Abstract
In this paper, the dynamical properties and the classification of single traveling wave solutions of the coupled nonlinear Schrödinger equations with variable coefficients are investigated by utilizing the bifurcation theory and the complete discrimination system method. Firstly, coupled nonlinear Schrödinger equations with variable coefficients are transformed into coupled nonlinear ordinary differential equations by the traveling wave transformations. Then, phase portraits of coupled nonlinear Schrödinger equations with variable coefficients are plotted by selecting the suitable parameters. Furthermore, the traveling wave solutions of coupled nonlinear Schrödinger equations with variable coefficients which correspond to phase orbits are easily obtained by applying the method of planar dynamical systems, which can help us to further understand the propagation of the coupled nonlinear Schrödinger equations with variable coefficients in nonlinear optics. Finally, the periodic wave solutions, implicit analytical solutions, hyperbolic function solutions, and Jacobian elliptic function solutions of the coupled nonlinear Schrödinger equations with variable coefficients are constructed.
1. Introduction
The coupled nonlinear Schrödinger (CNLS) equations [1–3] are a very important mathematical physical model in the fields of quantum mechanics, nonlinear optics, optical fiber communication, Bose-Einstein condensate, fluid mechanics, and so on. Because of its importance, many new methods and techniques have been proposed to study the CNLS equation, which include the Bäcklund transformation, the variational iteration method, the extended unified method, and the Hirota method [4–9]. However, the study of CNLS equations with variable coefficients [10–14] has more important theoretical significance and practical value than CNLS equations with constant coefficients because it usually describes the inhomogeneous effects of nonlinear optical pulse propagations in the real optical fiber communication system. The CNLS equations with variable coefficients are expressed as follows:where and denote the slowly varying amplitudes of two orthogonally polarized optical pluses, which are complex-valued functions with the retarded time and the normalized propagation distance . , , . The coefficients and are integrable real functions of , which represent the group velocity dispersion and the Kerr nonlinearity, respectively. Obviously, the integrability method, the modified sine-Gordon equation method, the unified method, the improved F-expansion method, and the Hirota bilinear method have used to establish the traveling wave solutions of the CNLS equation with variable coefficients in Ref. [10-14], respectively. In this paper, we investigate the dynamical properties and the classification of single traveling wave solutions of the CNLS equation with variable coefficients based on the bifurcation theory and the complete discrimination system method.
The synopsis of the article is organized as follows. In Section 2, the bifurcation theory is employed to investigate the CNLS equations with variable coefficients; phase portraits and some Jacobian elliptic function solutions are obtained. In Section 3, some single traveling wave solutions of the CNLS equations with variable coefficients are obtained by using the complete discrimination system method. In Section 4, we present some research results.
2. Bifurcations of Phase Portraits of System (9)
In order to analyze the dynamic behavior of Equation (1), we first introduce the following traveling wave transformation:where and are a real valued function, and are arbitrary constants, and and are arbitrary function of .
Substituting (2) into (1), we get the coupled nonlinear differential equations
Next, we decompose real parts and imaginary parts of (3) and put zero for imaginary parts of Equations (3), where the imaginary part of Equations (3) is . Then, we set , . Therefore, . Using the coefficients of (or ) as the normalization coefficient, we havewhere and are constants and is the integral constant. On the other hand, the real parts of Equations (3) are simplified as
Let , ; system (5) is transformed into a four-dimensional dynamical system as follows:
In order to remove the coupled relationship of (5), we consider the dynamical behavior of system (6) in the submanifold () of the 4-dimensional phase space . Substituting (or ) into (6), we obtain two completely uncoupled planar dynamical systems:
Systems (7) and (8) are the same if and only if they satisfy . Then, system (7) can be rewritten as the following planar dynamical systemwhich is equivalent to the following Hamiltonian system:
Suppose , and further assume that is an equilibrium points of system (9), then the eigenvalues of system (9) at the equilibrium points are expressed as . Thus, we can easily get three zeros of when , which are , and . We can easily obtain one zero of when , which is . By the bifurcation theory of planar dynamical systems, the equilibrium point is a saddle point when , the equilibrium point is a degraded saddle point when , and the equilibrium point is center point when . The phase portraits of (9) depending on different parameters and are shown in Figure 1.
Lemma 1 (see [15]). Let be a continuous solution of system (9) for and , :(1) is called a smooth solitary wave solution if . Usually, a solitary wave solution of Equation (1) corresponds to a smooth homoclinic orbit of system (7)(2) is called a smooth kink or antikink solution if . A smooth kink (or antikink) wave solution of Equation (1) corresponds to a smooth heteroclinic orbit (or a so-called connecting orbit) of system (7)
Remark 2. Obviously (see [16–20]), all traveling wave solutions of (1) can be obtained from the phase orbits of the Hamiltonian system (9) according to Lemma 1.
(a)
(b)
2.1. The Case and
(1)For , the system (10) can be written aswhere and .
Substituting (11) into the first equation of system (9), integrating them along the periodic orbits, then we obtain
Namely,(2)For , we easily obtain two kink solitary wave solutions
When , , , , and , the dark soliton solution of Equation (1) is drawn in Figure 2, where and , respectively.
(a)
(b)
2.2. The Case and
(1)For , the system (10) can be written aswhere , .
Substituting (16) into , we can obtain the smooth periodic wave solution(2)For , we obtain two bell solitary wave solutions:
When , , , , , the bright solitary wave solution of Equation (1) is drawn in Figure 3, where and , respectively.(3)For , the system (10) can be written aswhere and .
(a)
(b)
Substituting (20) into and integrating them, we obtain two families of periodic solutions
Namely,
Remark 3. In this section, we mainly discuss the solution of Equation (1) when . Moreover, we can easily obtain the solution of Equation (1) through their relationship .
3. The Classification of Single Traveling Wave Solutions via Complete Discrimination Systems
In order to remove the coupled relationship of (5), we assume (). Substituting into (5), we have
Multiplying both sides of (23) by and integrating once with respect to , we havewhere is the integral constant.
Make the transformation
Next, substituting (25) into (24), it becomeswhere , , and . The solution of (26) can be written as the integral formwhere is an arbitrary constant.
Let . Thus, we can obtain its complete discrimination system [21–23] as follows:
Case 1. When , , and . Then, admits a pair of conjugate complex roots of multiplicities two, namely,where . Substituting (29) into (27), we attainthen the solution of (23) is expressed as follows:Namely,
Case 2. , , and . has real roots of multiplicities of four, namely, . Substituting into (27), we can getthen the solutions of Equation (1) can be shown as
Case 3. , , , and . has two real roots of multiplicities of two, namely,where . Substituting (35) into (27), we can obtain
When or , we can obtain the solution of Equation (23) as follows:then we have
When , we can gain the solution of Equation (23) as follows:Similarly, we can have
Case 4. , , and . has two real roots and real roots with multiplicities of two, namely,where () are real numbers and .
When and , or when and , we can have the implicit analytical solution of Equation (26):
When and , or when and , we can obtain the implicit analytical solution of Equation (26):
When , we can have the implicit analytical solution of Equation (26):
Case 5. , , , and . has real roots of multiplicities of three and real roots with multiplicities of one, namely,where and are real numbers. Substituting (45) into (27), we can yield
When and , or when and , the solution of Equation (26) is given bythen we can obtain a rational solution of Equation (1) as
Case 6. and . has real roots of multiplicities of two and a pair of conjugate complex roots, namely,where are real numbers. Substituting (49) into (27), we can obtainwhere , . Thus, we obtain the solution of Equation (1):which is a solitary wave solution.
Case 7. , , and . has four distinct real roots, namely,where , , , and are real numbers, and .
When or , then we make the following transformation:By using (27), we obtainwhere .
From Equation (54) and the definition of Jacobian elliptic sine function, we obtain
Combining Equation (55) with the expressions (53), we can give the solution of Equation (1):where .
When , then we make the transformation
Similarly, we can obtain the solution of Equation (1):
Case 8. and . has two different real roots and a pair of conjugate complex roots, namely,where , , , and are real constants, and .
We make the following transformation:where , , , , , and .
By using (27), we obtainwhere .
From Equation (61) and the definition of Jacobian elliptic function, we obtain
Combining Equation (62) with the expression (60), we can give the solution of Equation (1):which is an elliptic double periodic function solution.
Case 9. and . has two pairs of conjugate complex roots, namely,where , , , and are real constants and .
We make the following transformation:where , , , , , and .
By using (27), we obtainwhere .
From (65) and the definition of Jacobian elliptic function, we obtain
Combining Equations (67) and (68) with expression (65), we can give the solution of Equation (1)which is an elliptic double periodic function solution, where .
Remark 4. In this section, we mainly discuss the solution of Equation (1). Similarly, we can easily obtain the solution of Equation (1) through their relationship .
4. Conclusion
In the paper, the planar dynamical system method and the complete discrimination system method are applied to the CNLS equations with variable coefficients for constructing different types of traveling wave solutions. The complex periodic wave solutions, implicit analytical solutions, complex hyperbolic function solutions, and complex Jacobian elliptic function solutions of the CNLS equations with variable coefficients are successfully investigated by two different methods. Moreover, phase portraits are drawn with the help of Maple software. New solutions of the CNLS equations with variable coefficients have not been reported in previous literatures, such as complex Jacobian elliptic function solutions and implicit analytical solutions.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there is no conflict of interest regarding the publication of this paper. The authors declare no conflict interest.
Acknowledgments
This work was supported by the Scientific Research Funds of Chengdu University under grant No. 2081920034 and Institutions of Higher Education of Sichuan Province under grant No. MSSB-2021-13.