Original articlesExact solitary wave and periodic-peakon solutions of the complex Ginzburg–Landau equation: Dynamical system approach
Introduction
Ginzburg–Landau equation is a mathematical model describing the phenomena of super-conductivity. It is a kind of nonlinear Schrödinger equation and has a wide range of applications. Many mathematical physicists have done a great deal of work on this equation. More recently, the complex Ginzburg–Landau equation (CGLE) has aroused a great deal of research interest among scholars [1], [3], [4], [5], [7], [12], [13], [20], [21], [22], [24], [25], [26], [27], [28], [29], [30]. Where is a complex-valued function representing the soliton molecule, are spatial and temporal coordinates respectively. is a nonlinear term that describes the general form of dependence on the refractive index of optical fibers. are parameters of Eq. (1). This model describes the dynamics of solitary wave propagation through optical fibers for long-distance. It is an interesting and widely applied topic, and many researchers are devoted to applying various powerful methods to construct the exact solution of CGLE. These methods include the traveling wave reduction method [12], [20], [21], the Jacobi elliptic function method [8], [18], the tanh-function method [6], [19], [23], Lie symmetry method [5], Kudryashov method [11], [29], - expansion method [21], [29], exp-function expansion method [1], F-expansion scheme [3], and so on [4], [9], [10], [21]. Moreover, in [21], the above methods are integrated to study optical solitons in CGLE with Kerr and power laws of nonlinearity. Four forms of optical solitons are recovered in this paper, and other wave solutions, as a by-product. However, most of these papers only obtain partial solutions of the CGLE, which are usually solitary wave solutions and periodic solutions. There are still many other solutions and abundant dynamic behaviors of the CGLE that remains to be discovered. Recently, Zhu and Xia [30] studied bifurcations and exact solutions of Eq. (1) when the term via dynamical system approach. Moreover, Kudryashov [12] studied Eq. (1) using traveling wave reduction and obtained the general solution of Eq. (1) when the parameters of the equation meet certain conditions. The dynamics behaviors of the solutions are shown only in the form of solitary wave and periodic wave.
Inspired by this, in this paper, we employ the dynamical systems method to investigate the corresponding traveling wave system of Eq. (1), under very general parametric conditions, to study the exact solutions of the traveling wave system and discuss the dynamical behavior of solutions.
To study the traveling wave solutions of Eq. (1), let Substituting (2) into Eq. (1), separating the imaginary part and real part, yield where . Multiplying Eq. (3) on and integrating the obtained equation once, we get where is an integral constant. If , then .
Substituting (5) into (4), we have where “” denote the differential with respect to .
Remark 1 We notice that the equation given by Kudryashov [12] is not correct. The last two items are miscalculated. Therefore, all results in [12] need to be made new studies. Eq. (6) is equivalent to the following system If we take is a positive integer, then, when , system (7) has the first integral where is a constant. Let be a positive integer. If we take , then . When , system (7) has the first integral When , system (7) has the first integral When , system (7) has the first integral When , we have . System (7) has the first integral Especially, when , system (7) has the first integral
System (7) is a singular traveling wave system of the first class (see Li and Chen [15], Li [14], Li and Qiao [16]) with the singular straight lines . It depends on seven-parameter group .
It is known that a wave profile defined by with some phase orbits of a planar dynamical system (7) has the following relationships. A smooth homoclinic orbit to a saddle point of a traveling wave system corresponds to a smooth solitary wave solution of a PDE. All periodic orbits which have segments close to the singular straight line in a period annulus, correspond to periodic cusp wave solutions (periodic peakon). A homoclinic orbit, which has a segment close to the singular straight line , corresponds to a pseudo-peakon solution. For a family of open orbits, when , which tend to a singular straight line, then this family of orbits corresponds to a family of compactons (see Li, Zhou and Chen [17]).
The present paper is built up as follows. In Section 2, we consider the equilibrium points and phase portraits of the system (7). In Sections 3, 4, 5, we derive all possible exact solutions of under different parameter conditions. Finally, the main theorem and conclusion are given in Section 6.
Section snippets
Phase portraits and equilibrium points of system (7)
Consider the regular system associated with the system (7) as follows, where , for . Systems (7), (14) have the same first integral, but in the phase plane, two systems define different vector fields in the two sides of the singular straight lines.
Obviously, an equilibrium point of system (14) satisfies , where .
When has three zeros at
Exact explicit solitary wave solutions and periodic wave solutions of system (7) when in (8)
Let be a positive integer. Assume that , then, . When , then . and , then . We discuss these cases in next sections.
Exact explicit solitary wave solutions, periodic wave solutions of system (7) when in (7) with
When in (7), we see from (9) that Thus, applying the first equation of (7), we have where , and
Firstly, assume that , we consider the phase portraits in Fig. 2(b).
(i) Corresponding to the two closed orbit families defined by . When , (27) can be written as . When , (27)
Exact explicit solitary wave solutions, periodic wave solutions of system (7) when in (7)
When . We see from (13) that if , then Under the parameter condition , system (7) has three equilibrium points and , where . We have . The phase portraits of system (7) are shown in Fig. 6.
Taking , then, using the first equation of system (7), we have where .
Firstly, assume that . we
Main results and conclusion
From the detailed analysis of bifurcations and the explicit expressions of solutions of Eq. (1), we summarize it as a theorem as follows.
Theorem 1 (1) The complex Ginzburg–Landau equation (1) has exact explicit solutions with the form where when the function is a solution of the corresponding traveling wave system (7) and . (2) Corresponding to some families of closed orbits of the system (7), has exact explicit
Acknowledgments
The authors wish to express their sincere appreciation to all those who made suggestions for improvements of this paper. This research was supported by the National Natural Science Foundation of China (11871231, 12071162, 11701191, 11371326, 11975145, 11901215).
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