Abstract
Using bifurcation method and numerical simulation approach of dynamical systems, we study a two-component Fornberg-Whitham equation. Two types of bounded traveling wave solutions are found, that is, the kink-like wave and compacton-like wave solutions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, two new phenomena are revealed; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. Exact implicit or parameter expressions of these solutions are given.
1. Introduction
The Fornberg-Whitham equation was used to study the qualitative behaviors of wave breaking [1]. It admits a wave of the greatest height, as a peaked limiting form of the traveling wave solution [2], , where is an arbitrary constant. Recently, Zhou and Tian found that (1) possess kink-like wave solutions in [3]. They obtained some solitons, peakons, and periodic cusp wave solutions in [4]. Further, they obtained the smooth periodic wave solutions and loop-soliton solutions by using elliptic integral [5]. Feng and Wu [6] considered the classification of single traveling wave solutions to (1). Chen et al. [7] gave some smooth periodic wave, smooth solitary wave, periodic cusp wave, and loop-soliton solutions of (1) and made the numerical simulation.
He et al. [8] studied the following modified Fornberg-Whitham equation: In some parametric conditions, some peakons and solitary waves were found and their exact parametric representations in explicit form were obtained.
Jiang and Bi [9] considered the Fornberg-Whitham equation with linear dispersion term given by where is a real constant. When , (3) reduces to (1). They investigated the existence of the smooth and nonsmooth traveling wave solutions and gave some analytic expressions of smooth solitary wave, periodic cusp wave, and peakon solutions for (3).
Fan et al. [10] presented a two-component Fornberg-Whitham equation given by where is the height of the water surface above a horizontal bottom and is related to the horizontal velocity field. When , (4) reduces to (1). Parametric conditions to smooth soliton solution, kink solution, antikink solution, and uncountable infinite many smooth periodic wave solutions of (4) were given. Later, Wen [11] further studied (4). He presented all possible phase portraits determinately and gave all the exact explicit parametric conditions for various solutions.
The concept of compacton: soliton with compact support or strict localization of solitary waves appeared in the work of Rosenau and Hyman [12], where a genuinely nonlinear dispersive equation defined by was subjected to experimental and analytical studies. 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 [13–18]. The kink-like wave or generalized kink wave is discovered by Liu et al. [14], which is defined on semifinal bounded domain and possesses some properties of the kink wave.
Many methods have been used to investigate traveling wave solutions to nonlinear equations, such as Jacobi elliptic function method [19, 20], F-expansion and extended F-expansion method [21, 22], and -expansion method [23, 24]. Here, our aim in this paper is to use the bifurcation method of dynamical systems [25–28] to investigate (4). We obtain the kink-like wave and compacton-like wave solutions with implicit or parameter expressions. The planar graphs of these solutions are simulated by using software Mathematica; meanwhile, we point out that the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively. To the best of our knowledge, these solutions and phenomena are new for (4). Our work may help people to know deeply the described physical process and possible applications of (4).
The remainder of this paper is organized as follows. In Section 2, we study the bifurcation phase portraits. In Section 3, we make the numerical simulation for bounded integral curves. In Section 4, we derive the exact implicit or parameter expressions of the kink-like wave and compacton-like wave solutions. A brief conclusion is given in Section 5.
2. Bifurcation Phase Portraits
We look for the traveling wave solutions of (4) in the form of where is the mean level and is the wave speed.
Substituting (6) into (4) and integrating once with respect to , it follows that where , are two integral constants and (if , then from the second equation of (7) and (6). In this case (4) reduces to (1), which was studied in [3–7]).
In order to study conveniently, we choose , and this only makes a translational movement of the singular line from to , so there is no essential difference for the results. Thus, substituting the second equation of (7) into the first equation of (7), we obtain
Letting , we obtain the following planar system: under the transformation , and system (9) becomes
Obviously, system (9) and system (10) have the same first integral where is an integral constant. Consequently, these two systems have the same topological phase portraits except for the straight line . Thus, we can understand the phase portraits of system (9) from those of system (10).
In order to state conveniently, for given constants and , let
Assume that , , and are three roots of equation , where
Meanwhile, we give conditions as follows.
Case 1. , or , .
Case 2. and .
Case 3. and .
Case 4. and .
Case 5. and .
Case 6. and .
Case 7. and .
Case 8. and .
Case 9. and .
Case 10. , or , .
Case 11. , or , .
Case 12. and .
Case 13. and .
Case 14. and .
Case 15. and .
Case 16. and .
According to the qualitative theory of differential equations and the above conditions, we have the results as Table 1.
3. Numerical Simulation for Bounded Integral Curves
In this section, we make the numerical simulation for bounded integral curves. For convenience, throughout the following work we only discuss the solution with respect to the first component and omit the second component of (4).
From the derivation in Section 2 we see that the bounded traveling waves of (4) correspond to the bounded integral curves of (8) and the bounded integral curves of (8) correspond to the orbits of system (9) in which is bounded. Therefore we can simulate the bounded integral curves of (8) by using the information of the phase portraits of system (9).
It follows from [14–18] that the open orbits = 1–5) of system (9) correspond to the compacton-like waves of (4), the heteroclinic orbits ( = 1,2) of system (9) correspond to the kink-like waves of (4), and the periodic orbits surrounding the center point correspond to the periodic waves of (4). Here we only make the numerical simulation for Cases 4 and 5 as Examples 1 and 2. The other cases are similar to Examples 1 and 2, so we omit them.
Example 1. For Case 4, taking , then , , , and .
From (11), the two heteroclinic orbits (see Figure 1(d)) passing through the saddle point have expressions, respectively,
where and . We assume that and are the initial values for the orbit of system (9). For any given (), then from the first equation of system (9) we have at . For example, setting , we have . Thus taking and as initial values, respectively, we simulate the integral curves of (8) as (a) and (b) in Figure 2.
From (11), the two heteroclinic orbits (see Figure 1(d)) passing through the saddle point have expressions, respectively,
where and . For any given (), then from the first equation of system (9) we have at . For example, setting , we have . Thus taking and as initial values, respectively, we simulate the integral curves of (8) as (c) and (d) in Figure 2.
From (11), the orbit (see Figure 1(d)) passing through has expressions
where , , and . Choosing and taking and as initial values, respectively, we simulate the integral curve of (8) as (e) in Figure 2.
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Example 2. For Case 5, taking , then , , , and . From (11), the orbit (see Figure 1(e)) passing through has expressions where , , and . Taking and as initial values, respectively, we simulate the integral curve of (8) as (f) in Figure 2.
Remark 3. The kink-like waves in Figures 2(a) and 2(b) are defined on and , respectively. The kink-like waves in Figures 2(c) and 2(d) are defined on and , respectively. The compacton-like wave in Figure 2(e) has peak form on , where , , and satisfy where , , and . Take the data of Example 1, that is, , , , , , , and , then from (18) we obtain , , and which are identical with the simulations (see Figures 2(a)–2(e)).
Remark 4. The compacton-like wave in Figure 2(f) is defined on , where satisfies where . Take the data of Example 2; that is, , , , and , and then from (19) we obtain which is identical with the simulation (see Figure 2(f)).
Remark 5. For Cases 4 and 5, there are a family of periodic orbits surrounding the center point , but the boundaries of the periodic orbits are different. For Case 4, the boundaries of the periodic orbits are the two heteroclinic orbits (see Figure 3(a)), while for Case 5, the boundary of the periodic orbits is the open orbit (see Figure 3(b)). Taking the data of Example 1 and a set of initial values , that is, , and , we simulate the periodic orbits of (8) as Figure 4. Similarly, taking the data of Example 2 and a set of initial values , that is, , , and ,, we simulate the periodic orbits of (8) as Figure 5. The simulations in Figure 4 imply that the periodic waves tend to two kink-like waves when the periodic orbits tend to the heteroclinic orbits . The simulations in Figure 5 imply that the periodic waves tend to the periodic compacton-like wave when the periodic orbits tend to the open orbit .
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4. The Expressions of Kink-Like and Compacton-Like Waves
In this section we derive the exact expressions of the kink-like and compacton-like waves in different cases ( 1–16). Assuming that is the initial point of an orbit of system (9). Let where , then form , the following equation: determines the orbit passing through .
4.1. Solutions of Kink-Like Wave
In Cases ( = 1–4) corresponding to phase portraits in Figures 1(a)–1(d), (20) becomes where , . Thus the orbits passing through saddle point have expressions where . Substituting (23) into and integrating along and for initial value , where , we obtain two kink-like wave solutions of implicit expression as follows: where
The derivations of other kink-like wave solutions are similar to the above case, so we omit the details and only list the results.
In Cases () corresponding to phase portraits in Figures 1(e) and 1(f), we obtain two kink-like wave solutions of implicit expression as follows: where
In Case 7 corresponding to phase portrait in Figure 1(g), we obtain two kink-like wave solutions of implicit expression as follows: where
In Case 11 corresponding to phase portraits in Figure 1(k), we obtain two kink-like wave solutions of implicit expression as follows: where
In Case 12 corresponding to phase portraits in Figure 1(l), we obtain two kink-like wave solutions of implicit expression as follows: where
In Cases () corresponding to phase portraits in Figures 1(m), 1(o), and 1(p), we obtain two kink-like wave solutions of implicit expression as follows: where
In Case 14 corresponding to phase portrait in Figure 1(n), we obtain two kink-like wave solutions of implicit expression as follows: where
In Case 4 corresponding to phase portrait in Figure 1(d), we obtain two kink-like wave solutions of implicit expression as follows: where and
In Cases ( = 5–12) corresponding to phase portraits in Figures 1(e)–1(l), we obtain two kink-like wave solutions of implicit expression as follows: where
4.2. Solutions of Compacton-Like Wave
In Cases ( = 1–4) corresponding to phase portraits in Figures 1(a)–1(d) and , (20) becomes where and are real roots and and are conjugate complex roots of and . Thus the orbit has expressions: where . By applying transformation to , we have
Substituting (43) into (44) and integrating along , we get where
Substituting (45) into and integrating once, we get where
Thus we obtain a compacton-like wave solution of parametric expression as follows: where is a parameter variable and , .
The derivations of other compacton-like solutions are similar to the above case, so we omit the details and only list the results.
In Cases ( = 7–10) corresponding to phase portraits in Figures 1(g)–1(j), we obtain a compacton-like wave solution of parametric expression as follows: where is a parameter variable, , and , ,
In Cases () corresponding to phase portraits in Figures 1(e) and 1(f), we obtain a compacton-like wave solution of implicit expression as follows: where
In Case 5 corresponding to phase portraits in Figure 1(e), we obtain a compacton-like wave solution of implicit expression as follows: where is a root of equation and
In Case 6 corresponding to phase portraits in Figure 1(f), we obtain a compacton-like wave solution of implicit expression as follows: where
In Case 16 corresponding to phase portraits in Figure 1(p), we obtain a compacton-like wave solution of implicit expression as follows: where
5. Conclusion
In this paper, we have found two types of bounded traveling wave solutions for a two-component Fornberg-Whitham equation (see (4)), that is, the compacton-like wave and kink-like wave solutions. Their planar graphs of these solutions are simulated by using software Mathematica (see Figure 2). Meanwhile we reveal two kinds of new phenomena; that is, the periodic wave solution can become the kink-like wave or compacton-like wave solution under some conditions, respectively (see Figures 4 and 5). In different Cases ( = 1–16), the exact implicit or parameter expressions of kink-like wave and compacton-like wave solutions are given.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
This paper is supported by the National Natural Science Foundation (no. 11361069), and Science Foundation of Shaoguan University (no. 201320501).