THE BIFURCATIONS OF SOLITARY AND KINK WAVES DESCRIBED BY THE GARDNER EQUATION

. In this paper, we investigate the bifurcations of nonlinear waves described by the Gardner equation u t + auu x + bu 2 u x + γu xxx = 0. We obtain some new results as follows: For arbitrary given parameters b and γ , we choose the parameter a as bifurcation parameter. Through the phase analysis and explicit expressions of some nonlinear waves, we reveal two kinds of important bifurcation phenomena. The ﬁrst phenomenon is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with elliptic expression and trigonometric expression respectively. The second phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.


1.
Introduction. It is well known that some phenomena in physics and engineering can be described by nonlinear partial differential equations. When we try to study the physical mechanism of the phenomena described by nonlinear partial differential equations, nonlinear wave solutions and their bifurcations often are investigated. Through study of bifurcation for a equation containing some parameters, we may know some varying process of the solutions when the parameters vary. Thus the study of bifurcation may give a good insight into the physical aspects of the problems.
In this paper, we are interested at the Gardner equation where u(x, t) is the amplitude of the relevant wave mode, the dispersion coefficient γ is always positive, but the nonlinear coefficients a and b are positive or negative [7]. Eq. (1) is widely applied in various branches of physics, such as fluid physics and plasma physics, and attract many scholars' attention [20], [21] and [9]. This equation arose as an auxiliary mathematical equation in the derivation of the infinite set of conservation laws of the KdV equation [15] and [16]. In [7], the authors pointed out the relationship between Eq. (1) and the mKdV equation v t + bv 2 v y + γv yyy = 0 on the basis of an invertible transformation v = u+a/2b, y = x+a 2 t/2b 2 . Furthermore, by means of the Mirura transformation the mKdV equation can be reduced to the KdV equation [17]. The exact solutions of Eq. (1) have been investigated by many researchers and some of powerful methods have been presented, such as the series expansion method [4], the mapping method [13], the Hirota methods [22], and so on [23], [14], [10], [6], [5] and [8]. The authors in [1] and [18] acquired some exact solution under the special parameters. However, the bifurcations of the nonlinear waves for Eq. (1) has been few discussed and understood. Hence it is the main investigation of our paper by using phase analysis which was used in some lectures, for instance [12], [19], [11], [2] and [3]. In this paper, for arbitrary given parameters b and γ, choosing a as bifurcation parameter, we show that in Eq. (1) there exist two kinds of important bifurcation phenomena. One of the bifurcation phenomena is that the solitary waves with fractional expressions can be bifurcated from three types of nonlinear waves which are solitary waves with hyperbolic expression and two types of periodic waves with trigonometric expression and elliptic expression respectively. Another phenomenon is that the kink waves can be bifurcated from the solitary waves and the singular waves.
This paper is organized as follows. In Sec. 2, our main results are stated in a propasition. In Sec. 3, we give proof to our main results. A short conclusion is put in Sec. 4.

2.
Main results. In this section, we state our main results. For given parameter b and constant c, let and Obviously, a 1 , a 2 , a 3 and a 4 satisfy inequality In the following proposition, we will show that a 2 , a 3 are two bifurcation parametric values for solitary wave bifurcation, and a 1 , a 4 are two bifurcation parametric values for kink wave bifurcation.
Proposition 1. For given parameters b, γ (γ > 0) and constant c, let ξ and a i (i = 1, 2, 3, 4) be in (2) and (3). Choosing a as bifurcation parameter, then in Eq. (1), there exist solitary wave and kink wave bifurcations as follow: A. If b > 0 and c < 0, then there is the following solitary wave bifurcation. (A 1 ) When a 2 < a < a 1 and a → a 2 + 0, the peak solitary wave with fractional expression can be bifurcated from the following three types of nonlinear waves: (A 1 ) a Solitary wave with hyperbolic expresstion where where where For the varying processes of u • 2 (ξ) and u • 3 (ξ), see Fig. 1. (A 2 ) When 0 < a < a 2 and a → a 2 − 0, the peak solitary wave with fractional expression u 1 (ξ) (see (5)) can be bifurcated from only one type of periodic wave which has elliptic expression where For the varying process of u • 4 (ξ), see Fig. 2. (A 3 ) When a 3 < a < 0 and a → a 3 + 0, the valley solitary wave with fractional expression can be bifurcated from the periodic wave with elliptic expression u • 4 (ξ) (see (32)). For the varying process of u • 4 (ξ), see Fig. 3. (A 4 ) When a 4 < a < a 3 and a → a 3 − 0, the valley solitary wave with fractional expression u 2 (ξ) (see (44)) can be bifurcated from following three types of nonlinear waves.
bifurcation parameter a tends to these values, the kink waves can be bifurcated from the solitary waves and the singular waves listed in proposition 1.B.