2))𝜙^𝑛−14𝑛𝜙^𝑛−2. (3.25) It is easy to know that the function (2𝑛𝜎/(4𝑛−2))(𝑎−𝜙^𝑛−1)(𝜙^𝑛−1−0)(𝜙^𝑛−1−𝑐)(𝜙^𝑛−1−𝑐)=(2𝑛𝜎/(4𝑛−2))(𝑎−𝜙^𝑛−1)(𝜙^𝑛−1−0)[(𝜙^𝑛−1−𝑏₁)²+𝑎²₁], where 𝑏₁=(𝑐+𝑐)/2=−𝑎/2,𝑎²₁=−(𝑐−𝑐)²/4=3𝑎²/4. Using (𝑎^1/(𝑛−1),0) and (0,0) as initial values, respectively, substituting (3.25) into the first expression in (2.3), we obtain four elliptic integral equations as follows.(1)When 𝜎>0,𝑣>0,^𝜙₀𝑑𝜙^𝑛−1𝑎−𝜙^𝑛−1𝜙^𝑛−1−0𝜙^𝑛−1−𝑏₁²+𝑎²₁=±𝑛−1𝑛2𝑛𝜎4𝑛−2^𝜉₀𝑑𝜉. (3.26)(2)When 𝜎>0,𝑣<0,^𝜙_{𝑎1/(𝑛−1)}𝑑𝜙^𝑛−1𝜙^𝑛−1−𝑎𝜙^𝑛−1−0𝜙^𝑛−1−𝑏₁²+𝑎²₁=±𝑛−1𝑛2𝑛𝜎4𝑛−2^𝜉₀𝑑𝜉. (3.27)(3)When 𝜎<0,𝑣<0,^𝜙₀𝑑𝜙^𝑛−1𝑎−𝜙^𝑛−1𝜙^𝑛−1−0𝜙^𝑛−1−𝑏₁²+𝑎²₁=±𝑛−1𝑛2𝑛𝜎2−4𝑛^𝜉₀𝑑𝜉. (3.28)(4)When 𝜎<0,𝑣>0,^𝜙_{𝑎1/(𝑛−1)}𝑑𝜙^𝑛−1𝜙^𝑛−1−𝑎𝜙^𝑛−1−0𝜙^𝑛−1−𝑏₁²+𝑎²₁=±𝑛−1𝑛2𝑛𝜎2−4𝑛^𝜉₀𝑑𝜉. (3.29) Corresponding to (3.26), (3.27), (3.28), and (3.29), respectively, we obtain four periodic solutions of elliptic function type for 𝐾(3𝑛−2,𝑛) equation as follows:𝑢()(𝜉)=𝑥,𝑡=𝜙𝑎𝐵1−cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(4𝑛−2)𝜉,√63−√3/12𝐴+𝐵+(𝐴−𝐵)cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(4𝑛−2)𝜉,√63−√3/12^1/(𝑛−1),𝑢()(𝜉)=𝑥,𝑡=𝜙𝑎𝐵1+cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(4𝑛−2)𝜉,√6−√2/4𝐵−𝐴+(𝐴+𝐵)cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(4𝑛−2)𝜉,√6−√2/4^1/(𝑛−1),𝑢()(𝜉)=𝑥,𝑡=𝜙𝑎𝐵1−cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(2𝑛−4)𝜉,√63−√3/12𝐴+𝐵+(𝐴−𝐵)cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(2𝑛−4)𝜉,√63−√3/12^1/(𝑛−1),𝑢()(𝜉)=𝑥,𝑡=𝜙𝑎𝐵1+cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(2−4𝑛)𝜉,√6−√2/4𝐵−𝐴+(𝐴+𝐵)cn(𝑛−1)/𝑔𝑛2𝑛𝜎/(2−4𝑛)𝜉,√6−√2/4^1/(𝑛−1), (3.30)(3.31)(3.32)(3.33) where 𝐴=(𝑎−𝑏₁)²+𝑎²₁=√3𝑎,𝐵=(0−𝑏₁)²+𝑎²₁=𝑎,and𝑔=1/√𝐴𝐵=√⁴27/3𝑎 with 𝑎=³((4𝑛−2)𝑣)/((𝑛+1)𝜎) given previously.

(a) Periodic blowup wave

(b) Periodic cusp wave

(c) Periodic blowup wave

Figure 3 

Three periodic waves of solutions (3.22), (3.23), and (3.24) for parameters 𝑛=4,𝑣=2,and𝜎=1.

The solution (3.30) shows a shape of periodic wave with blowup form, which is shown in Figure 4(a). The solution (3.31) shows s shape of compacton-like periodic wave, which is shown in Figure 4(b). The profile of solution (3.32) is similar to that of solution (3.30). Also the profile of solution (3.33) is similar to that of solution (3.31). So we omit the graphs of their profiles here.(v) When 𝑚=(𝑘−1)𝑛−𝑘+2,𝑛>1,𝑘>4, (3.1) can be directly reduced to𝑦=±(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(𝑘(𝑛−1)+2))𝜙^{𝑘(𝑛−1)}𝑛𝜙^𝑛−2. (3.34) Suppose that 𝜙₀=𝜙(0) is one of roots for equation (2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(𝑘(𝑛−1)+2))𝜙^{𝑘(𝑛−1)}=0. Clearly, the 0 is its one root. Anyone solution of 𝐾((𝑘−1)𝑛−𝑘+2,𝑛) equation can be obtained theoretically from the following integral equations:^𝜙_𝜙0𝑑𝜙^𝑛−1(2𝑛𝑣/(𝑛+1))𝜙^𝑛−1−(2𝑛𝜎/(𝑘(𝑛−1)+2))𝜙^𝑛−1𝑘=±𝑛−1𝑛𝜉. (3.35) The left integral of (3.35) is called hyperelliptic integral for 𝜙^𝑛−1 when the degree 𝑘 is greater than four. Let 𝜙^𝑛−1=𝑧. Thus, (3.35) can be reduced to^𝑧_{𝑧01/(𝑛−1)}𝑑𝑧(2𝑛𝑣/(𝑛+1))𝑧−(2𝑛𝜎/(𝑘(𝑛−1)+2))𝑧^𝑘=±𝑛−1𝑛𝜉. (3.36) In fact, we cannot obtain exact solutions by (3.36) when the degree 𝑘 is grater than five. But we can obtain exact solutions by (3.36) when 𝑘=5,𝑣=−𝜎(𝑛+1)𝑘(𝑛−1)+2, and 𝜎<0. Under these particular conditions, taking 𝜙₀=𝑧₀^1/(𝑛−1)=0 as initial value, (3.36) becomes^𝑍₀𝑑𝑧√𝑧+𝑧⁵=±𝑛−1𝑛−𝜎(𝑛+1)5𝑛−3𝜉. (3.37) Let 1𝑧=2[𝜌−𝜌²−4],and𝑧=(1+𝑍²)/𝑍². We obtain −𝑑𝑧/𝑧√𝑧=(1/2)[1/𝜌+2+1/𝜌−2] and 0<𝑍≤1. Thus, (3.37) can be transformed to12^∞_𝑧𝑑𝜌𝜌+2𝜌²+−2^∞𝑧𝑑𝜌𝜌−2𝜌²−2=±𝑛−1𝑛−𝜎(𝑛+1)5𝑛−3𝜉. (3.38) Completing (3.38) and refunded the variable 𝑧=𝜙^𝑛−1, we obtain two implicit solutions of elliptic function type for 𝐾(4𝑛−3,𝑛) equation as follows:sn⁻¹√2+2𝜙^𝑛−1,+22−√22+√2+sn⁻¹√2+2𝜙^𝑛−1+√2,2√22+√2=Ω_1,2𝜉, (3.39) where Ω_1,2=±((𝑛−1)/𝑛)(2+√2)−𝜎(𝑛+1)/(5𝑛−3). The solutions also can be rewritten as𝐹sin⁻¹√2+2𝜙^𝑛−1,+22−√22+√2+𝐹sin⁻¹√2+2𝜙^𝑛−1+√2,2√22+√2=Ω_1,2𝜉, (3.40) where the function 𝐹(𝜑,𝑘)=Elliptic𝐹(𝜑,𝑘) is the incomplete Elliptic integral of the first kind.

(a) 𝑛=10,𝜎=1,𝑣=2

(b) 𝑛=9,𝜎=3,𝑣=0.1

Figure 4 

Two different periodic waves on solutions (3.30) and (3.31) for given parameters.

The two solutions in (3.40) are asymptotically stable. Under Ω₁=((n−1)/n)(2+√2)−𝜎(𝑛+1)/(5𝑛−3),𝜙→0 as 𝜉→∞. Under Ω₂=−((n−1)/n)(2+√2)−𝜎(𝑛+1)/(5𝑛−3),  𝜙→0 as 𝜉→−∞. The graphs of their profiles are shown in Figure 5.

(a) Asymptotically stable wave

(b) Asymptotically stable wave

Figure 5 

Waveforms of two asymptotically stable solutions in (3.40) when 𝑛=4,𝜎=−1,and𝑡=1.

3.2. Exact Solutions and Their Properties of (1.1) under ℎ≠0

In this subsection, under the conditions ℎ=ℎ_𝐴0,andℎ=ℎ_𝐴1,ℎ=ℎ_𝐴2, we will investigate exact solutions of (1.1) and discuss their properties. When ℎ≠0, (2.6) can be reduced to𝑦=±ℎ+(2𝑛𝑣/(𝑛+1))𝜙^𝑛+1−(2𝑛𝜎/(𝑛+𝑚))𝜙^𝑛+𝑚𝑛𝜙^𝑛−1. (3.41) Substituting (3.41) into the first expression of (2.3) yields^𝜙_𝜙∗𝑑𝜙^𝑛ℎ+(2𝑛𝑣/(𝑛+1))𝜙^𝑛+1−(2𝑛𝜎/(𝑛+𝑚))𝜙^𝑛+𝑚=±𝜉, (3.42) where 𝜙_∗ is one of roots for equation ℎ+(2𝑛𝑣/(𝑛+1))𝜙^𝑛+1−(2𝑛𝜎/(𝑛+𝑚))𝜙^𝑛+𝑚=0. However we cannot obtain any exact solutions by (3.42) when the degrees 𝑚and𝑛 are more great, because we cannot obtain coincidence relationship among different degrees 𝑛,𝑛+1and𝑛+𝑚. But, we can always obtain some exact solutions when the degree 𝑚+𝑛 is not greater than four. For example, by using (3.42) directly, we can also obtain many exact solutions of 𝐾(2,1) and 𝐾(3,1) equations; see the next computation and discussion.(i) If 𝑚=𝑛=2, then (3.41) can be reduced to𝑦=±ℎ+(4𝑣/3)𝜙³−𝜎𝜙⁴.2𝜙 (3.43) Taking ℎ=ℎ_𝐴0|_𝑚=𝑛=2=−𝑣⁴/6𝜎³ as Hamiltonian quantity, substituting (3.43) and 𝑚=𝑛=2 into the first expression of (2.5) yields𝑑𝜙−𝑣⁴/6𝜎³+(4𝑣/3)𝜙³−𝜎𝜙⁴=±𝑑𝜏. (3.44) Then −(𝑣⁴/6𝜎³)+(4𝑣/3)𝜙³−𝜎𝜙⁴=0 has four roots, two real roots, and two complex roots as follows:𝑣𝑎,𝑏=𝜎13+𝜇6±168−34+2√2^1/3−64+2√2^−1/3+16𝜇,𝑐,𝑣𝑐=𝜎13−𝜇61±𝑖6−8+34+2√2^1/3+64+2√2^−1/3+16𝜇, (3.45)

with 𝜇=4+3(4+2√2)^1/3+6(4+2√2)^−1/3.(1)When 𝜎>0 and 𝑎>𝜙>𝑏, taking 𝑏 as initial value, then integrating (3.44) yields^𝜙_𝑏𝑑𝜙𝑎−𝜙𝜙−𝑏𝜙−𝑐𝜙−𝑐=±√𝜎^𝜏₀𝑑𝜏. (3.46) Solving the aforementioned integral equation yields𝜙=𝑎𝐵+𝑏𝐴𝐴+𝐵1+𝛼₁cn√𝐴𝐵𝜎𝜏,𝑘1+𝛼cn√,𝐴𝐵𝜎𝜏,𝑘 (3.47) where 𝛼₁=(𝑏𝐴−𝑎𝐵)/(𝑎𝐵+𝑏𝐴),𝛼=(𝐴−𝐵)/(𝐴+𝐵)and 𝑘=(1/2)((𝑎−𝑏)²−(𝐴−𝐵)²)/𝐴𝐵 with 𝐴=(𝑎−((𝑐+𝑐)/2))²−((𝑐−𝑐)²/4)  and 𝐵=(𝑏−((𝑐+𝑐)/2))²−((𝑐−𝑐)²/4) Substituting (3.47) and 𝑛=2 into (2.4) yields2()𝜉=𝑎𝐵+𝑏𝐴(𝐴+𝐵)√𝛼𝐴𝐵𝜎𝛼¹𝑢₁+𝛼−𝛼₁𝛼1−𝛼²Π𝛼𝜑,²𝛼²−1,𝑘−𝛼𝑓₁, (3.48) where 𝑢₁=sn⁻¹(√𝐴𝐵𝜎𝜏,𝑘)=𝐹(𝜑,𝑘),𝜑=am𝑢₁=arcsin(√𝐴𝐵𝜎𝜏),𝛼²≠1, the  Π(𝜑,𝛼²/(𝛼²−1),𝑘) is an elliptic integral of the third kind, and the function 𝑓₁ satisfies the following three cases, respectively:

𝑓₁=1−𝛼²𝑘²+𝑘^′2𝛼²arctan𝑘²+𝑘^′2𝛼²1−𝛼²sd(√𝐴𝐵𝜎𝜏,𝑘),if𝛼²(𝛼²−1)<𝑘²,=sd(√𝐴𝐵𝜎𝜏,𝑘),if𝛼²(𝛼²−1)=𝑘²,=12𝛼²−1𝑘²+𝑘^′2𝛼²×ln𝑘²+𝑘^′2𝛼²dn(√𝐴𝐵𝜎𝜏,𝑘)+√𝛼²−1sn(√𝐴𝐵𝜎𝜏,𝑘)𝑘²+𝑘^′2𝛼²dn(√𝐴𝐵𝜎𝜏,𝑘)−√𝛼²−1sn(√,𝐴𝐵𝜎𝜏,𝑘)if𝛼²𝛼²−1>𝑘².

In the previous three cases, 𝑘^′2=1−𝑘². Thus, by using (3.47) and (3.48), we obtain a parametric solution of Jacobian elliptic function for 𝐾(2,2) equation as follows:𝜙=𝑎𝐵+𝑏𝐴𝐴+𝐵1+𝛼₁cn√𝐴𝐵𝜎𝜏,𝑘1+𝛼cn√,2()𝐴𝐵𝜎𝜏,𝑘𝜉=𝑎𝐵+𝑏𝐴(𝐴+𝐵)√𝛼𝐴𝐵𝜎𝛼¹𝑢₁+𝛼−𝛼₁𝛼1−𝛼²Π𝛼𝜑,²𝛼²−1,𝑘−𝛼𝑓₁. (3.49)(2)When 𝜎<0 and 𝑏<𝑎<𝜙<∞, taking 𝑎 as initial value, integrating (3.44) yields^𝜙_𝑎𝑑𝜙𝜙−𝑎𝜙−𝑏𝜙−𝑐𝜙−𝑐=±√−𝜎^𝜏₀𝑑𝜏. (3.50) Solving the aforementioned integral equation yields𝜙=𝑎𝐵−𝑏𝐴𝐵−𝐴1+𝛼₁cn√−𝐴𝐵𝜎𝜏,𝑘1+𝛼cn√𝑘,−𝐴𝐵𝜎𝜏, (3.51) where 𝛼₁=(𝑎𝐵+𝑏𝐴)/(𝑎𝐵−𝑏𝐴),𝛼=(𝐴+𝐵)/(𝐵−𝐴),𝑘=(1/2)(𝐴+𝐵)²(𝑎−𝑏)²/𝐴𝐵, and 𝐴and𝐵 are given in case (1). Substituting (3.51) and 𝑛=2 into (2.4) yields𝜉=𝑎𝐵−𝑏𝐴(𝐵−𝐴)√×−𝐴𝐵𝜎𝛼₁𝛼𝑢₁+𝛼−𝛼₁𝛼1−𝛼²Π𝜑,𝛼²𝛼²,𝑘𝑓−1−𝛼₁, (3.52) where 𝑢₁=sn⁻¹(√−𝐴𝐵𝜎𝜏,𝑘)=𝐹(𝜑,𝑘),𝜑=am𝑢₁=arcsin(√−𝐴𝐵𝜎𝜏),𝛼²≠1,Π(𝜑,𝛼²/(𝛼²−1),𝑘) is an elliptic integral of the third kind, and the function 𝑓₁ satisfies the following three cases, respectively:

𝑓₁=1−𝛼²𝑘²+𝑘^′2𝛼²arctan𝑘²+𝑘^′2𝛼²1−𝛼²sd(√−𝐴𝐵𝜎𝜏,𝑘),if𝛼²(𝛼²<𝑘−1)²,=sd(√−𝐴𝐵𝜎𝜏,𝑘),if𝛼²(𝛼²=𝑘−1)²,=12𝛼²−1𝑘²+𝑘^′2𝛼²×ln𝑘²+𝑘^′2𝛼²dn(√−𝐴𝐵𝜎𝜏,𝑘)+√𝛼²−1sn(√−𝐴𝐵𝜎𝜏,𝑘)𝑘²+𝑘^′2𝛼²dn(√−𝐴𝐵𝜎𝜏,𝑘)−√𝛼²−1sn(√,−𝐴𝐵𝜎𝜏,𝑘)if𝛼²𝛼²>𝑘−1².

In the previous three cases, 𝑘^′2=1−𝑘². Thus, by using (3.51) and (3.52), we obtain another parametric solution of Jacobian elliptic function for 𝐾(2,2) equation as follows:𝑢=𝜙=𝑎𝐵−𝑏𝐴𝐵−𝐴1+𝛼₁cn√−𝐴𝐵𝜎𝜏,𝑘1+𝛼cn√𝑘,−𝐴𝐵𝜎𝜏,𝜉=𝑎𝐵−𝑏𝐴(𝐵−𝐴)√−𝐴𝐵𝜎𝛼₁𝛼𝑢₁+𝛼−𝛼₁𝛼1−𝛼²Π𝜑,𝛼²𝛼²,𝑘𝑓−1−𝛼₁. (3.53) In addition, when ℎ<−𝑣⁴/6𝜎³,  ℎ+(4𝑣/3)𝜙³−𝜎𝜙⁴=0 has four complex roots; in this case, we cannot obtain any useful results for 𝐾(2,2) equation. When ℎ>−𝑣⁴/6𝜎³, the case is very similar to (3.52); that is, the equation ℎ+(4𝑣/3)𝜙³−𝜎𝜙⁴=0 has two real roots and two complex roots. So we omit the discussions for these parts of results.

In order to describe the dynamic properties of the traveling wave solutions (3.49) and (3.53) intuitively, as an example, we draw profile figure of solution (3.53) by using the software Maple, when 𝑣=4,and𝜎=−2, see Figure 6(a).

(a) Peculiar compacton wave

(b) Bounded region of independent variable 𝜉

Figure 6 

Peculiar compacton wave and its bounded region of independent variable 𝜉.

Figure 6(a) shows a shape of peculiar compacton wave; its independent variable 𝜉 is bounded region (i.e., |𝜉|<𝛼₁+1); see Figure 6(b). From Figure 6(a), we find that its shape is very similar to that of the solitary wave, but it is not solitary wave because when |𝜉|≥𝛼₁+1,   𝑢≡0. So, this is a new compacton.(ii) Under 𝑚=2,𝑛=1, taking ℎ=ℎ_𝐴0|_{𝑚=2,𝑛=1}=−𝑣³/3𝜎² as Hamiltonian quantity, (3.42) can be reduced to^𝜙_𝜙∗𝑑𝜙−𝑣³/3𝜎²+𝑣𝜙²−(2𝜎/3)𝜙³=±𝜉, (3.54) where 𝜙_∗ is one of roots for the equation −(𝑣³/3𝜎²)+𝑣𝜙²−(2𝜎/3)𝜙³=0. Clearly, this equation has three real roots, one single root −𝑣/2𝜎 and two double roots 𝑣/𝜎,𝑣/𝜎. If 𝜎>0, then the function −(𝑣³/3𝜎²)+𝑣𝜙²−(2𝜎/3)𝜙³=(2𝜎/3)|𝜙−(𝑣/𝜎)|−(𝑣/2𝜎)−𝜙; if 𝜎<0, then the function (−𝑣³/3𝜎²)+𝑣𝜙²−(2𝜎/3)𝜙³=−(2𝜎/3)|𝜙−(𝑣/𝜎)|(𝑣/2𝜎)+𝜙. In these two conditions, taking 𝜙_∗=−(𝑣/2𝜎) as initial value and completing the (3.54), we obtain a periodic solution and a solitary wave solutions for 𝐾(2,1) as follows:𝑢()(𝜉)𝑣𝑥,𝑡=𝜙=−+2𝜎3𝑣2𝜎tan²12√𝑢()(𝜉)𝑣𝑣𝜉,𝑣>0,𝑥,𝑡=𝜙=−−2𝜎3𝑣2𝜎tanh²12√−𝑣𝜉,𝑣<0. (3.55) Similarly, taking 𝜙_∗=𝑣/𝜎 as initial value, we obtain two periodic solutions for 𝐾(2,1) as follows:𝑢()(𝜉)𝑣𝑥,𝑡=𝜙=−−2𝜎3𝑣2𝜎tan²𝜋4±12√𝑣𝜉,𝑣>0. (3.56)(iii) Under 𝑚=2,𝑛=1, taking arbitrary constant ℎ as Hamiltonian quantity, (3.42) can be reduced to^𝜙_𝜙∗𝑑𝜙−(2𝜎/3)𝜙³+𝑝𝜙²+𝑞=±𝜉, (3.57) where 𝑝=−3𝑣/2𝜎,  𝑞=−3ℎ/2𝜎.Write Δ=(𝑞²/4)+(𝑝³/27)=(9𝑣⁴/64𝜎⁴)−((𝑣³+6ℎ𝜎²)³/1728𝜎⁹). It is easy to know that Δ=0 as ℎ=ℎ_𝐴0|_{𝑚=2,𝑛=1}=−𝑣³/3𝜎²; this case is same as case (ii). So, we only discuss the case Δ<0 in the next.

When ℎ,𝜎,and𝑣 satisfy Δ<0,  𝜙³+𝑝𝜙²+𝑞=0 has three real roots 𝑧₁,𝑧₂,and𝑧₃ such as 𝑣/2𝜎cos(𝜃/3),𝑣/2𝜎cos(𝜃/3+2𝜋/3),and𝑣/2𝜎cos(𝜃/3+4𝜋/3) with 𝜃=arccos[(3ℎ/2𝜎)2𝜎³/𝑣³] and 𝑣/𝜎>0. Under these conditions, taking the 𝑧₁,𝑧₂,and𝑧₃ as initial values replacing 𝜙_∗, respectively, (3.57) can be reduced to the following three integral equations:^𝜙_𝑧1𝑑𝜙𝜙−𝑧₁𝜙−𝑧₂𝜙−𝑧₃=±−2𝜎3𝜉𝜎<0,𝑧₃<𝑧₂<𝑧₁<𝜙<∞_𝜙𝑧2,𝑑𝜙𝑧₁−𝜙𝜙−𝑧₂𝜙−𝑧₃=±2𝜎3𝜉𝜎>0,𝑧₃<𝑧₂<𝜙<𝑧₁,_𝜙𝑧3𝑑𝜙𝑧₁−𝜙𝜙−𝑧₂𝜙−𝑧₃=±2𝜎3𝜉𝜎>0,𝑧₃<𝜙<𝑧₂<𝑧₁, (3.58)

Integrating the (3.58), then solving them, respectively, we obtain three periodic solutions of elliptic function type for 𝐾(2,1) as follows:𝑢()(𝜉)=𝑧𝑥−𝑣𝑡=𝜙¹−𝑧₂sn²𝜔₁𝜉,𝑘₁cn²𝜔₁𝜉,𝑘₁, (3.59)𝑢()(𝜉)=𝑧𝑥−𝑣𝑡=𝜙²−𝑧₃𝑘²₂sn²𝜔₂𝜉,𝑘₂dn²𝜔₂𝜉,𝑘₂, (3.60)𝑢(𝑥−𝑣𝑡)=𝜙(𝜉)=𝑧₃+𝑧₂−𝑧₃sn²𝜔₂𝜉,𝑘₁, (3.61) where 𝜔₁=(1/2)−(2𝜎/3)(𝑧₁−𝑧₃),  𝑘₁=(𝑧₂−𝑧₃)/(𝑧₁−𝑧₃),  𝜔₂=(1/2)(2𝜎/3)(𝑧₁−𝑧₃), and 𝑘₂=(𝑧₁−𝑧₂)/(𝑧₁−𝑧₃).(iv) When 𝑚=3,𝑛=1, taking the constant ℎ=ℎ_𝐴1=ℎ_𝐴2|_{𝑚=3,𝑛=1}=−𝑣/2𝜎 as Hamiltonian quantity, (3.42) can be reduced to^𝜙_𝜙∗𝑑𝜙(𝑣/𝜎)−𝜙²=±−𝜎2𝜉(𝜎<0,𝑣<0). (3.62) Clearly, (𝑣/𝜎)−𝜙²=0 has two real roots 𝑣/𝜎and−𝑣/𝜎. Taking 𝜙_∗=(𝑣/𝜎+(−𝑣/𝜎))/2=0 as initial value, solving (3.62), we obtain a kink wave solution and an antikink wave solution for 𝐾(3,1) as follows:𝑢(𝑥−𝑣𝑡)=𝜙(𝜉)=±𝑣𝜎tanh−𝑣2𝜉, (3.63) where 𝑣<0 shows that the waves defined by (3.63) are reverse traveling waves.(v) Under 𝑚=3,𝑛=1, taking arbitrary constant ℎ as Hamiltonian quantity and ℎ≠−(𝑣²/2𝜎), (3.42) can be reduced to^𝜙_𝜙∗𝑑𝜙𝜙⁴−(2𝑣/𝜎)𝜙²−(2ℎ/𝜎)=±−𝜎2𝜉(𝜎<0,𝑣<0), (3.64)

or^𝜙_𝜙∗𝑑𝜙−𝜙⁴−(2𝑣/𝜎)𝜙²−(2ℎ/𝜎)=±𝜎2𝜉(𝜎>0,𝑣>0). (3.65) Clearly, 𝜙⁴−(2𝑣/𝜎)𝜙²−(2ℎ/𝜎)=0 has four real roots 𝑟_1,2,3,4=±𝑣/𝜎±𝑣²/𝜎²+2ℎ/𝜎 if 𝜎<0,𝑣<0,and0<ℎ<−(𝑣²/2𝜎) or 𝜎>0,𝑣>0,and−(𝑣²/2𝜎)<ℎ<0; it has two real roots 𝑠_1,2=±𝑣/𝜎±𝑣²/𝜎²+2ℎ/𝜎 and two complex roots 𝑠,𝑠=±𝑖|𝑣/𝜎−𝑣²/𝜎²+2ℎ/𝜎| if 𝜎<0,𝑣<0,andℎ<0 or 𝜎>0,𝑣>0,andℎ>0; it has not any real roots if 𝜎<0,𝑣<0,andℎ>−𝑣²/2𝜎 or 𝜎>0,𝑣>0,andℎ<−𝑣²/2𝜎.(1)Under the conditions 𝜎<0,𝑣<0,and0<ℎ<−𝑣²/2𝜎 or 𝜎>0,𝑣>0,and−𝑣²/2𝜎<ℎ<0, taking 𝜙_∗=𝑟₁ as an initial value, (3.64) and (3.65) can be reduced to^𝜙_𝑟1𝑑𝜙𝜙−𝑟₁𝜙−𝑟₂𝜙−𝑟₃𝜙−𝑟₄=±−𝜎2𝜉,_𝑟1𝜙𝑑𝜙𝑟₁−𝜙𝜙−𝑟₂𝜙−𝑟₃𝜙−𝑟₄=±𝜎2𝜉, (3.66) where 𝑟₁>𝑟₂>𝑟₃>𝑟₄. Solving the integral equations (3.66), we obtain two periodic solutions of Jacobian elliptic function for 𝐾(3,1) equation as follows:𝑢()(𝜉)=𝑟𝑥−𝑣𝑡=𝜙¹𝑟₂−𝑟₄−𝑟₂𝑟₁−𝑟₄sn²Ω₁𝜉,𝑘₁𝑟₂−𝑟₄−𝑟₁−𝑟₄sn²Ω₁𝜉,𝑘₁𝜙<𝑟₁, (3.67) where Ω₁=(1/2)−(𝜎/2)(𝑟₁−𝑟₃)(𝑟₂−𝑟₄),

𝑘₁=(𝑟₂−𝑟₃)(𝑟₁−𝑟₄)/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄), 𝑢()(𝜉)=𝑟𝑥−𝑣𝑡=𝜙¹𝑟₂−𝑟₄+𝑟₄𝑟₁−𝑟₂sn²Ω₂𝜉,𝑘₂𝑟₂−𝑟₄−𝑟₁−𝑟₂sn²Ω₂𝜉,𝑘₂𝑟₂<𝜙<𝑟₁, (3.68) where Ω₂=(1/2)(𝜎/2)(𝑟₁−𝑟₃)(𝑟₂−𝑟₄),and𝑘₂=(𝑟₁−𝑟₂)(𝑟₃−𝑟₄)/(𝑟₁−𝑟₃)(𝑟₂−𝑟₄). The case for taking 𝜙_∗=𝑟₂,𝑟₃,𝑟₄ as initial values can be similarly discussed; here we omit these discussions because these results are very similar to the solutions (3.67) and (3.68).(2)Under the conditions 𝜎<0,𝑣<0,andℎ<0 or 𝜎>0,𝑣>0,andℎ>0, respectively taking 𝜙_∗=𝑠₁,𝑠₂ as initial value, (3.64) and (3.65) can be reduced to^𝜙_𝑠1𝑑𝜙𝜙−𝑠₁𝜙−𝑠₂𝜙−𝑠𝜙−𝑠=±−𝜎2𝜉,_𝜙𝑠2𝑑𝜙𝑠₁−𝜙𝜙−𝑠₂𝜙−𝑠𝜙−𝑠=±𝜎2𝜉. (3.69) Solving the aforementioned two integral equations, we obtain two periodic solutions of Jacobian elliptic function for 𝐾(3,1) equation as follows:𝑢()(𝜉)=𝑠𝑥−𝑣𝑡=𝜙¹𝐵−𝑠₂𝐴+𝑠₁𝐵+𝑠₂𝐴cn1/𝑔(−𝜎/2)𝜉,𝑘₃𝐵−𝐴+𝐴+𝐵cn1/𝑔(−𝜎/2)𝜉,𝑘₃,𝑢()(𝜉)=𝑠𝑥−𝑣𝑡=𝜙¹𝐵+𝑠₂𝐴+𝑠₂𝐴−𝑠₁𝐵cn1/𝑔(𝜎/2)𝜉,𝑘₄𝐵+𝐴+𝐴−𝐵cn1/𝑔(𝜎/2)𝜉,𝑘₄, (3.70) where 𝑔=(1/𝐴𝐵),  𝑘₃=((𝐴+𝐵)²−(𝑠₁−𝑠₂)²)/4𝐴𝐵,  𝑘₄=((𝑠₁−𝑠₂)²−(𝐴−𝐵)²)/4𝐴𝐵 with 𝐴=(𝑠₁−𝑏̃₁)²+𝑎²₁,   𝐵=(𝑠₂−𝑏̃₁)²+𝑎²₁,   𝑎²₁=−(𝑠−𝑠)²/4=|𝑣/𝜎−𝑣²/𝜎²+2ℎ/𝜎|,𝑏̃₁=(𝑠+𝑠)/2=0, and 𝑠₁and𝑠₂ are given previously.

Among these aforementioned solutions, (3.59) shows a shape of solitary wave for given parameters 𝑣=4,and𝜎=1 which is shown in Figure 7(a). Equation (3.60) shows a shape of smooth periodic wave for given parameters 𝑣=2,𝜎=1,andℎ=4 which is shown in Figure 7(b). Also (3.61) shows a shape of smooth periodic wave for given parameters 𝑣=2,𝜎=1,andℎ=0.4 which is shown in Figure 7(c). Equation (3.63) shows two shapes of kink wave and antikink wave for given parameters 𝑣=−4,and𝜎=−2 which are shown in Figures 7(d)–7(e). Equation (3.68) shows a shape of singular periodic wave for given parameters 𝑣=−10,𝜎=−1,andℎ=48 which is shown in Figure 7(f).

(a) Bright soliton

(b) Smooth periodic wave

(c) Smooth periodic wave

(d) Antikink wave

(e) Kink wave

(f) Singular periodic wave

Figure 7 

The graphs of six kinds of waveforms for solutions (3.59), (3.60), (3.61), (3.63), and (3.68).

4. Conclusion

In this work, by using the integral bifurcation method, we study the nonlinear 𝐾(𝑚,𝑛) equation for all possible values of 𝑚 and 𝑛. Some travelling wave solutions such as normal compactons, peculiar compacton, smooth solitary waves, smooth periodic waves, periodic blowup waves, singular periodic waves, compacton-like periodic waves, asymptotically stable waves, and kink and antikink waves are obtained. In order to show their dynamic properties intuitively, the solutions of 𝐾(𝑛,𝑛), 𝐾(2𝑛−1,𝑛), 𝐾(3𝑛−2,𝑛), 𝐾(4𝑛−3,𝑛), and 𝐾(𝑚,1) equations are chosen to illustrate with the concrete features; using software Maple, we display their profiles by graphs; see Figures 1–7. These phenomena of traveling waves are different from those in existing literatures and they are very interesting. Although we do not know how they are relevant to the real physical or engineering problem for the moment, these interesting phenomena will attract us to study them further in the future works.

Acknowledgments

The authors thank the reviewers very much for their useful comments and helpful suggestions. This work was financially supported by the Natural Science Foundation of China (Grant no. 11161038). It was also supported by the Natural Science Foundations of Yunnan Province (Grant no. 2011FZ193) and Zhejiang Province (Grant no. Y2111160).