When Fujimoto and Watanabe^[1] classified the third-order polynomial evolution equations of uniform rank with non-constant separants, which admit generalized symmetries (non-trivial Lie–Bäcklund algebras), they derived eight third-order differential equations and two fifth-order differential equations. Sakovich^[2] showed that these differential equations can be connected with the famous KdV equation, Qiao equation,^[3] and so on. Shi and Wen^[4] studied the bifurcation and dynamics of traveling wave solutions to one of the Fujimoto–Watanabe equations. It is well known that traveling wave solution is an important type of solution to nonlinear wave equations. Many methods have been employed to find exact traveling wave solutions of nonlinear wave equations, such as, Jacobi elliptic function method,^[5] F-expansion method,^[6] (G′/G)-expansion method,^[7] Bäcklund transformation method,^[8] the simplified Hirota’s method,^[9] and so on. Our goal in this paper is to seek traveling wave solutions to the following Fujimoto–Watanabe equation:

More precisely, we look for the traveling wave solutions to Eq. (1) through the transformation u(x,t) = φ(ξ) with ξ = x – ct, where c is the wave speed. Substituting it into Eq. (1) yields

Dividing both sides of Eq. (2) by φ² and integrating it once, we obtain

Letting y = φ′, we obtain a three-parameter planar system

The purpose of this paper is to investigate the dynamical behaviors of traveling wave solutions to system (4) in the three-dimensional parameter space (α,c,g) and to show traveling wave solutions to Eq. (1).

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

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

We suppose that g > 0. Let , , and . Now we summarize the singular points of system (7) and their relative positions when α > 0 in the following lemma. We just omit the other case when α < 0, since it can be analyzed similarly.

Through qualitative analysis, we obtain all possible bifurcations of phase portraits of system (4) as shown in Figs. 1 and 2.

Fig. 1.

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	Fig. 1. (color online) The phase portraits of system (4) when α > 0.

Fig. 2.

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	Fig. 2. (color online) The phase portraits of system (4) when α < 0.

To state conveniently, we introduce some marks , , , , .

Our main results will be stated in the following theorem, and the proof follows. Note that we only focus our attention on the case when α > 0, because the other case when α < 0 can be considered similarly.

Fig. 3.

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	Fig. 3. The (a) solitary wave solution and (b), (c) kink-like (antikink-like) wave solutions to Eq. (1).

The stable and unstable manifolds on the left side of singular point (φ^*,0) in Fig. 1(b) can be expressed as

(iv) We only illustrate the cases when and in Figs. 1(a) and 1(c), since the other cases can be obtained similarly.

Corresponding to the one family of orbit, passing through the point (φ₀,0) with φ₀ ∈ (0,+∞) in Fig. 1(a), equation (1) has a family of compactons shown in Fig. 4(a).

Fig. 4.

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	Fig. 4. (color online) The graphics of compactons to Eq. (1).

Corresponding to the two family of orbits, passing through the point (φ₀,0) with φ₀ ∈ (0,φ₋)∪(a²,+ ∞) in Fig. 1(c), equation (1) has two families of compactons shown in Figs. 4(b) and 4(c).

Through all possible bifurcations for the system in three-dimensional 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.