† Corresponding author. E-mail:
In this paper, we study soliton–cnoidal wave solutions for the reduced Maxwell–Bloch equations. The truncated Painlevé analysis is utilized to generate a consistent Riccati expansion, which leads to solving the reduced Maxwell–Bloch equations with solitary wave, cnoidal periodic wave, and soliton–cnoidal interactional wave solutions in an explicit form. Particularly, the soliton–cnoidal interactional wave solution is obtained for the first time for the reduced Maxwell–Bloch equations. Finally, we present some figures to show properties of the explicit soliton–cnoidal interactional wave solutions as well as some new dynamical phenomena.
It is well known that the Maxwell–Bloch equations[1] describe the propagation of short (
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn2.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn3.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn4.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn5.gif)
Apparently, adopting the following transformation
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The research of the RMB equations was initiated by Eilbeck et al.[2] in 1973, where slowly varying envelope approximation was avoided and the backscattered wave was neglected due to a weaker assumption. The RMB equations have a wide usage in describing phenomena in nonlinear optics, including the theory of optical self-induced transparency.[3,4] One of their significant features is the propagation of a short laser pulse in a rarefied medium of two level atoms. Other important applications include electrical engineering experiments with essential algorithms and simulations.[5–7] During the past few decades, many effective methods, such as the inverse scattering transform (IST),[8,9] the Hirota bilinear method,[10–12] and the Darboux transformation (DT),[13–15] have been utilized to investigate the explicit N-soliton solutions of the RMB equations. The RMB equations are one of the integrable systems as shown in Refs. [16] and [17] and of course admit other integrable properties including Hamitonian structure and recursion operator[18] as well as the N-degenerate periodic, N-rational solutions, and rogue waves.[19]
To find interactional solutions of nonlinear systems is a difficult and tedious but very meaningful and important work. Fortunately, the truncated Painlevé expansion and the tanh function expansion methods, are effective to derive interactional solutions between solitary waves and other kinds of nonlinear waves, such as Airy waves, cnoidal waves, and Bessel waves. Recently, the consistent Riccati expansion (CRE) method and consistent tanh expansion (CTE) method were proposed to investigate the interactional wave solutions of the nonlinear systems.[20] These methods play an important role in solving nonlinear systems, whether an integrable system or not. Then many nonlinear systems are proved CRE solvable and own interactional wave solutions between a solitary wave and a cnoidal wave.[21–33]
In this paper, we concentrate on constructing soliton-cnoidal interactional wave solutions of the RMB equations. The truncated Painlevé analysis is utilized to generate a CRE and the RMB equations are proved to be CRE solvable and CTE solvable. By the CTE method, solitary wave, cnoidal periodic wave, and soliton–cnoidal interactional wave solutions of the RMB equations are obtained in an explicit form. Finally, we present some figures to show properties of the explicit soliton–cnoidal interactional wave solutions as well as some new dynamical phenomena, which could be helpful to better understand the dynamic properties of the interactional wave solutions in optical and electromagnetic fields.
The paper is constructed as follows. In Section
Based on the CRE method, the CRE solution for the RMB equations (
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn6.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn7.gif)
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn8.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn9.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn10.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn11.gif)
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn12.gif)
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From the above results, it is clear that the RMB equations (
By using a dependent variable transformation
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn15.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn16.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn17.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn18.gif)
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It is evident that the solutions (
To search for the explicit solutions of the RMB equations (
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Substituting Eq. (
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn19.gif)
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Substituting Eq. (
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Then the cnoidal periodic wave solutions can be derived as
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn20.gif)
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According to the definition of an elliptic function, the explicit solution of an equation can be expressed by the terms of a Jacobian elliptic function. It is indicated that equation (
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![](cpb_27_2_020201/cpb_27_2_020201_ieqn22.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn23.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn24.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn25.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn26.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn27.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn28.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn29.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn30.gif)
![](cpb_27_2_020201/cpb_27_2_020201_ieqn31.gif)
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Finally, substituting Eq. (
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The solutions of Eq. (
![]() | Fig. 1. (color online) The soliton–cnoidal wave interactional solution of the RMB equations for the components u, v, ω, and E expressed by Eq. (![]() ![]() ![]() ![]() |
![]() | Fig. 3. (color online) Density plots of the components u, v, ω, and E for the values described in Fig. |
As a matter of fact, there are plenty of such interactional solutions in our real physics world.
(i) For Figs.
(ii) For Figs.
(iii) For Figs.
(iv) From analysis of the density graph between Figs.
![]() | Fig. 5. (color online) Density plots of the components u, v, ω, and E for the values described in Fig. |
![]() | Fig. 7. (color online) Density plots of the components u, v, ω, and E for the values described in Fig. |
These dynamical behaviors of the RMB equations for the components u, v, ω, and E have not been shown before, and the physical significance of these solutions are well worth further study. Our research results may have a significant contribution to investigations of the dynamics of the distinct nonlinear waves, such as rogue waves, breather solutions, and dark solitary waves for nonlinear systems in optics, electromagnetic field, plasma physics, and Bose–Einstein condensates.
In summary, the RMB equations (
Due to the important physics significance of the RMB equations, the physical properties of these new dynamical behaviors with interactions are needed to do a further investigation. In addition, from the viewpoint of mathematics it is worth studying the relationship between the CRE method and the associated Schwarzian z functions.
We would like to express our sincere thanks to other members of our research group for their valuable suggestions.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] |