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 () optical pulses in a resonant two-level media. The propagation of ultra-short optical pulses is usually governed by the following reduced Maxwell–Bloch (RMB) equations[2]
where u is microscopic polarization, v is phase information, ω is the atomic inversion, is the electric field, is the atomic resonant frequency, p is the dipole operator, c is the speed of light in a vacuum, n is the atomic dipole density, the angular bracket represents the summation over all the media characterized by the frequency, and is the energy separation for the two levels.
Apparently, adopting the following transformation
sends the RMB equations (1) to the following system
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 2, the RMB equations are verified CRE solvable and CTE solvable. In Section 3, the solitary wave, cnoidal periodic wave, and soliton–cnoidal wave solutions are constructed. The last section gives a short summary and discussion.
2. CRE solvable and CTE solvable
2.1. CRE solvable of the RMB equations
Based on the CRE method, the CRE solution for the RMB equations (3) can be mapped into
where u0, u1, v0, v1, v2, ω0, ω1, ω2, E0, and z are functions of x and t to be determined. The function satisfies the following Riccati equation
and b0, b1, and b2 are arbitrary constants. After substituting Eq. (4) with Eq. (5) into Eqs. (3), all the expansion coefficients of need to be equal to zero. Fourteen overdetermined equations are obtained, that only eleven undetermined functions u0, u1, v0, v1, v2, ω0, ω1, ω2, E0, E1, and z need to be satisfied. Fortunately, these overdetermined functions can be consistent. From the coefficients of , , and , u0, u1, v0, v1, v2, ω0, ω1, ω2, E0, and E1 can be solved as,
Then from the overdetermined functions, the associated compatibility condition of function z is
with and . Finally, it can be verified that all the coefficients of the overdetermined equations are identically satisfied by using Eqs. (6) and (7).
From the above results, it is clear that the RMB equations (3) really possess the truncated Painlevé expansion solutions, which is corresponding to the Riccati equation (5). Therefore, the RMB equations (3) have CRE solvable property.[20] It should be point out that equation (7) is a generalization of the Schwarzian form for the RMB equations, as equation (7) is nothing but the standard Schwarzian form if dropping out the δ term. In order to make the result clear, we arrive at the following non-auto BT theorem.
2.2. CTE solvable of the RMB equations
By using a dependent variable transformation
it is obvious that satisfies Riccati equation (5). With the parameters , , and , equation (4) can be mapped into
where u0, u1, v0, v1, v2, ω0, ω1, ω2, E0, E1, and z are decided by Eqs. (5), (6), and (7).
It is evident that the solutions (10) are consistent with the above Theorem 1. Since CTE (10) is an exceptional situation of CRE, CTE solvable is equivalent to CRE solvable. Based on the CTE solvable property, we can obtain some interesting interactional solitary wave solutions, such as the solitary wave and the interactions between a solitary wave and an elliptic cosine wave, whose dynamical characteristics can be shown visually via the following expressions. Through the non-auto BT theorem below, we can clearly understand the above relation and take advantage of it to derive exact solutions.
3. Soliton–cnoidal interactional wave solutions
To search for the explicit solutions of the RMB equations (3), z is assumed by the following form
where g is an arbitrary function with x and t. The interactional solutions between solitary waves and other nonlinear waves will be obtained. According to Theorem 2, solitary wave solutions, cnoidal wave solutions, and soliton–cnoidal interactional wave solutions can be derived as follows.
Substituting Eq. (15) into Eq. (8), it yields the elliptic equation as follows:
where , a1, a2, and a3 are arbitrary constants. It is apparent that equation (16) has the general Jacobian elliptic functions solutions. To show this type of solution clearly, we choose solutions of Eq. (16) to be expressed by
Substituting Eq. (17) into Eq. (16), the following relations can be obtained
Then the cnoidal periodic wave solutions can be derived as
where , S, C, and D respectively express Jacobi elliptic functions sn, cn, and dn with modulus n and
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 (23) demonstrates interactional solutions between solitary waves and other nonlinear waves. For a more intuitive demonstration of the above soliton–cnoidal interactional solutions, we give the following ansatz for solutions of Eq. (21)
where is the general Jacobian elliptic sine function. The modulus n satisfies the condition: . If , degenerates into hyperbolic function . If , it degenerates into trigonometric function . If substituting Eq. (24) with Eq. (22) into Eq. (21), gathering the coefficients of , dn , sn and setting them to zero, results in the following relations
Finally, substituting Eq. (24) and
with the parameters satisfied by Eq. (25) into the general solutions (23), special interactional solutions with respect to solitary waves and cnoidal waves are obtained.
The solutions of Eq. (23) with Eq. (22), namely the interactional solutions between the solitary waves and the cnoidal waves. In Figs. 1, 2, 4, and 6, the interactional solutions of u, v, ω, and E are visually shown with the condition modulus . This type of solution has a wide application value in analyzing some interesting physical phenomena.
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. (23). The parameters are m = −1, n = 1/2, , , , and .
Fig. 2. (color online) The soliton–cnoidal wave interactional solution of the RMB equations for the components u, v, ω, and E expressed by Eq. (23). The parameters are m = 1, n = 1/2, , , μ = 1, and .
Fig. 3. (color online) Density plots of the components u, v, ω, and E for the values described in Fig. 2.
As a matter of fact, there are plenty of such interactional solutions in our real physics world.
(i) For Figs. 1 and 2, the waves generate in pairs, and the solutions are symmetrical at x = 0. The periodic waves are periodic in x directions and localized in t directions. The background waves can be clearly presented from the components u and E in Fig. 2, while the background periodic waves of the components v and ω are not clearly presented.
(ii) For Figs. 4(a), 4(c), 4(d), 6(a), 6(c), and 6(d), dark solitary waves reside on cnoidal waves instead of on constant backgrounds.
Fig. 4. (color online) The soliton–cnoidal wave interactional solution of the RMB equations for the components u, v, ω, and E expressed by eq. (23. The parameters are m = 1/2, n = 1/2, , , μ = 2, and .
(iii) For Figs. 4(b) and 6(b), a bright solitary wave travelling on a cnoidal wave.
(iv) From analysis of the density graph between Figs. 5 and 7, the direction of the cnoidal wave propagation is inconsistent, while the direction of the solitary wave is consistent.
Fig. 6. (color online) The soliton–cnoidal wave interactional solution of the RMB equations for the components u, v, ω, and E expressed by Eq. (23). The parameters are m = 1/2, n = 1/2, , , μ = 1, and .
Fig. 7. (color online) Density plots of the components u, v, ω, and E for the values described in Fig. 6.
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.
4. Summary and discussions
In summary, the RMB equations (3) are studied through utilizing the truncated Painlevé expansion and the consistent Riccati expansion method. Based on the CRE approach, the soliton–cnoidal wave interactional solutions of the RMB equations are explicitly expressed in terms of the Jacobi elliptic and the corresponding elliptic integral. Some interesting physical phenomena with respect to the soliton–cnoidal interactional solutions are shown in the figures above. Compared with other traditional methods, the CRE method and CTE method are more effective in obtaining the above soliton–cnoidal wave interactional solutions. Through the mathematical formulations, we plot the soliton–cnoidal wave interactional solutions, and discover some new dynamical phenomena. Those dynamical behaviors (as shown in Figs. 1, 2, 4, and 6 for the components u, v, ω, and E) had not been investigated before.
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.
Acknowledgment
We would like to express our sincere thanks to other members of our research group for their valuable suggestions.
