International Journal of Differential Equations

International Journal of Differential Equations / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 582512 | https://doi.org/10.1155/2011/582512

Jiangbo Zhou, Lixin Tian, "Periodic and Solitary-Wave Solutions for a Variant of the K(3,2) Equation", International Journal of Differential Equations, vol. 2011, Article ID 582512, 16 pages, 2011. https://doi.org/10.1155/2011/582512

Periodic and Solitary-Wave Solutions for a Variant of the K(3,2) Equation

Academic Editor: Mayer Humi
Received05 May 2011
Accepted16 Aug 2011
Published31 Oct 2011

Abstract

We employ the bifurcation method of planar dynamical systems and qualitative theory of polynomial differential systems to derive new bounded traveling-wave solutions for a variant of the 𝐾(3,2) equation. For the focusing branch, we obtain hump-shaped and valley-shaped solitary-wave solutions and some periodic solutions. For the defocusing branch, the nonexistence of solitary traveling wave solutions is shown. Meanwhile, some periodic solutions are also obtained. The results presented in this paper supplement the previous results.

1. Introduction

To study the role of nonlinear dispersion in the formation of patterns in the liquid drops, in 1993, Rosenau and Hyman [1] suggested a family of KdV-like equations,𝑢𝑡+ğ‘Ž(𝑢𝑚)𝑥+(𝑢𝑛)𝑥𝑥𝑥=0,𝑚,𝑛≥1,(1.1) which is the so-called 𝐾(𝑚,𝑛) equation. In [2], the authors obtained topological soliton solution of the 𝐾(𝑚,𝑛) equation by using the ğºî…ž/𝐺 method.

For the focusing branch (ğ‘Ž=1), the 𝐾(𝑛,𝑛) has compacton solutions [3–9] given byâŽ§âŽªâŽ¨âŽªâŽ©îƒ¯î‚™ğ‘¢(𝑥,𝑡)=2𝑐𝑛𝑛+1cos𝑛−12𝑛(𝑥−𝑐𝑡)2/(𝑛−1),|𝑥−𝑐𝑡|≤𝑛𝜋𝑛−1,𝑛>1,0,otherwise,âŽ§âŽªâŽ¨âŽªâŽ©îƒ¯î‚™ğ‘¢(𝑥,𝑡)=2𝑐𝑛𝑛+1sin𝑛−12𝑛(𝑥−𝑐𝑡)2/(𝑛−1),|𝑥−𝑐𝑡|≤𝑛𝜋𝑛−1,𝑛>1,0,otherwise.(1.2)

For the defocusing branch (ğ‘Ž=−1), the 𝐾(𝑛,𝑛) equation has solutions [7, 9] given by𝑢(𝑥,𝑡)=2𝑐𝑛𝑛+1cosh𝑛−12𝑛(𝑥−𝑐𝑡)2/(𝑛−1),𝑢(𝑥,𝑡)=2𝑐𝑛𝑛+1sinh𝑛−12𝑛(𝑥−𝑐𝑡)2/(𝑛−1).(1.3)

Later, in 2000, Rosenau [5] introduced a variant of 𝐾(𝑚,𝑛) equation defined by𝑢𝑡+ğ‘Ž(𝑢𝑚)𝑥+𝑢(𝑢𝑛)𝑥𝑥𝑥=0,𝑚,𝑛≥1,(1.4) which emerges in nonlinear lattices and was used to describe the dispersion of dilute suspensions for 𝑚=2 and 𝑛=1 [5]. In [10], (1.4) was investigated to support the feature of the dispersion-velocity method for ğ‘Ž=1, 𝑚=2, and 𝑛=1.

For the focusing branch (ğ‘Ž>0) and 𝑚=𝑛+1, Rosenau [5] obtained a compacton solution of (1.4) given byâŽ§âŽªâŽ¨âŽªâŽ©îƒ¯ğ‘¢(𝑥,𝑡)=2ğ‘ğ‘Žcos2îƒ¬âˆšğ‘Ž2(𝑥−𝑐𝑡)1/𝑛,|𝑥−𝑐𝑡|≤𝜋,0,otherwise.(1.5) Later, Wazwaz [11] found another compacton solution given byâŽ§âŽªâŽ¨âŽªâŽ©îƒ¯ğ‘¢(𝑥,𝑡)=2ğ‘ğ‘Žsin2îƒ¬âˆšğ‘Ž2(𝑥−𝑐𝑡)1/𝑛,|𝑥−𝑐𝑡|≤𝜋,0,otherwise.(1.6)

For the defocusing branch (ğ‘Ž<0) and 𝑚=𝑛+1, Wazwaz [11] showed that (1.4) has solutions given by−𝑢(𝑥,𝑡)=2ğ‘ğ‘Žcosh2îƒ¬âˆšâˆ’ğ‘Ž2(𝑥−𝑐𝑡)1/𝑛,−𝑢(𝑥,𝑡)=2ğ‘ğ‘Žsinh2îƒ¬âˆšâˆ’ğ‘Ž2(𝑥−𝑐𝑡)1/𝑛.(1.7) Here we note that (1.3) and (1.7) are unbounded solutions and have no physical meaning. For more works regarding other variants of the 𝐾(𝑚,𝑛) equation, we refer to [12, 13].

Rosenau and Wazwaz's work did not consider the periodic and general solitary-wave solutions for (1.4). So, in this paper, we attempt to employ the bifurcation method of planar dynamical systems and qualitative theory of polynomial differential systems to obtain such two types of solutions of (1.4) for ğ‘Ž=±1, 𝑚=3, and 𝑛=2, which are not obtained in [5, 11]. The equation reads𝑢𝑡±𝑢3𝑥+𝑢𝑢2𝑥𝑥𝑥=0.(1.8) From Rosenau's point of view [5], we call (1.8) a variant of the 𝐾(3,2) equation. ğ‘Ž=1 and ğ‘Ž=−1 correspond to the focusing branch and defocusing branch of (1.8), respectively. It is known that periodic solution and solitary-wave solution are two important types of solutions for physical systems. They describe practical physical processes and can explain many physical phenomena. Here we must note that the bifurcation method was first used by Li and Liu [14] to obtain smooth and nonsmooth traveling-wave solutions to a nonlinearly dispersive equation and was later proved to be a powerful tool to solve a large number of integrable and nonintegrable partial differential equations [15–22].

The remainder of the paper is organized as follows. In Section 2, using the traveling wave transformation, we transform (1.8) into planar dynamical systems and then discuss bifurcations and phase portraits of the systems corresponding to the focusing and defocusing branch of (1.8), respectively. In Section 3, We obtain the solitary-wave solutions for the focusing branch of (1.8) and demonstrate the nonexistence of solitary-wave solutions for the defocusing branch of (1.8). In Section 4, we obtain some periodic solutions for both the focusing branch and the defocusing branch of (1.8). A short conclusion is given in Section 5.

2. Plane Phase Analysis

Let 𝜉=𝑥−𝑐𝑡, where 𝑐 is the wave speed. By using the traveling wave transformation 𝑢(𝑥,𝑡)=𝜑(𝑥−𝑐𝑡)=𝜑(𝜉), we can reduce (1.8) to the following ordinary differential equations:âˆ’ğ‘ğœ‘î…žÂ±î€·ğœ‘3+𝜑𝜑2=0,(2.1) where ′ denotes the derivative of the function with respect to 𝜉.

Integrating (2.1) once with respect to 𝜉, we have−𝑐𝜑±𝜑3+2𝜑𝜑′2+2𝜑2ğœ‘î…žî…ž=𝑔,(2.2) where 𝑔 is a constant of integration.

Let 𝜑′=𝑦, then (2.2) can be transformed into the following planar dynamical system:𝑑𝜑𝑑𝜉=𝑦,𝑑𝑦=𝑑𝜉𝑔+𝑐𝜑∓𝜑3−2𝜑𝑦22𝜑2.(2.3) We call it the traveling wave system of (1.8). It is a planar dynamical system with Hamiltonian function𝐻±1(𝜑,𝑦)=±4𝜑4−𝑐2𝜑2−𝑔𝜑+𝜑2𝑦2=ℎ,(2.4) where ℎ is a constant.

Note that (2.3) has a singular line 𝜑=0. To avoid the line temporarily, we make transformation 𝑑𝜉=2𝜑2𝑑𝜁. Under this transformation, system (2.3) becomes𝑑𝜑𝑑𝜉=2𝜑2𝑦,𝑑𝑦𝑑𝜉=𝑔+𝑐𝜑∓𝜑3−2𝜑𝑦2.(2.5)

System (2.5) and system (2.3) have the same first integral as (2.4). Consequently, system (2.5) has the same topological phase portraits as system (2.3) except for the straight line 𝜑=0.

For a fixed ℎ, (2.4) determines a set of invariant curves of system (2.5). As ℎ is varied, (2.4) determines different families of orbits of system (2.5) having different dynamical behaviors. Let 𝑀(𝜑𝑒,𝑦𝑒) be the coefficient matrix of the linearized version of system (2.5) at the equilibrium point (𝜑𝑒,𝑦𝑒), then𝑀𝜑𝑒,𝑦𝑒=⎛⎜⎜⎝4𝜑𝑒𝑦𝑒2𝜑2𝑒𝑐∓3𝜑2𝑒−2𝑦2𝑒−4ğœ‘ğ‘’ğ‘¦ğ‘’âŽžâŽŸâŽŸâŽ (2.6) and, at this equilibrium point, we have𝐽𝜑𝑒,𝑦𝑒𝜑=det𝑀𝑒,𝑦𝑒=−12𝜑2𝑒𝑦2𝑒−2𝜑2𝑒𝑐∓3𝜑2𝑒,𝑝𝜑𝑒,𝑦𝑒=trace𝑀𝜑𝑒,𝑦𝑒=0.(2.7)

By the theory of planar dynamical system (see [23]), for an equilibrium point of a planar dynamical system, if 𝐽<0, then this equilibrium point is a saddle point; it is a center point if 𝐽>0 and 𝑝=0; if 𝐽=0 and the Poincaré index of the equilibrium point is 0, then it is a cusp.

2.1. The Focusing Branch

The traveling wave system corresponding to the focusing branch of (1.8) is𝑑𝜑𝑑𝜉=2𝜑2𝑦,𝑑𝑦𝑑𝜉=𝑔+𝑐𝜑−𝜑3−2𝜑𝑦2.(2.8)

If the wave speed 𝑐<0, then system (2.8) has only one equilibrium point, which is a center point (𝑔≠0) or a degenerate center point (𝑔=0).

If the wave speed 𝑐>0, then, by using the first integral value and properties of equilibrium points, we can obtain three bifurcation curves of system (2.8): they are 𝑔1√(𝑐)=(2𝑐/9)3𝑐,𝑔2(𝑐)=0, and 𝑔3√(𝑐)=(−2𝑐/9)3𝑐. Obviously, these three curves have no intersection point and 𝑔3(𝑐)<𝑔2(𝑐)<𝑔1(𝑐) for arbitrary constant 𝑐>0. They partition the (𝑐,𝑔) parameter plane into four regions. Using the bifurcation method of vector fields (e.g., [23–25]), we have the following result which describes the properties of the equilibrium points of system (2.8).

Proposition 2.1. For given arbitrary constant 𝑐>0, one has the following.(1)If 𝑔>𝑔1(𝑐) or 𝑔<𝑔3(𝑐), then system (2.8) has one equilibrium point. It is a center point.(2)If 𝑔=𝑔1(𝑐) or 𝑔=𝑔3(𝑐), then system (2.8) has two equilibrium points. One is a cusp, and the other is a center point.(3)If 𝑔2(𝑐)<𝑔<𝑔1(𝑐) or 𝑔3(𝑐)<𝑔<𝑔2(𝑐), then system (2.8) has three equilibrium points. One is a saddle point, and the other two are all center points.(4)If 𝑔=𝑔2(𝑐), then system (2.8) has three equilibrium points. One is a degenerate center point, and the other two are all center points.

The phase portraits of system (2.8) in each region and on the bifurcation curves for the wave speed 𝑐<0 and 𝑐>0 are shown in Figures 1 and 2, respectively.

2.2. The Defocusing Branch

The traveling wave system corresponding to the defocusing branch of (1.8) is𝑑𝜑𝑑𝜉=2𝜑2𝑦,𝑑𝑦𝑑𝜉=𝑔+𝑐𝜑+𝜑3−2𝜑𝑦2.(2.9)

If the wave speed 𝑐>0, then one can easily see that system (2.9) has only one equilibrium point, which is a saddle point (𝑔≠0) or a degenerate center point (𝑔=0).

If the wave speed 𝑐<0, then, by using the first integral value (2.4) and properties of equilibrium points, we can obtain three bifurcation curves of system (2.9) for 𝑐<0: they are 𝑔4√(𝑐)=(−2𝑐/9)−3𝑐, 𝑔5(𝑐)=0, and 𝑔6√(𝑐)=(2𝑐/9)−3𝑐. Obviously, the three curves have no intersection point and 𝑔6(𝑐)<𝑔5(𝑐)<𝑔4(𝑐) for arbitrary 𝑐<0. They partition the (𝑐,𝑔) parameter plane into four regions. Again using the bifurcation method of vector fields (e.g., [23–25]), we have the following result which describes the properties of the equilibrium points of system (2.9) for the wave speed 𝑐<0.

Proposition 2.2. For arbitrary constant 𝑐<0, one has the following.(1)If 𝑔>𝑔4(𝑐) or 𝑔<𝑔6(𝑐), then system (2.9) has one equilibrium point. It is a saddle point.(2)If 𝑔=𝑔4(𝑐) or 𝑔=𝑔6(𝑐), then system (2.9) has two equilibrium points. One is a cusp, and the other is a saddle point.(3)If 𝑔5(𝑐)<𝑔<𝑔4(𝑐) or 𝑔6(𝑐)<𝑔<𝑔5(𝑐), then system (2.9) has three equilibrium points. One is a center point, and the other two are all saddle points.(4)If 𝑔=𝑔5(𝑐), then system (2.9) has three equilibrium points. One is a degenerate center point, and the other two are all saddle points.

The phase portraits of system (2.9) in each region and on the bifurcation curves for the wave speed 𝑐>0 and 𝑐<0 are given in Figures 3 and 4, respectively.

3. Solitary-Wave Solutions for (1.8)

Usually, a solitary-wave solution for (1.8) corresponds to a homoclinic orbit of system (2.8). From Figure 2, we can see that, when the parameter 𝑔 satisfies 𝑔2(𝑐)<𝑔<𝑔1(𝑐) and 𝑔3(𝑐)<𝑔<𝑔2(𝑐) for 𝑐>0, system (2.8) has homoclinic orbits. So the focusing branch of (1.8) has solitary-wave solutions accordingly.

Now, take √𝑔=(11𝑐/72)3𝑐 for 𝑐>0, which satisfies 𝑔2(𝑐)<𝑔<𝑔1(𝑐). Then we can figure out three equilibrium points of system (2.8). Denote them as (𝜑1−,0), (𝜑2,0), and (𝜑1+,0), where 𝜑1±√=(√3𝑐±315𝑐)/12 and 𝜑2√=−(3𝑐/6).(𝜑1−,0) and (𝜑1+,0) are two center points, while (𝜑2,0) is a saddle point. There is a homoclinic orbit connecting with the saddle point (𝜑2,0), which lies on the left side of the singular line 𝜑=0 (see Figure 2(e)). This homoclinic orbit is determined by 𝐻+(𝜑,𝑦)=𝐻+(𝜑2,0) and can be expressed as𝑦=±𝜑−𝜑2𝜑−𝜑−2𝜑+2−𝜑2𝜑for𝜑−2≤𝜑≤𝜑2<0,(3.1) where𝜑±2=√√3𝑐±66𝑐6.(3.2)

Substituting (3.1) into the first equation of system (2.3) and integrating along the corresponding homoclinic orbit, we have𝜑𝜑−2𝑠𝑠−𝜑2𝑠−𝜑−2𝜑+21−𝑠𝑑𝑠=2||𝜉||,𝜑−2≤𝜑≤𝜑2.(3.3) It follows from (3.3) that𝛼arcsin1𝛼(𝜑)−arcsin1𝜑−2−2𝜑−2+𝜑+23𝜑−2+𝜑+2𝜑−2+3𝜑+2×𝛽arctan1𝛽(𝜑)−arctan1𝜑−2=12||𝜉||,𝜑−2≤𝜑≤𝜑2,(3.4) where𝛼1(𝜑)=2𝜑−𝜑−2−𝜑+2𝜑+2−𝜑−2,𝛽1𝜑(𝜑)=−22−𝜑+22𝜑+2−2−𝜑+2𝜑𝜑+4−2+𝜑+2𝜑−𝜑−2𝜑+2−𝜑−2𝜑+𝜑−2+𝜑+23𝜑−2+𝜑+2𝜑−2+3𝜑+2.(3.5) Equation (3.4) is the implicit expression of a valley-shaped solitary-wave solution for the focusing branch of (1.8). In physics, it can be called dark soliton (see [26]). A typical such solution is shown in Figure 5(a).

Similarly, if taking √𝑔=(−11𝑐/72)3𝑐 for 𝑐>0, which satisfies 𝑔3(𝑐)<𝑔<𝑔2(𝑐), then we can obtain a hump-shaped solitary-wave solution for the focusing branch of (1.8) given by𝛼arcsin2−𝜑−2𝛼−arcsin2+2𝜑(𝜑)−2+𝜑+23𝜑−2+𝜑+2𝜑−2+3𝜑+2×𝛽arctan2−𝜑−2𝛽−arctan2=1(𝜑)2||𝜉||,−𝜑2≤𝜑≤−𝜑−2,(3.6) where 𝛼2(𝜑)=2𝜑+𝜑−2+𝜑+2𝜑+2−𝜑−2,𝛽2𝜑(𝜑)=−22−𝜑+22𝜑−2−2−𝜑+2𝜑𝜑+4−2+𝜑+2𝜑+𝜑+2−𝜑−2−𝜑2𝜑+𝜑−2+𝜑+23𝜑−2+𝜑+2𝜑−2+3𝜑+2,(3.7) and 𝜑±2 are as in (3.2).

In physics, (3.6) can be called bright soliton (see [26]). A typical such solution is shown in Figure 5(b).

For the defocusing branch of (1.8), one can see from Figures 3 and 4 that system (2.9) has no homoclinic orbits. So the defocusing branch of (1.8) has no solitary-wave solutions of traveling-wave type accordingly.

4. Periodic Solutions for (1.8)

Usually, a periodic solution for (1.8) corresponds to a periodic orbit of system (2.8). From Figures 1 and 2, we can see that system (2.8) has periodic orbits except for the case 𝑐<0, 𝑔=0. So the focusing branch of (1.8) has periodic solutions accordingly. In the following we take some parametric conditions to obtain the exact expressions of some periodic solutions.

Firstly, take √𝑔=(11𝑐/72)3𝑐 for 𝑐>0, which satisfies 𝑔2(𝑐)<𝑔<𝑔1(𝑐). Then 𝐻+(𝜑,𝑦)=𝐻+(𝜑1−,0) determines a periodic orbit of system (2.8) on the right-hand side of the singular line 𝜑=0 (see Figure 2(e)). This periodic orbit can be expressed as𝑦=±𝜑−𝜑1−𝜑−𝜑−3𝜑+3−𝜑2𝜑for0<𝜑−3≤𝜑≤𝜑+3,(4.1) where𝜑±3=1−√12√3𝑐+315𝑐±2√3𝑐+9.5𝑐(4.2) Substituting (4.1) into the first equation of system (2.3) and integrating along the corresponding periodic orbit, we have𝜑𝜑−32𝑠𝑠−𝜑1−𝑠−𝜑−3𝜑+3=||𝜉||−𝑠𝑑𝑠,𝜑−3≤𝜑≤𝜑+3.(4.3) It follows from (4.3) that2𝛼arcsin3𝛼(𝜑)−arcsin3𝜑−3−4𝜑−3+𝜑+33𝜑−3+𝜑+3𝜑−3+3𝜑+3×𝛽arctan3𝛽(𝜑)−arctan3𝜑−3=±𝜉+𝑛𝑇1,𝑛=0,±1,±2,…,(4.4) where 𝛼3(𝜑)=2𝜑−𝜑−3−𝜑+3𝜑+3−𝜑−3,𝛽3𝜑(𝜑)=−32−𝜑+32𝜑+2−3−𝜑+3𝜑𝜑+4−3+𝜑+3𝜑−𝜑−3𝜑+3−𝜑−2𝜑+𝜑−3+𝜑+33𝜑−3+𝜑+3𝜑−3+3𝜑+3,(4.5)𝑇1âŽ›âŽœâŽœâŽœâŽğœ‘=4𝜋1−−3+𝜑+33𝜑−3+𝜑+3𝜑−3+3𝜑+3⎞⎟⎟⎟⎠,(4.6) and 𝜑±3 is as (4.2).

Equation (4.4) is the implicit expression of a periodic solution with the period 𝑇1 for the focusing branch of (1.8). A typical such solution is shown in Figure 6(a).

Next take √𝑔=(−11𝑐/72)3𝑐 for 𝑐>0, which satisfies 𝑔3(𝑐)<𝑔<𝑔2(𝑐). Similar to the above discussion, we can obtain another periodic solution with the same period 𝑇1 for the focusing branch of (1.8) as follows:2𝛼arcsin4𝛼(𝜑)−arcsin4−𝜑−3+2𝜑−3+𝜑+33𝜑−3+𝜑+3𝜑−3+3𝜑+3×𝛽arctan4𝛽(𝜑)−arctan4−𝜑+3=±𝜉+𝑛𝑇1,𝑛=0,±1,±2,…,(4.7) where 𝛼4(𝜑)=2𝜑+𝜑+3+𝜑−3𝜑+3−𝜑−3,(4.8)𝛽4𝜑(𝜑)=−32−𝜑+32𝜑−2−3−𝜑+3𝜑𝜑+4−3+𝜑+3𝜑+𝜑+3−𝜑−3−𝜑2𝜑−𝜑−3−𝜑+33𝜑−3+𝜑+3𝜑−3+3𝜑+3,(4.9)𝜑±3 and 𝑇1 are as (4.2) and (4.6), respectively. A typical such solution is shown in Figure 6(b).

Last, take 𝑔=𝑔2(𝑐). Then 𝐻+(𝜑,𝑦)=0 determines a periodic orbit of system (2.9) (see Figure 2(g)). It can be expressed as 1𝑦=±2√2𝑐−𝜑2for−√√2𝑐≤𝜑≤2𝑐.(4.10)

Substituting (4.10) into the first equation of system (2.3) and integrating along the corresponding periodic orbit, we have√𝜑2𝑐1√2𝑐−𝑠21𝑑𝑠=2||𝜉||√,−√2𝑐≤𝜑≤2𝑐.(4.11) It follows from (4.11) that√𝜑=12𝑐cos2𝜉.(4.12) Equation (4.12) is obviously a periodic solution with the period 𝑇=4𝜋 for the focusing branch of (1.8). A typical such solution is shown in Figure 6(c).

For the defocusing branch of (1.8), we can see from Figures 4(e) and 4(f) that, in the parametric regions 𝑐<0, 𝑔5(𝑐)<𝑔<𝑔4(𝑐) and 𝑐<0, 𝑔6(𝑐)<𝑔<𝑔5(𝑐), system (2.9) has periodic orbits. So the defocusing branch of (1.8) has periodic solutions accordingly.

Now, take √𝑔=(−11𝑐/72)(−3𝑐) for 𝑐<0, which satisfies 𝑔5(𝑐)<𝑔<𝑔4(𝑐). Then system (2.9) has three equilibrium points (𝜑∗1,0), (𝜑∗2,0), and (𝜑∗3,0)(𝜑∗1<0<𝜑∗2<𝜑∗3). For fixed ℎ(0<ℎ<𝐻−(𝜑∗2,0)), the curve 𝐻+(𝜑,𝑦)=ℎ intersects the 𝜑-axis with four points ğ‘Ž, 𝑏, 𝑑, and 𝑓, which satisfy that 𝑑<0<𝑓<ğ‘Ž<𝑏. This curve determines a periodic orbit of system (2.9) (see Figure 4(e)). It can be expressed as√𝑦=±(𝜑−𝑑)(𝜑−𝑓)(ğ‘Žâˆ’ğœ‘)(𝑏−𝜑)2𝜑forğ‘“â‰¤ğœ‘â‰¤ğ‘Ž.(4.13)

Substituting (4.13) into the first equation of system (2.3) and integrating along the corresponding periodic orbit, we have𝜑𝑓2𝑠√||𝜉||(𝑠−𝑑)(𝑠−𝑓)(ğ‘Žâˆ’ğ‘ )(𝑏−𝑠)𝑑𝑠=,ğ‘“â‰¤ğœ‘â‰¤ğ‘Ž.(4.14)

It follows from (4.14) that−4√(ğ‘Žâˆ’ğ‘‘)(𝑏−𝑓)𝑑𝐹(arcsin(𝛾(𝜑))∣𝑚)+(𝑓−𝑑)Π𝑙;𝑠𝑛−1(𝛾(𝜑)∣𝑚)∣𝑚=±𝜉+𝑛𝑇,𝑛=0,±1,±2,…,(4.15) where îƒŽğ›¾(𝜑)=(ğ‘Žâˆ’ğ‘‘)(𝜑−𝑓),(ğ‘Žâˆ’ğ‘“)(𝜑−𝑑)(4.16)𝑚=(𝑏−𝑑)(ğ‘Žâˆ’ğ‘“)(,𝑏−𝑓)(ğ‘Žâˆ’ğ‘‘)𝑙=ğ‘Žâˆ’ğ‘“,ğ‘‡ğ‘Žâˆ’ğ‘‘2=8√(ğ‘Žâˆ’ğ‘‘)(𝑏−𝑓)(𝑑𝐾(𝑚)+(𝑓−𝑑)Π(𝑛∣𝑚)),(4.17) and 𝐹(⋅|⋅) is the elliptic integral of the first kind, and the notation is as used in Section  17.2.6 of [27]. Π(⋅;⋅|⋅) is the elliptic integral of the third kind, and the notation is as used in Section  17.2.15 of [27]. 𝑠𝑛−1(⋅|⋅) is the inverse of the Jacobian elliptic function 𝑠𝑛(⋅|⋅), where the notation is as used in Chapter 16 of [27]. 𝐾(⋅) is the complete elliptic integral of the first kind, and Π(⋅|⋅) is the complete elliptic integral of the third kind.

Equation (4.15) is a periodic solution with the period 𝑇2 for the defocusing branch of (1.8). A typical such solution is shown in Figure 7(a).

Next take √𝑔=(11𝑐/72)3𝑐 for 𝑐>0, which satisfies 𝑔6(𝑐)<𝑔<𝑔5(𝑐). For the same fixed ℎ as above, we can obtain another periodic solution with the same period 𝑇2 as follows:4√(ğ‘Žâˆ’ğ‘‘)(𝑏−𝑓)𝑑𝐹(arcsin(𝛾(𝜑))∣𝑚)+(𝑓−𝑑)Π𝑛;𝑠𝑛−1(𝛾(𝜑)∣𝑚)∣𝑚=±𝜉+𝑛𝑇2,𝑛=0,±1,±2,…,(4.18) where 𝛾(𝜑), 𝑚, 𝑙, and 𝑇2 are as (4.16)-(4.9), respectively.

A typical such solution is shown in Figure 7(b).

5. Conclusion

In summary, by using the bifurcation method, we investigate two new types of bounded traveling-wave solutions for a variant of the 𝐾(3,2) equation. For the focusing branch, we obtain its solitary-wave solutions and some periodic solutions. For the defocusing branch, we demonstrate the nonexistence of solitary-wave solutions of traveling type and obtain some periodic solutions in terms of elliptic integrals. Our work supplements the results in [5, 11] and may help people to know deeply the described physical process and possible applications of this nonlinear dispersive equation.

Acknowledgments

The authors are very grateful to the anonymous reviewers for their valuable comments and suggestions. This work was supported by the Startup Fund for Advanced Talents of Jiangsu University (no. 09JDG013), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 09KJB110003), the Jiangsu Planned Projects for Postdoctoral Research Funds (no. 0902107C), the Jiangsu Government Scholarship for Overseas Studies, and the Priority Academic Program Development of Jiangsu Higher Education Institutions.

References

  1. P. Rosenau and J. M. Hyman, “Compactons: solitons with finite wavelength,” Physical Review Letters, vol. 70, no. 5, pp. 564–567, 1993. View at: Publisher Site | Google Scholar
  2. G. Ebadi and A. Biswas, “The G′/G method and topological soliton solution of the K(m,n) equation,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 6, pp. 2377–2382, 2011. View at: Publisher Site | Google Scholar
  3. P. Rosenau, “Nonlinear dispersion and compact structures,” Physical Review Letters, vol. 73, no. 13, pp. 1737–1741, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  4. P. Rosenau, “On nonanalytic solitary waves formed by a nonlinear dispersion,” Physics Letters A, vol. 230, no. 5-6, pp. 305–318, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  5. P. Rosenau, “Compact and noncompact dispersive patterns,” Physics Letters A, vol. 275, no. 3, pp. 193–203, 2000. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  6. A. M. Wazwaz, “New solitary-wave special solutions with compact support for the nonlinear dispersive K(m,n) equations,” Chaos, Solitons and Fractals, vol. 13, no. 2, pp. 321–330, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  7. A. M. Wazwaz, “Exact special solutions with solitary patterns for the nonlinear dispersive K(m,n) equations,” Chaos, Solitons and Fractals, vol. 13, no. 1, pp. 161–170, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  8. A. M. Wazwaz, “General compactons solutions for the focusing branch of the nonlinear dispersive K(n,n) equations in higher-dimensional spaces,” Applied Mathematics and Computation, vol. 133, no. 2-3, pp. 213–227, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  9. A. M. Wazwaz, “General solutions with solitary patterns for the defocusing branch of the nonlinear dispersive K(n,n) equations in higher dimensional spaces,” Applied Mathematics and Computation, vol. 133, no. 2-3, pp. 229–244, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  10. A. Chertock and D. Levy, “Particle methods for dispersive equations,” Journal of Computational Physics, vol. 171, no. 2, pp. 708–730, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  11. A. M. Wazwaz, “Compact structures for variants of the generalized KdV and the generalized KP equations,” Applied Mathematics and Computation, vol. 149, no. 1, pp. 103–117, 2004. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  12. A. Biswas, “1-soliton solution of the K(m,n) equation with generalized evolution,” Physics Letters A, vol. 372, no. 25, pp. 4601–4602, 2008. View at: Publisher Site | Google Scholar
  13. A. Biswas, “1-soliton solution of the K(m,n) equation with generalized evolution and time-dependent damping and dispersion,” Computers & Mathematics with Applications, vol. 59, no. 8, pp. 2536–2540, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  14. J. Li and Z. Liu, “Smooth and non-smooth traveling waves in a nonlinearly dispersive equation,” Applied Mathematical Modelling, vol. 25, no. 1, pp. 41–56, 2000. View at: Publisher Site | Google Scholar
  15. T. Qian and M. Tang, “Peakons and periodic cusp waves in a generalized Camassa-Holm equation,” Chaos, Solitons and Fractals, vol. 12, no. 7, pp. 1347–1360, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  16. J. Shen and W. Xu, “Bifurcations of smooth and non-smooth travelling wave solutions in the generalized Camassa-Holm equation,” Chaos, Solitons and Fractals, vol. 26, no. 4, pp. 1149–1162, 2005. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  17. L. Zhang, L. Q. Chen, and X. Huo, “Bifurcations of smooth and nonsmooth traveling wave solutions in a generalized Degasperis-Procesi equation,” Journal of Computational and Applied Mathematics, vol. 205, no. 1, pp. 174–185, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  18. Z. Liu and Z. Ouyang, “A note on solitary waves for modified forms of Camassa-Holm and Degasperis-Procesi equations,” Physics Letters A, vol. 366, no. 4-5, pp. 377–381, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  19. J. Zhou and L. Tian, “Soliton solution of the osmosis K(2,2) equation,” Physics Letters A, vol. 372, no. 41, pp. 6232–6234, 2008. View at: Publisher Site | Google Scholar
  20. J. Zhou and L. Tian, “A type of bounded traveling wave solutions for the Fornberg-Whitham equation,” Journal of Mathematical Analysis and Applications, vol. 346, no. 1, pp. 255–261, 2008. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  21. A. Chen, W. Huang, and S. Tang, “Bifurcations of travelling wave solutions for the Gilson-Pickering equation,” Nonlinear Analysis: Real World Applications, vol. 10, no. 5, pp. 2659–2665, 2009. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  22. B. He and Q. Meng, “New exact explicit peakon and smooth periodic wave solutions of the K(3,2) equation,” Applied Mathematics and Computation, vol. 217, no. 4, pp. 1697–1703, 2010. View at: Publisher Site | Google Scholar | Zentralblatt MATH
  23. D. Luo, X. Wang, D. Zhu, and M. Han, Bifurcation Theory and Methods of Dynamical Systems, vol. 15 of Advanced Series in Dynamical Systems, World Scientific Publishing, London, UK, 1997.
  24. J. Guckenheimer and P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, vol. 42 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1983.
  25. L. Perko, Differential Equations and Dynamical Systems, vol. 7 of Texts in Applied Mathematics, Springer, New York, NY, USA, 1991.
  26. H. Triki and A. M. Wazwaz, “Bright and dark soliton solutions for a K(m,n) equation with t-dependent coefficients,” Physics Letters A, vol. 373, no. 25, pp. 2162–2165, 2009. View at: Publisher Site | Google Scholar
  27. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1972.

Copyright © 2011 Jiangbo Zhou and Lixin Tian. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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