Eight third-order differential equations and two fifth-order differential equations are derived by Fujimoto and Watanabe,^[1] and among these equations, one highly nonlinear Fujimoto–Watanabe equation possesses the following form 1 where α is a parameter, when they classified the third-order polynomial evolution equations of uniform rank with non-constant separants, which admit generalized symmetries (non-trivial Lie–Bäcklund algebras). Fujimoto and Watanabe^[1] have shown that equation (1) possesses the hereditary recursion operator of Lie–Bäcklund symmetries, which means that the Lie algebras of its Lie–Bäcklund symmetries are infinite-dimensional and commutative. However, they did not prove its integrability.^[2] Sakovich^[3] further showed that these differential equations can be connected with the famous KdV equation, Qiao equation,^[4] and so on, via chains of differential substitutions. Note that there have little studies about Eq. (1), because of its high nonlinearity (precisely, quartic nonlinearity), in our opinion. Here we find a way to handle its high nonlinearity, according to its specific structures. Shi and Wen^[5,6] studied traveling wave solutions to two of these Fujimoto–Watanabe equations and showed that these traveling wave solutions possess abundant dynamical behaviors. One may consider what about the dynamics of traveling wave solutions to other Fujimoto–Watanabe equations. Driven by this motivation, in this work, we aim at dynamics of traveling wave solutions to Eq. (1) from the the perspective of the theory of dynamical systems.^[7–14]

More precisely, we first transform Eq. (1) into a planar system by substituting with into Eq. (1), which yields 2 where the prime stands for the derivative with respect to ξ.

Dividing both sides of Eq. (2) by and then integrating it once, we obtain 3 where g is the integral constant.

Letting , we obtain a three-parameter planar system 4 with first integral 5

Note that if c = 0, system (4) becomes a different system 6 the phase portraits of which should be analyzed individually.

Of particular interest of this paper is to study complex dynamics of traveling wave solutions to system (4) in the three-dimensional parameter space (α, c, g) and to show their exact expressions and simulations, which can fascinate the underlying physical behaviors of traveling wave solutions to Eq. (1).

Note that system (4) is not well-defined on the line l:φ=0. Therefore, we consider the associated system of system (4) as follows: 7 where .

We can analyze the phase portraits of system (4) from those of system (7), since system (7) has the same level curves as system (4).

To study the singular points and their properties of system (7), let 8

We can easily obtain the graphics of the function f(φ) in Figs. 1 and 2 in corresponding parametric space.

Fig. 1.

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	Fig. 1. The graphics of f(φ) when and . (a) , (b) , (c) , (d) c = 0, (e) , (f) , and (g) .

Fig. 2.

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	Fig. 2. The graphics of f(φ) when and . (a) , (b) , (c) , (d) c = 0, (e) , (f) , and (g) .

Let λ(φ,y) be the characteristic value of the linearized system of system (7) at the singular point (φ,y). We easily obtain 9

From Eq. (9), we see that the sign of can determine the dynamical properties (saddle, center, and degenerate singular point) of the singular point according to the theory of planar dynamical systems.

Now we summarize the singular points of system (7) and their relative positions when , in the following lemma. We just omit the other case when , , since it can be analyzed similarly.

Lemma 1 When , , the singular points of system (7) are as follows.

(i) When , system (7) has only one singular point with .

(ii) When , system (7) has two singular points and with .

(iii) When , system (7) has three singular points , and with .

(iv) When c = 0, system (7) has three singular points , and with .

(v) When , system (7) has three singular points , and with .

(vi) When , system (7) has two singular points and with .

(vii) When , system (7) has only one singular point with .

Through qualitative analysis, we obtain all possible bifurcations of phase portraits of system (4) in Figs. 3 and 4, respectively.

Fig. 3.

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	Fig. 3. The phase portraits of system (5) when and . (a) , (b) , (c) , (d) c = 0, (e) , (f) , and (g) .

Fig. 4.

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	Fig. 4. The phase portraits of system (5) when and . (a) , (b) , (c) , (d) c = 0, (e) , (f) , and (g) .

Our main results will be stated in the following theorems, and the proofs follow.

Theorem 1 When and , one has

I) When or , there exist a family of periodic wave solutions to Eq. (1). When , there exist two families of periodic wave solutions to Eq. (1).

II) When or , there exist a pair of kink-like (antikink-like) wave solutions to Eq. (1). Specially, taking , the pair of kink-like (antikink-like) wave solutions possess the expression 10 where .

III) When or , there exists a solitary wave solution to Eq. (1). Specifically, when , the solitary wave solution possesses the expression 11 where and with .

IV) When or , there exist a family of compactons to Eq. (1). When or , there exist two families of compactons to Eq. (1).

Proof

(I) The statements follow from the fact that periodic orbit gives rise to periodic wave solution.

(II) The stable and unstable manifolds, on the left side of singular point in Figs. 3(b) and 3(c), and on the right side of singular point in Figs. 3(e) and 3(f), respectively, give rise to kink-like (antikink-like) wave solutions to Eq. (1).^[15–19] Here we only illustrate the case when .

The stable and unstable manifolds on the left side of singular point in Fig. 3(b) can be expressed as 12

Substituting Eq. (12) into and integrating them along the stable and unstable manifolds with , it follows that 13

From Eq. (13), we gain the kink-like (antikink-like) wave solutions as Eq. (10), the graphics of which are shown in Figs. 5(a) and 5(b).

Fig. 5.

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	Fig. 5. The graphics of kink-like (antikink-like) wave solutions to Eq. (1): (a) and (b) .

(III) When , from Fig. 3(c), we see that there is one homoclinic orbit for system (4), the expressions of which are given as 14

Substituting Eq. (14) into and integrating them along the homoclinic orbit, it follows that 15

From Eq. (15), we get solitary wave solution Eq. (11).

(IV) We only illustrate the cases when in Fig. 3(a) and in Fig. 3(b), since the other cases can be obtained similarly.

Corresponding to the one family of orbits, passing through the point with in Fig. 3(a), equation (1) has a family of compactons as shown in Fig. 6(a).

Fig. 6.

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	Fig. 6. The graphics of compactons to Eq. (1): (a) , (b) , and (c) .

Corresponding to the two family of orbits, passing through the point with in Fig. 3(b), equation (1) has two families of compactons as shown in Figs. 6(b) and 6(c).

Similarly, we can get the following results.

Theorem 2 When and , one has

I) When or , there exist a family of periodic wave solutions and a solitary wave solution to Eq. (1).

II) When or , there exist a pair of kink-like (antikink-like) wave solutions and a family of compactons to Eq. (1).

Remark 1 It seems that the form of Eq. (1) is simpler than the equations studies in Refs. [5] and [6]. However, the traveling wave system Eq. (7) is more complicated than those in Refs. [5 and 6] and possesses more abundant dynamics, such as the the appearance of nodes in the phase portraits.

In this paper, through all possible bifurcations for the system in parameter space, we not only show the existence of traveling wave solutions including solitary wave solutions, periodic wave solutions, kink-like (antikink-like) wave solutions, and compactons, under corresponding parameters conditions, but also give their exact expressions and simulations.