Solitons and Bifurcations for the Generalized Tzitzéica Type Equation in Nonlinear Fiber Optics ()
1. Introduction
The Tzitzéica type equations contain the following four nonlinear evolution equations [1] - [6]
(1.1)
and
(1.2)
and
(1.3)
and
(1.4)
which play an important role in nonlinear fiber optics. Equations (1.1), (1.2), (1.3) and (1.4) present the Tzitzéica, Liouville, Dodd-Bullough-Mikhailov and Tzitzéica-Dodd-Bullough equations, respectively. For instance, Kumar [1] studied the Tzitzéica type equations by the method of sine-Gordon expansion. In [7] , with the help of generalized exponential rational function method, the authors researched Equations (1.1) and (1.4). Zafar [8] discussed Equations (1.1), (1.3) and (1.4) by the Painlevé transformation and the simplest equation method. Hosseini [9] studied the Tzitzéica type equations using the expα function method. In [10] , the
-expansion method was applied for constructing more general exact solutions of the Tzitzéica type equations. In [11] , the improved
-expansion method was used to study the Tzitzéica type equations and the dispersive dark optical solitons were obtained. For more literature, we can refer to [12] - [19] .
Obviously, there is no single way to solve the soliton solutions of all nonlinear evolution equations, which leads to the generation of many new methods, such as the
-expansion method and the sine-cosine method [20] , the functional variable method and first integral approach [21] , the semi-inverse variational principle [22] , Lie symmetry approach [23] , the method of F-expansion [24] [25] , the modified simple equation method and the trial equation method [26] [27] , the asymmetric method [28] , the Gaussian ansatz [29] [30] , the soliton perturbation theory [31] [32] , the simplest equation method and the
-expansion method [33] [34] , the theory of dynamical systems [35] [36] [37] [38] .
In this paper, We will devote ourselves to research the following generalized Tzitzéica type equation
(1.5)
where
and
are real constants. As far as we know, Equation (1.5) has not been studied. In particular, we notice that some well-known equations are included in Equation (1.5). When
,
,
and
, Equation (1.5) reduces to the Tzitzéica equation. When
and
, Equation (1.5) reduces to the Liouville equation. When
and
, Equation (1.5) reduces to the Dodd-Bullough-Mikhailov equation. When
,
and
, Equation (1.5) reduces to the Tzitzéica-Dodd-Bullough equation. Using the theory of dynamical systems, the bounded solitons and bifurcations of Equation (1.5) will be discussed. These conclusions are new and have important applications in mathematics and physics. It can help us understand many experiments in physics, and can reveal the laws of motion and the objective nature of physical experiments.
2. Traveling Wave Transformation and First Integral
Introducing the following function transformation
(2.1)
where
. Without loss of generality, we only discuss the traveling wave solutions when
.
Substituting Equation (2.1) into Equation (1.5), we get
(2.2)
In order to seek traveling wave solutions of Equation (2.2), we assume that
(2.3)
where c is the wave speed. Substituting Equation (2.3) into Equation (2.2), we get
(2.4)
Let
, then Equation (2.4) is equivalent to the following two dimensional singular traveling wave system
(2.5)
where
,
,
.
System (2.5) has the following Hamilton function
(2.6)
where h is a real constant.
3. Bifurcation of Parameters and Phase Portraits
Firstly, we study the following associated regular system of system (2.5)
(3.1)
where
. System (2.5) and system (3.1) have the same Hamiltonian function. Consequently, system (2.5) and system (3.1) have the same dynamic properties except for the straight line
. Near the straight line
, the dynamics of the solutions for system (2.5) usually change abruptly.
Define
Discussing the equilibrium points of the system (3.1), we get the following proposition.
Proposition 3.1
(i) When
, there exists a equilibrium point
on the v-axis.
(ii) When
,
is a monotone function. Therefore there is only one equilibrium point on the v-axis.
(iii) When
,
, there is one equilibrium point on the v-axis.
(iv) When
,
, there exist three different equilibrium points on the v-axis.
(v) When
and
, there exist two equilibrium points on the v-axis, one of which is a double equilibrium point.
(vi) When
,
, there is a triple equilibrium point
on the v-axis.
(vii) When
,
, there is a equilibrium point
on the v-axis.
(vii) When
, there are two equilibrium points
on the y-axis. When
, there is no equilibrium point on the y-axis. When
, there is a double equilibrium point
on the y-axis.
Suppose
is the Jacobian matrix of the linearized system (3.1) and defines
(3.2)
From (3.2), we know that
According to the above analysis, the parameter condition
is obtained from
,
. Let
,
. Thus there are three bifurcation curves
,
and
in the
-parameter plane, as shown in Figure 1. Using the bifurcation theory of differential equation [35] [36] [37] [38] , we obtain the phase portraits of system (2.5), as shown in Figure 2.
4. Exact Traveling Wave Solutions of Equation (2.4)
In this section, we will use (2.6) and the Maple software to obtain the bounded traveling wave solutions of Equation (2.4). According to the relationship between
Figure 1. The parameter regions partitioned by bifurcation curves in the
-plane.
the solutions of Equation (1.5) and Equation (2.4), we know that
. Thus we can obtain traveling wave solutions of Equation (1.5). We agree to take only the part of
as the solutions of Equation (2.4), and we will not repeat it.
From (2.6), we obtain
(4.1)
Substituting into the first equation of system (2.5), we get
(4.2)
(i) When
,
, corresponding to Figure 2(2). When
, there exist a family of periodic orbits defined by
. From Equation (4.2), we obtain
(4.3)
Thus we get the following solutions, as shown in Figure 3 and Figure 4.
(4.4)
and
(4.5)
(ii) When
,
, corresponding to Figure 2(4). When
, the curves defined by
are a family of periodic orbits. From Equation (4.2), it follows
(4.6)
From Equation (4.6), we get the parameter expressions for
and
as follows
(4.7)
and
(4.8)
(iii) When
,
,
, corresponding to Figure 2(8). There exist a center point
on the v-axis. When
, the curves defined by
is a family of periodic orbits surrounding the center point
. From Equation (4.2), then we get
(4.9)
where
,
.
From Equation (4.9), we get
(4.10)
and
(4.11)
(iv) When
,
,
, corresponding to Figure 2(10). There exist a saddle point
on the v-axis. When
, there exist a family of homoclinic orbits corresponding to a family of bright solitons of Equation (2.4). From Equation (4.2), we obtain
(4.12)
From Equation (4.12), we get the following parameter expressions for
and
, as shown in Figure 5 and Figure 6.
(4.13)
and
(4.14)
(v) When
,
,
or
,
,
or
,
,
or
,
,
or
,
,
, corresponding to Figure 2(12), Figure 2(21), Figure 2(23), Figure 2(25), Figure 2(34), respectively. There exists a center equilibrium point
on the right of y-axis, respectively. When
, there exists a family of periodic orbits defined by
, respectively. Especially, when
,
,
, then
. From Equation (4.2), we have
(4.15)
where
, and satisfies
,
.
From (4.15), we get the following solutions for
and
, as shown in Figure 7 and Figure 8.
(4.16)
and
(4.17)
where
is an elliptic integral of the first kind.
(vi) When
,
,
, corresponding to Figure 2(16). When
, there exist a family of periodic orbits defined by
. We obtain the parameter expressions for
and
as follows
(4.18)
and
(4.19)
where
,
.
(vii) When
,
,
, corresponding to Figure 2(17). There are two saddle points
and
, where
is a degenerate saddle point.
(a) When
, there exists a homoclinic orbit connecting
and
. From Equation (4.2), we get
(4.20)
From Equation (4.20), we get the parameter expressions for
and
as follows
(4.21)
and
(4.22)
(b) When
, there exist a family of periodic orbits passing the saddle point
. The parameter expressions for
and
as follows
(4.23)
and
(4.24)
where
,
.
(viii) When
,
,
, corresponding to Figure 2(18). There exist a center point
on v-axis. When
, there exist a family of periodic orbits defined by
. We obtain
and
as follows
(4.25)
and
(4.26)
where
,
.
(ix) When
,
,
, corresponding to Figure 2(27). There exist two saddle points
,
and a center point
, where
.
(a) When
, there exists a homoclinic orbit connecting the point
. From Equation (4.2), we have
(4.27)
where
. From (4.27), we obtain
(4.28)
and
(4.29)
(b) When
, there exists a family of periodic orbits defined by
. The parameter expressions of
and
are the same as Equation (4.16) and Equation (4.17).
(x) When
,
,
, corresponding to Figure 2(28). There exist two center points
,
and a saddle point
, where
.
(a) When
, there exists a homoclinic orbit connecting the point
. From Equation (4.2), we have
(4.30)
where
. From (4.30), we get
and
, as shown in Figure 9 and Figure 10.
(4.31)
and
(4.32)
where
is a bright soliton and
is a dark soliton.
(b) When
, there exists a family of periodic orbits defined by
. From Equation (4.2), it follows
(4.33)
where
, and satisfies
,
.
From (4.33), then we get the periodic wave solutions as follows
(4.34)
and
(4.35)
5. Conclusion
To summarize, with the help of differential equation dynamical systems theory and methods, we obtain all bifurcation phase diagrams with directional fields. These directional fields can help us better grasp its dynamic behavior. By the same energy value of the Hamiltonian function corresponding to the same orbit, we get a lot of periodic wave solutions and bright soliton, dark soliton solutions.