BIFURCATIONS AND EXACT TRAVELLING WAVE SOLUTIONS FOR A SHALLOW WATER WAVE MODEL WITH A NON-STATIONARY

We consider a shallow water wave model with a non-stationary bottom surface. By applying dynamical system approach to the model problem, we are able to obtain all possible bounded solutions (compactons, solitary wave solutions and periodic wave solutions) under different parameter conditions. More than 19 exact parametric representations are provided explicitly.


Introduction
It is well known that the equations for shallow water waves were developed using approximations, and there have been a number of different formulations that were developed in the literature. The interesting history of various formulations was discussed in [2]. Recently, in [4], the author pointed out that the Serre's nonlinear shallow water wave equations developed in ( [9,10]) for uniform depth and later generalized by Seabra-Santos et al. [11] for non-uniform depth are limited to a stationary bottom surface and a uniform pressure applied to the top surface, while the Green-Naghdi nonlinear shallow water wave equations developed by Green and Naghdi [3] are valid for a non-stationary, non-uniform bottom surface and a nonuniform pressure on the top surface. To be specific, in 2019, Kogelbauer and Rubin [4] developed a class of exact nonlinear traveling wave solutions of the Green-Naghdi equations for a non-stationary and non-uniform bottom surface. The traveling wave equation is given by (see [4], (3.9)): where φ is the current depth of the fluid depending on ξ = x − ct, the constants a,H parameterize the class, with a specified andH determined by a critical value of the depth φ, the Froude number F defined by F 2 = k 2 1 gH 3 . For convenience, we writeβ = (1+a) F 2H 3 . Expanding equation (1.1) and dividing the result by 2φ, one has (see [4], (3.11)): Integrating equation (2) once with respect to ξ, we obtain an equivalent planar dynamical system where g is an integral constant and β = 6β. Clearly, system (1.3) is a singular traveling wave system of the first class of the same type as [6][7][8]12], with the singular straight line φ = 0. system (1.3) has a first integral: when a = −1, i.e., β = 0, and when a = ± 1 √ 3 . Especially, at the time that a = 0, − 1 2 and a =ã ± = 1 2 (1 ± √ 5), for the first integral becomes and Hã + (φ, y) = φ 2 y 2 + (7 − 3 √ 5) 6 (βφ 4 − 9gφ 2 + 9) = h, (1.10) (βφ 4 − 9gφ 2 + 9) = h, (1.11) respectively. We would like to point out that in [4], Kogelbauer and Rubin did not investigate the dynamical behavior of system (1.3), while is the main focus in this work. More precisely, by considering the dynamics of the travelling wave solutions determined by system (1.3), we shall generally give all possible exact travelling wave solutions explicitly for equation (1.1) under different parameter conditions (see, e.g., [5][6][7][8]12]). More than 19 exact parametric representations are obtained by using the elliptic functions and hyperbolic functions.
The rest of this paper is organized as follows. In section 2, we discuss bifurcations of phase portraits of system (1.3). In section 3 and section 4, corresponding to all bounded orbits given in section 2, we give all possible exact parametric representations of the travelling wave solutions for equation (1.1) explicitly.

then it is a node;
(iv) if J = 0 and the Poincare index of the equilibrium point is 0, then it is a cusp.
Thus, if there exist two equilibrium points S ∓ (0, ∓ y s ), then, they are both nodes because of (1 − 3a 2 ) > 0. We write that h j = H(φ j , 0) for j = 1, · · · , 4. Taking the above fact into account, we know that for all g > 0 satisfying ∆ > 0, the parameter a can be taken as a bifurcation parameter such that a = −1, − 2 3 , ∓ 1 √ 3 are bifurcation values. As a increases, we have the bifurcations of phase portraits of system (1.3) shown in Fig.1 it gives rise to the two families of closed orbits (see Fig.1

5)
In this section, we consider the exact parametric representations of traveling wave solutions of equation (1.1) given by the bounded orbits in Fig.1. 3.1. The case a = −1 (see Fig.1

(c))
In this case, it deduces from equation (1.2) that Thus, from β = 6β and first equation of system (1.3), equation (3.2) gives rise to the curve equation Clearly, for β > 0, g > 0, equation (3.2) defines two closed curves enclosing the points E ± ± 3g±∆ β , 0 , respectively. Equation (3.2) can be rewritten as . Hence, we have the exact parametric representations of the two families of periodic wave solutions of equation (1.1) as follows: Fig.1
We see from equation (1.11) that By using the first equation of system (1.3), we have where ω 0 = (7+3 √ 5)β 6 . We can use equation (3.5) to calculate all parametric representations of the bounded orbits of system (1.3) in Fig.2.
(i) Corresponding to the two periodic orbit families defined by where r 3 < 0 < r 2 < φ 2 4 < r 1 . Hence, we obtain the exact parametric representations of the two families of periodic wave solutions of equation (1.1) (see Fig.3 (a)): where . We notice that the existence of uncountably infinitely many bounded breaking wave solutions is an important property of a singular travelling wave system. For example, we consider the two families of open curves defined by Hã − (φ, y) = h with h ∈ (0, h 3 ), (h 3 , ∞) (see Fig.2 (b), (d)), which lie on the two sides of the singular straight line φ = 0, respectively. Obviously, along each open curve as |ξ| increases, φ(ξ) approaches to φ = 0 and φ(ξ) → 0, |y(ξ)| → ∞. By the theory of singular traveling wave systems developed in [6], we know that these open curves give rise to uncountably infinitely many bounded two-sided breaking wave solutions of φ(ξ). In other words, these traveling wave solutions have compact supports, that is, they vanish identically outside finite core regions. Such compact support solutions are called compactons.
(ii) Corresponding to the two periodic orbit families and the two open curve families which approach the straight line φ = 0 when |y| → ∞ (see Fig.2  For the two open curve families, we obtain the two families of compacton solution of equation (1.1) as follows (see Fig.3 (c)):

2A1
,  Similarly, for a = 1 2 (1 +   In this section, we consider the exact parametric representations of traveling wave solutions of equation (1.1) given by the bounded orbits in Fig.1 (f). In this case, we have When h varies, the changes of the level curves defined by H 0 (φ, y) = h are shown in Fig.4 (a)-(g). It follows from equation (1.8) that y 2 = 1 3 β 9 β + 3h β φ + 18g β φ 2 − φ 4 . Thus, it deduces from the first equation of system (1.3) that By using equation (4.1), we can obtain all parametric representations of the bounded orbits of system (1.3). We see from Fig.4 that there exist two node points of the associated regular system (2.1) on the singular straight line φ = 0, then the singular system has no peakon, periodic peakon and compacton solutions. In this case, the traveling wave system has no curve triangle surrounding a periodic annulus of a center. Instead, there exist smooth periodic waves, solitary waves of the singular system, because the singular system and its associated regular system define different vector fields respectively. As shown by Fig.1 (f), on the left-hand side of the singular straight line φ = 0, the direction of the orbits of the vector field defined by the singular system (1.3) is the inverse direction of the orbits of the vector field defined by its associated regular system (2.1).
(i) Corresponding to the level curves defined by H 0 (φ, y) = h, h ∈ (−∞, h 4 ], there exists a family of periodic orbits of system (1.3) (see Fig.4 (a)), with its boundary passing through two equilibrium points S ± (0, ± y s ). Now, equation (4.1) can be written as β . Therefore, we obtain the following exact parametric representation of a periodic wave solution family of equation (1.1): and (ii) Corresponding to the level curves defined by H 0 (φ, y) = h, h ∈ (h 4 , h 3 ), there exist the two families of periodic orbits of system (1.3), enclosing the equilibrium points E 1 (φ 1 , 0) and E 4 (φ 4 , 0), respectively (see Fig.4 (b)). For the right family of periodic orbits, equation (4.1) can be written as β .
(v) Corresponding to the level curves defined by H 0 (φ, y) = h 2 , there are two homoclinic orbits of system (1.3) enclosing two equilibrium points E 4 (φ 4 , 0) and E 1 (φ 1 , 0), respectively (see Fig.4 (e)). Now, equation (4.1) can be written as , respectively. Thus, we have the following solitary wave solutions of equation (1.1): and where As for a = − 1 2 , we have similar results as the above conclusions. With the above estimates at hand, we give the following main results. (ii) Equation (1.1) has the exact solitary wave solutions given by (4.5)-(4.8).