Periodic Wave Solutions and Their Asymptotic Property for a Modiﬁed Fornberg–Whitham Equation

: Recently, periodic traveling waves, which include periodically symmetric traveling waves of nonlinear equations, have received great attention. This article uses some bifurcations of the traveling wave system to investigate the explicit periodic wave solutions with parameter α and their asymptotic property for the modiﬁed Fornberg–Whitham equation. Furthermore, when α tends to given parametric values, the elliptic periodic wave solutions become the other three types of nonlinear wave solutions, which include the trigonometric periodic blow-up solution, the hyperbolic smooth solitary wave solution, and the hyperbolic blow-up solution.

Since the appearance of the Camassa-Holm equation Equation (2), a huge amount of work has been carried out to study the dynamic properties of Equation (2). Equation (2) has been proved to possess the global existence, the precise blow-up scenario, the blow-up set and the blow-up rate for the strong solutions [7][8][9][10]. It has also been confirmed that the peakon of Equation (2) are orbitally stable [11,12].
In [13], Liu and Qian suggested a generalized Camassa-Holm equation Similarly, by softening the nonlinear term, He et al. [14] studied the peakons and solitary waves for the modified Fornberg-Whitham equation However, little attention has been given to the periodic traveling wave solutions in their study.
Recently, periodic traveling waves of nonlinear equations have received great attention. For instance, Angulo et al. [15] mentioned that the cnoidal waves of KdV equation converge to the limit soliton when the period tends to infinity. The detailed study was presented by Neves [16]. In [17], the periodic asymptotics of a class of stationary nonlinear Schrödinger equations has been studied together with the existence of dark soliton. The authors [18] showed that the limit forms of the periodic loop solutions of the Kudryashov-Sinelshchikov equation contained loop soliton solutions, smooth periodic wave solutions, and periodic cusp wave solutions.
In this paper, we study the explicit periodic wave solutions and their asymptotic property for Equation (4) using bifurcation analysis [19][20][21][22][23][24][25][26]. Also, some periodic wave solutions are symmetric [27]. First, we obtain two types of explicit periodic wave solutions, elliptic smooth periodic wave solutions and periodic blow-up solutions with a parameter α. Secondly, we reveal that there exist four parametric values. When α tends to these parametric values, these elliptic periodic wave solutions can become other three types of nonlinear wave solutions, the hyperbolic smooth solitary wave solutions, the hyperbolic blow-up solutions and the trigonometric periodic blow-up solutions.
This paper is organized as follows. In Section 2, we give some preliminaries. Our main results are listed in Section 3. In Section 4, we provide derivation to our main results. A short conclusion is given in Section 5.

Preliminaries
To derive our results, we give some preliminaries in this section. For given constant c, substituting u = ϕ(ξ) with ξ = x − ct into Equation (4), it follows that Integrating (5) once, we have where g is an integral constant. Letting y = ϕ , we get a planar system with the first integral where h is another integral constant.
Assuming that α and c (double root) are two real roots of the equation we get its other two roots β and γ of forms and Solving equations β = γ, α = β, and α = γ respectively, we get the four numbers α i (i = 1 − 4) of forms where From above expressions we get the following lemma.

Our Main Results
In this section, we state our main results. The pictures of Proposition 1 are included in the Appendix A.

The Derivation of Main Results
In this section, we give the derivation for our main results listed in Proposition 1. First, we derive u i (ξ, α) (i = 1, 2) and their limit forms.
Since the period of the function sn(τ, k) is 4K, it follows that the period of the function sn(η 2 ξ, k 2 ) is 4K/η 2 . Now we derive the limit forms. First, we derive the limit forms 2 a . From the expressions (25)-(27), we have the following limits.

Conclusions
In this paper, we have studied the explicit smooth periodic wave solutions and periodic blow-up solutions and their asymptotic property for Equation (4). In Proposition 1, the explicit expressions of these solutions and their limits have been shown. Based on these results, Equation (4) possesses explicit periodic wave solutions, and solitary wave solution has been exposed. Furthermore, we have found that the periodic blow-up solution u 1 (ξ, α) can converge to the smooth solitary wave solution u • 2 (ξ). On the other hand, this example shows that not only the cnoidal wave solution but also the periodic blow-up solution can converge to the smooth solitary wave solution.