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]:ut+σ(um)x+(un)xxx=0,m>1,n≥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 (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>) 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′+σ(ϕm)′+(ϕn)′′′=0.
Integrating (2.1) once and setting the integral constant as zero yields-vϕ+σϕm+n(n-1)ϕn-2(ϕ′)2+nϕn-1ϕ′′=0.
Let ϕ′=(dϕ/dξ)=y. The equation (2.2) can be reduced to a 2D planar system:dϕdξ=y,dydξ=vϕ-σϕm-n(n-1)ϕn-2y2nϕn-1,
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=nϕn-1dτ,
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=nϕn-1y,dydτ=vϕ-σϕm-n(n-1)ϕn-2y2.
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))ϕn+1-(2σ/(n+m))ϕn+mnϕ2n-2,
where h is an integral constant. From (2.6), we define a function as follows:H(ϕ,y)=nϕ2n-2y2+2σn+mϕm+n-2vn+1ϕn+1=h.
It is easy to verify that (2.5) satisfiesdϕdτ=12ϕn-1∂H∂y,dydτ=-12ϕn-1∂H∂ϕ.
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))ϕm+n-(2v/(n+1))ϕn+1 with M=2nϕ2n-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
hO=H(0,0)=0,hA0=-2v(m-1)(m+n)(n+1)(vσ)(n+1)/(m-1).

Case 2.

When m is odd number and σv>0, (2.5) has three equilibrium points O(0,0) and A1,2(±(v/σ)1/(m-1),0). From (2.7), we also obtain hO=H(0,0)=0 and
hA1=-2(m-1)v(m+n)(n+1)(vσ)(n+1)/(m-1),hA2=(-1)n+22(m-1)v(m+n)(n+1)(vσ)(n+1)/(m-1).
Obviously, if n is odd, then hA1=hA2. If n is even, then hA1≠hA2. Then hO=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 Properties3.1. Exact Solutions and Their Properties of (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>) under <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M97"><mml:mi>h</mml:mi><mml:mo>=</mml:mo><mml:msub><mml:mrow><mml:mi>h</mml:mi></mml:mrow><mml:mrow><mml:mi>O</mml:mi></mml:mrow></mml:msub></mml:math></inline-formula>

Taking h=hO=0, (2.6) can be reduced toy2=(2v/(n+1))ϕn+1-(2σ/(n+m))ϕn+mnϕ2n-2.

When m=n>1, (3.1) can be rewritten as

y=±(2nv/(n+1))ϕn+1-σϕ2nnϕn-1.
Substituting (3.2) into the first expression in (2.5) yieldsdϕdτ=±ϕ2nvn+1ϕn-1-σ(ϕn-1)2.
Noticing that equation (2nv/(n+1))ϕn-1-σ(ϕn-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∫[2nv/(n+1)σ]1/(n+1)ϕdϕϕ(2nv/(n+1))ϕn-1-σ(ϕn-1)2=±∫0τdτ.
After completing the aforementioned integral, we solve this equation; thus we obtainϕ=[2n(n+1)vn2(n-1)2v2τ2+(n+1)2σ]1/(n-1).
Substituting (3.5) into (2.4), then integrating it yieldsξ=2n(n-1)σarctan[n(n-1)v(n+1)στ],σ>0,ξ=-2n(n-1)-σtanh-1[n(n-1)v(n+1)-στ],σ<0.
Thus, we respectively obtain a periodic wave solution and solitary wave solution of parametric type for the equation K(n,n) as follows:u=ϕ(τ)=[2n(n+1)vn2(n-1)2v2τ2+(n+1)2σ]1/(n-1),ξ=2n(n-1)σarctan[n(n-1)v(n+1)στ],σ>0,u=ϕ(τ)=[2n(n+1)vn2(n-1)2v2τ2+(n+1)2σ]1/(n-1),ξ=-2n(n-1)-σtanh-1[n(n-1)v(n+1)-στ],σ<0.
On the other hand, (3.1) can be rewritten asy=±(2nv/(n+1))ϕn-1-σϕ2(n-1)nϕn-2.
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:∫[2nv/(n+1)σ]1/(n-1)ϕnϕn-2dϕ(2nv/(n+1))ϕn-1-σϕ2(n-1)=±∫0ξdξ.
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)σ2nξ]1/(n-1),σ>0,u(x,t)=ϕ(ξ)=[2nv(n+1)σcosh2(n-1)-σ2nξ]1/(n-1),σ<0.
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)σ2nξ]1/(n-1),σ>0,u(x,t)=ϕ(ξ)=[2nv(n+1)σsinh2(n-1)-σ2nξ]1/(n-1),σ<0.
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)σ2nξ]1/(n-1),σ>0,-nπn-1≤ξ≤nπn-1,0,otherwise,{u(x,t)=ϕ(ξ)=[2nv(n+1)σsin2(n-1)σ2nξ]1/(n-1),σ>0,0≤ξ≤2nπn-1,0,otherwise.
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.

When n=1,m>1, (3.1) can be directly reduced to

y=±ϕv-2σm+1ϕm-1.
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(x,t)=ϕ(ξ)=[(m+1)v2σsech2(m-1)v2ξ]1/(m-1),v>0,u(x,t)=ϕ(ξ)=[(m+1)v2σsec2(m-1)-v2ξ]1/(m-1),v<0.
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.

When n is even number and m=2n-1, (3.1) can be reduced to

y=±(2nv/(n+1))ϕn-1-(2nσ/(3n-1))ϕ3(n-1)nϕn-2.
It is easy to know that (2nv/(n+1))ϕn-1-(2nσ/(3n-1))ϕ3(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:∫αϕnϕn-2dϕ(2nv/(n+1))ϕn-1-(2nσ/(3n-1))ϕ3(n-1)=±∫0ξdξ,∫ϕ0nϕn-2dϕ(2nv/(n+1))ϕn-1-(2nσ/(3n-1))ϕ3(n-1)=±∫0ξdξ.∫ϕγnϕn-2dϕ(2nv/(n+1))ϕn-1-(2nσ/(3n-1))ϕ3(n-1)=±∫0ξdξ,
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)2α2nξ,12)]1/(n-1),n=evennumber,u(x,t)=ϕ(ξ)=[-αsn2((((n-1)2α)/2n)ξ,1/2)2dn2((((n-1)2α)/2n)ξ,1/2)]1/(n-1),n=evennumber,u(x,t)=ϕ(ξ)=[γnc2((((n-1)2α)/2n)ξ,1/2)]1/(n-1),n=evennumber.

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

n=2

n=15

n=400

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

m=2

m=20

n=200

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).

When m=3n-2,n>1, (3.1) can be directly reduced to

y=±(2nv/(n+1))ϕn-1-(2nσ/(4n-2))(ϕn-1)4nϕn-2.
It is easy to know that the function (2nσ/(4n-2))(a-ϕn-1)(ϕn-1-0)(ϕn-1-c)(ϕn-1-c̅)=(2nσ/(4n-2))(a-ϕn-1)(ϕn-1-0)[(ϕn-1-b1)2+a12], where b1=(c+c̅)/2=-a/2,a12=-(c-c̅)2/4=3a2/4. Using (a1/(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.

∫a1/(n-1)ϕdϕn-1(ϕn-1-a)(ϕn-1-0)[(ϕn-1-b1)2+a12]=±n-1n2nσ2-4n∫0ξdξ.
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(x,t)=ϕ(ξ)=[aB[1-cn(((n-1)/gn)2nσ/(4n-2)ξ,6(3-3)/12)]A+B+(A-B)cn(((n-1)/gn)2nσ/(4n-2)ξ,6(3-3)/12)]1/(n-1),u(x,t)=ϕ(ξ)=[aB[1+cn(((n-1)/gn)2nσ/(4n-2)ξ,(6-2)/4)]B-A+(A+B)cn(((n-1)/gn)2nσ/(4n-2)ξ,(6-2)/4)]1/(n-1),u(x,t)=ϕ(ξ)=[aB[1-cn(((n-1)/gn)2nσ/(2n-4)ξ,6(3-3)/12)]A+B+(A-B)cn(((n-1)/gn)2nσ/(2n-4)ξ,6(3-3)/12)]1/(n-1),u(x,t)=ϕ(ξ)=[aB[1+cn(((n-1)/gn)2nσ/(2-4n)ξ,(6-2)/4)]B-A+(A+B)cn(((n-1)/gn)2nσ/(2-4n)ξ,(6-2)/4)]1/(n-1),
where A=(a-b1)2+a12=3a,B=(0-b1)2+a12=a,andg=1/AB=274/3a with a=((4n-2)v)/((n+1)σ)3 given previously.

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

Periodic blowup wave

Periodic cusp wave

Periodic blowup wave

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.

When m=(k-1)n-k+2,n>1,k>4, (3.1) can be directly reduced to

y=±(2nv/(n+1))ϕn-1-(2nσ/(k(n-1)+2))ϕk(n-1)nϕn-2.
Suppose that ϕ0=ϕ(0) is one of roots for equation (2nv/(n+1))ϕn-1-(2nσ/(k(n-1)+2))ϕk(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:∫ϕ0ϕdϕn-1(2nv/(n+1))ϕn-1-(2nσ/(k(n-1)+2))(ϕn-1)k=±n-1nξ.
The left integral of (3.35) is called hyperelliptic integral for ϕn-1 when the degree k is greater than four. Let ϕn-1=z. Thus, (3.35) can be reduced to∫z01/(n-1)zdz(2nv/(n+1))z-(2nσ/(k(n-1)+2))zk=±n-1nξ.
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=z01/(n-1)=0 as initial value, (3.36) becomes∫0Zdzz+z5=±n-1n-σ(n+1)5n-3ξ.
Let z=(12)[ρ-ρ2-4],andz=(1+Z2)/Z2. We obtain -dz/zz=(1/2)[1/ρ+2+1/ρ-2] and 0<Z≤1. Thus, (3.37) can be transformed to12[∫z∞dρ(ρ+2)(ρ2-2)+∫z∞dρ(ρ-2)(ρ2-2)]=±n-1n-σ(n+1)5n-3ξ.
Completing (3.38) and refunded the variable z=ϕn-1, we obtain two implicit solutions of elliptic function type for K(4n-3,n) equation as follows:sn-1(2+2ϕn-1+2,2-22+2)+sn-1(2+2ϕn-1+2,222+2)=Ω1,2ξ,
where Ω1,2=±((n-1)/n)(2+2)-σ(n+1)/(5n-3). The solutions also can be rewritten asF(sin-12+2ϕn-1+2,2-22+2)+F(sin-12+2ϕn-1+2,222+2)=Ω1,2ξ,
where the function F(φ,k)=EllipticF(φ,k) is the incomplete Elliptic integral of the first kind.

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

n=10,σ=1,v=2

n=9,σ=3,v=0.1

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.

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

Asymptotically stable wave

Asymptotically stable wave

3.2. Exact Solutions and Their Properties of (<xref ref-type="disp-formula" rid="EEq1">1.1</xref>) under <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M214"><mml:mi>h</mml:mi><mml:mo>≠</mml:mo><mml:mn>0</mml:mn></mml:math></inline-formula>

In this subsection, under the conditions h=hA0,andh=hA1,h=hA2, 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))ϕn+1-(2nσ/(n+m))ϕn+mnϕn-1.
Substituting (3.41) into the first expression of (2.3) yields∫ϕ*ϕdϕnh+(2nv/(n+1))ϕn+1-(2nσ/(n+m))ϕn+m=±ξ,
where ϕ* is one of roots for equation h+(2nv/(n+1))ϕn+1-(2nσ/(n+m))ϕn+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.

If m=n=2, then (3.41) can be reduced to

y=±h+(4v/3)ϕ3-σϕ42ϕ.
Taking h=hA0|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.
Then -(v4/6σ3)+(4v/3)ϕ3-σϕ4=0 has four roots, two real roots, and two complex roots as follows:a,b=vσ[13+μ6±168-3(4+22)1/3-6(4+22)-1/3+16μ],c,c̅=vσ[13-μ6±i16-8+3(4+22)1/3+6(4+22)-1/3+16μ],

with μ=4+3(4+22)1/3+6(4+22)-1/3.

When σ>0 and a>ϕ>b, taking b as initial value, then integrating (3.44) yields

∫bϕdϕ(a-ϕ)(ϕ-b)(ϕ-c)(ϕ-c̅)=±σ∫0τdτ.
Solving the aforementioned integral equation yieldsϕ=aB+bAA+B[1+α1cn(ABστ,k)1+αcn(ABστ,k)],
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+c̅)/2))2-((c-c̅)2/4) and B=(b-((c+c̅)/2))2-((c-c̅)2/4) Substituting (3.47) and n=2 into (2.4) yieldsξ=2(aB+bA)(A+B)ABσ[α1αu1+α-α1α(1-α2)(Π(φ,α2α2-1,k)-αf1)],
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:

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+B[1+α1cn(ABστ,k)1+αcn(ABστ,k)],ξ=2(aB+bA)(A+B)ABσ[α1αu1+α-α1α(1-α2)(Π(φ,α2α2-1,k)-αf1)].

When σ<0 and b<a<ϕ<∞, taking a as initial value, integrating (3.44) yields

∫aϕdϕ(ϕ-a)(ϕ-b)(ϕ-c)(ϕ-c̅)=±-σ∫0τdτ.
Solving the aforementioned integral equation yieldsϕ=aB-bAB-A[1+α̃1cn(-ABστ,k̃)1+α̃cn(-ABστ,k̃)],
where α̃1=(aB+bA)/(aB-bA),α̃=(A+B)/(B-A),k̃=(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)-ABσ×[α̃1α̃ũ1+α̃-α̃1α̃(1-α̃2)(Π(φ̃,α̃2α̃2-1,k̃)-α̃f̃1)],
where ũ1=sn-1(-ABστ,k̃)=F(φ̃,k),φ̃=amũ1=arcsin(-ABστ),α̃2≠1,Π(φ̃,α̃2/(α̃2-1),k̃) is an elliptic integral of the third kind, and the function f̃1 satisfies the following three cases, respectively:

In the previous three cases, k̃′2=1-k̃2. 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-A[1+α̃1cn(-ABστ,k̃)1+α̃cn(-ABστ,k̃)],ξ=aB-bA(B-A)-ABσ[α̃1α̃ũ1+α̃-α̃1α̃(1-α̃2)(Π(φ̃,α̃2α̃2-1,k̃)-α̃f̃1)].
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).

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

Peculiar compacton wave

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.

Under m=2,n=1, taking h=hA0|m=2,n=1=-v3/3σ2 as Hamiltonian quantity, (3.42) can be reduced to

∫ϕ*ϕdϕ-(v3/3σ2)+vϕ2-(2σ/3)ϕ3=±ξ,
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(x,t)=ϕ(ξ)=-[v2σ+3v2σtan2(12vξ)],v>0,u(x,t)=ϕ(ξ)=-[v2σ-3v2σtanh2(12-vξ)],v<0.
Similarly, taking ϕ*=v/σ as initial value, we obtain two periodic solutions for K(2,1) as follows:u(x,t)=ϕ(ξ)=-v2σ-3v2σtan2(π4±12vξ),v>0.

Under m=2,n=1, taking arbitrary constant h as Hamiltonian quantity, (3.42) can be reduced to

∫ϕ*ϕdϕ-(2σ/3)(ϕ3+pϕ2+q)=±ξ,
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=hA0|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:∫z1ϕdϕ(ϕ-z1)(ϕ-z2)(ϕ-z3)=±-2σ3ξ(σ<0,z3<z2<z1<ϕ<∞),∫z2ϕdϕ(z1-ϕ)(ϕ-z2)(ϕ-z3)=±2σ3ξ(σ>0,z3<z2<ϕ<z1),∫z3ϕdϕ(z1-ϕ)(ϕ-z2)(ϕ-z3)=±2σ3ξ(σ>0,z3<ϕ<z2<z1),

Integrating the (3.58), then solving them, respectively, we obtain three periodic solutions of elliptic function type for K(2,1) as follows:u(x-vt)=ϕ(ξ)=z1-z2sn2(ω1ξ,k1)cn2(ω1ξ,k1),u(x-vt)=ϕ(ξ)=z2-z3k22sn2(ω2ξ,k2)dn2(ω2ξ,k2),u(x-vt)=ϕ(ξ)=z3+(z2-z3)sn2(ω2ξ,k1),
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).

When m=3,n=1, taking the constant h=hA1=hA2|m=3,n=1=-v/2σ as Hamiltonian quantity, (3.42) can be reduced to

∫ϕ*ϕdϕ(v/σ)-ϕ2=±-σ2ξ(σ<0,v<0).
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),
where v<0 shows that the waves defined by (3.63) are reverse traveling waves.

Under m=3,n=1, taking arbitrary constant h as Hamiltonian quantity and h≠-(v2/2σ), (3.42) can be reduced to

∫ϕ*ϕdϕϕ4-(2v/σ)ϕ2-(2h/σ)=±-σ2ξ(σ<0,v<0),

or∫ϕ*ϕdϕ-(ϕ4-(2v/σ)ϕ2-(2h/σ))=±σ2ξ(σ>0,v>0).
Clearly, ϕ4-(2v/σ)ϕ2-(2h/σ)=0 has four real roots r1,2,3,4=±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 s1,2=±v/σ±v2/σ2+2h/σ and two complex roots s,s̅=±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.

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

∫r1ϕdϕ(ϕ-r1)(ϕ-r2)(ϕ-r3)(ϕ-r4)=±-σ2ξ,∫ϕr1dϕ(r1-ϕ)(ϕ-r2)(ϕ-r3)(ϕ-r4)=±σ2ξ,
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(x-vt)=ϕ(ξ)=r1(r2-r4)-r2(r1-r4)sn2(Ω1ξ,k̃1)r2-r4-(r1-r4)sn2(Ω1ξ,k̃1)(ϕ<r1),
where Ω1=(1/2)-(σ/2)(r1-r3)(r2-r4),

k̃1=(r2-r3)(r1-r4)/(r1-r3)(r2-r4),
u(x-vt)=ϕ(ξ)=r1(r2-r4)+r4(r1-r2)sn2(Ω2ξ,k̃2)r2-r4-(r1-r2)sn2(Ω2ξ,k̃2)(r2<ϕ<r1),
where Ω2=(1/2)(σ/2)(r1-r3)(r2-r4),andk̃2=(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).

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

∫s1ϕdϕ(ϕ-s1)(ϕ-s2)(ϕ-s)(ϕ-s̅)=±-σ2ξ,∫s2ϕdϕ(s1-ϕ)(ϕ-s2)(ϕ-s)(ϕ-s̅)=±σ2ξ.
Solving the aforementioned two integral equations, we obtain two periodic solutions of Jacobian elliptic function for K(3,1) equation as follows:u(x-vt)=ϕ(ξ)=s1B̃-s2Ã+(s1B̃+s2Ã)cn((1/g̃)(-σ/2)ξ,k̃3)B̃-Ã+(Ã+B̃)cn((1/g̃)(-σ/2)ξ,k̃3),u(x-vt)=ϕ(ξ)=s1B̃+s2Ã+(s2Ã-s1B̃)cn((1/g̃)(σ/2)ξ,k̃4)B̃+Ã+(Ã-B̃)cn((1/g̃)(σ/2)ξ,k̃4),
where g̃=(1/ÃB̃), k̃3=((Ã+B̃)2-(s1-s2)2)/4ÃB̃, k̃4=((s1-s2)2-(Ã-B̃)2)/4ÃB̃ with Ã=(s1-b̃1)2+ã12, B̃=(s2-b̃1)2+ã12, ã12=-(s-s̅)2/4=|v/σ-v2/σ2+2h/σ|,b̃1=(s+s̅)/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).

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

Bright soliton

Smooth periodic wave

Smooth periodic wave

Antikink wave

Kink wave

Singular periodic wave

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).

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