The dynamics of the smooth positon and b-positon solutions for the NLS-MB equations
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ORIGINAL PAPER
The dynamics of the smooth positon and b-positon solutions for the NLS-MB equations Feng Yuan
Received: 23 July 2020 / Accepted: 21 September 2020 © Springer Nature B.V. 2020
Abstract In this paper, the smooth order-n positon solutions and b-positon solutions of the NLS-MB equation are constructed by the degenerate Darboux transformation (DT). The decomposition of the order-2, order-3, and order-4 positon solutions is discussed based on the decomposition of modulus square, and its approximate orbits and variable “phase shift” are also exhibited. Furthermore, we discuss the degeneration progress of the order-2 b-positon solution which means the central area of b-positon solution is a very good approximation of the rogue wave solution. The approximate trajectories of the order-2 b-positon solution are also discussed. Keywords NLS-MB equations · Darboux transformation · Positon solution · b-Positon solution
1 Introduction For the past few years, long haul optical communication through fibers has aroused great interest of scientists around the world. Especially in ultra-fast communication systems, the soliton pulse transmission plays an important role and it is considered to be the tools of the future to achieve low loss, efficient, high-speed F. Yuan (B) School of Science, Nanjing University of Posts and Telecommunications, Nanjing 210023, People’s Republic of China e-mail: [email protected]; [email protected]
communications. Many equations as optical fiber communication models have been studied by mathematicians and physicists [1–6]. The nonlinear schrödinger (NLS) equation describes the propagation of optical pulses through nonlinear optical fibers in the picosecond range [7]. It is indispensable to be compensated for attenuation in a fiber so that the soliton-based communication systems are more highly competitive, reliable, and economical than the conventional systems. The Maxwell–Bloch (MB) equations describe a kind of pulse propagation called self-induced transparency (SIT) soliton [8]. This kind of pulse reaches a stable state where the width, energy, and shape of the pulse remain unchanged after several classical absorption lengths, and the pulse speed is much lower than the speed of light in this medium. To consider the effects of the large width pulses further, the system dynamics are controlled by the coupled system of the NLS equation and the MB equation (NLS-MB system) [9]. The NLS-MB equations are written as [9,10] ⎧ ⎨ E t = i 21 E x x + |E|2 E + 2P, P = 2iω0 P + 2Eη, ⎩ x ηx = −E P ∗ − E ∗ P.
(1.1)
The Lax pair of the NLS-MB equations is given by [11]
x = U , t = V ,
(1.2)
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where 1 V−1 , U = λσ3 + U0 , V = i σ3 λ2 + λV1 + V0 + 2 λ − iω0
ψ1 1 0 0 E = , U0 = , σ3 = , ∗ 0 −1 −E 0 ψ2 2
0 E |E| Ex V1 = , , V0 = −E ∗ 0 E x∗ −|E|2
η −P . (1.3) V−1 = −P ∗ −η
Recently, a lot of researches have been done about the NLS-MB equations. The multi-soliton solution is given by Ref. [11]. The single soliton and the single breather solutions of t
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