PHYSICAL REVIEW E 91, 023210 (2015) Bilinearization of the generalized coupled nonlinear Schr ¨ odinger equation with variable coefficients and gain and dark-bright pair soliton solutions Sushmita Chakraborty, Sudipta Nandy, * and Abhijit Barthakur Department of Physics, Cotton College Guwahati, Guwahati-781001, India (Received 14 August 2014; revised manuscript received 22 December 2014; published 24 February 2015) We investigate coupled nonlinear Schr¨ odinger equations (NLSEs) with variable coefficients and gain. The coupled NLSE is a model equation for optical soliton propagation and their interaction in a multimode fiber medium or in a fiber array. By using Hirota’s bilinear method, we obtain the bright-bright, dark-bright combinations of a one-soliton solution (1SS) and two-soliton solutions (2SS) for an n-coupled NLSE with variable coefficients and gain. Crucial properties of two-soliton (dark-bright pair) interactions, such as elastic and inelastic interactions and the dynamics of soliton bound states, are studied using asymptotic analysis and graphical analysis. We show that a bright 2-soliton, in addition to elastic interactions, also exhibits multiple inelastic interactions. A dark 2-soliton, on the other hand, exhibits only elastic interactions. We also observe a breatherlike structure of a bright 2-soliton, a feature that become prominent with gain and disappears as the amplitude acquires a minimum value, and after that the solitons remain parallel. The dark 2-soliton, however, remains parallel irrespective of the gain. The results found by us might be useful for applications in soliton control, a fiber amplifier, all optical switching, and optical computing. DOI: 10.1103/PhysRevE.91.023210 PACS number(s): 05.45.Yv, 42.65.Tg, 42.81.Dp I. INTRODUCTION After the first real breakthrough in optical communication in 1970s with the development of the InGaAsP semiconductor laser and low loss optical fibers, the most significant devel- opment was the demonstration of an optical-soliton-based communication system. It has been demonstrated both theoret- ically [1] as well as experimentally [2] that a suitable balance between the dispersion and the nonlinear effect can generate a stable pulse, which can propagate through a fiber as a soliton. In comparison to bright solitons, dark solitons are found to be less affected by the background noise and perturbations, and their interactions are weaker [3]. Over the past two decades there have been many significant contributions to the experimental and theoretical development of dark and bright optical soliton (see Refs. [4,5], and the references therein). Dispersion and nonlinear effect in some cases, for example in a mode-locked fiber laser, can be so strong that the pulse parameters, namely the width, chirp, phase, and position, vary significantly from their initial values. To deal with such a problem, the concept of soliton dispersion management and soliton control in a fiber has been recently developed, that is, with a suitable combination of fibers exhibiting normal and anomalous dispersion, a stretched fiber laser can be realized that can produce a dispersion-managed pulse in the form of a soliton [4,6]. Dispersion management of a soliton is described by the standard nonlinear Schr¨ odinger equation (NLSE) model with varying dispersion and nonlinear coefficients along with a gain or loss coefficient [7], i q z + β D(z) 2 q tt + γR(z)|q| 2 q = iŴ(z)q, (1) * Author to whom all correspondence should be addressed: sudiptanandy@gmail.com where q(z,t ) is a slowly varying pulse envelope in a reference frame, moving with the group velocity of the pulse, D(z) and R(z) represent the group velocity dispersion and nonlinearity parameters, and Ŵ(z) = ∂ z R(z)D(z)−R(z)∂ z D(z) 2R(z)D(z) is the gain param- eter. In [7], the authors reported for the first time the existence of a dispersion managed (DM) soliton for such a system. The fundamental bright and dark soliton solutions of Eq. (1) with β =±1, respectively, are q = ⎧ ⎨ ⎩ η  D(z) γR(z) sech(ηt )e 0.5iη 2  z 0 D(ζ )dζ (bright), η  D(z) γR(z) tanh(ηt )e iη 2  z 0 D(ζ )dζ (dark). (2) The fact that a DM soliton can not only be accelerated but also be amplified preserving its shape and elastic character makes it more suitable for physical applications compared to a conventional soliton. Recently, there have been many important publications based on the model (1)[7,8] and also based on a more recently developed nonautonomous soliton model [9,10], where researchers showed many new results and predicted various applications of DM solitons. For example, in [11] the authors analyzed the control of soliton velocity with a dispersion-decreasing fiber profile in the framework of a variable coefficient NLSE, and through asymptotic analysis they verified the elastic character of two soliton interaction. In [12] the authors studied the dynamics of nonlinear pulse propagation in an averaged DM soliton system by using a variable coefficient NLSE, and they showed that the Hirota bilinear method, which is a well-known method for conventional NLSEs, is also applicable to a variable coefficient NLSE. In [13] the authors reported the interaction of chirped solitons based on models described in [8]. In more recent studies, authors reported on the dynamics of a bright soliton [14,15] and a dark soliton [16] in a generalized nonautonomous NLSE model. It may be noted here that Eq. (1) also serves as a model for the Bose-Einstein condensates (BECs) but with the roles of space and time interchanged 1539-3755/2015/91(2)/023210(11) 023210-1 ©2015 American Physical Society