Akhmediev breathers, Ma solitons, and general breathers from rogue waves: A case study in the Manakov system

Abstract

We present explicit forms of general breather (GB), Akhmediev breather (AB), Ma soliton (MS) and rogue wave (RW) solutions of the two component nonlinear Schrödinger (NLS) equation, namely Manakov equation. We derive these solutions through two different routes. In the forward route we first construct a suitable periodic envelope soliton solution to this model from which we derive GB, AB, MS and RW solutions. We then consider the RW solution as the starting point and derive AB, MS and GB in the reverse direction. The second approach has not been illustrated so far for the two component NLS equation. Our results show that the above rational solutions of the Manakov system can be derived from the standard scalar nonlinear Schrödinger equation with a modified nonlinearity parameter. Through this two way approach we establish a broader understanding of these rational solutions which will be of interest in a variety of situations.

On the characterization of vector rogue waves in two-dimensional two coupled nonlinear Schrödinger equations with distributed coefficients

“…Once the constraint (6) has been accommodated the solutions of (1) can be picked up from the solutions of the two component NLS equation (3). Equation (3) is the well-known Manakov equation whose integrability properties have been extensively studied in the literature [33,34,35,36,37,38,39,40,41,42]. A detailed investigation on the soliton solutions of two coupled NLS equation with same nonlinearity (nonlinear coefficients have the same signs '+' and '-') and mixed nonlinearity (nonlinear coefficients have opposite signs) gave birth to three different vector soliton solutions, namely (i) bright-bright [33,34], (ii) bright-dark [56,57] and (iii) dark-dark solitons [34,58].…”

“…(3) have been constructed through Hirota's bilinearization method [59]. Very recently, a new form of localized solution, called RW solution, has attracted considerable attention in the two coupled NLS equation (3) [37,38,40,39]. In [40,41,42], explicit RW solutions of two coupled NLS equation have been constructed through modified Darboux transformation method.…”

“…The vector NLS equations yield potentially rich and significant results for optical fiber communication systems [12]. Motivated by this, several studies have also been undertaken to identify the localized solutions in coupled NLS equations with constant coefficients [33,34,35,36,37,38,39,40,41,42,43]. Subsequently attempts have been made to identify the localized solutions such as bright-bright, bright-dark, dark-dark soliton and RW solutions in the coupled vcNLS equations (in one-dimensions) [44,45,46,47,48,49,50,51].…”

Rogue waves for a system of coupled derivative nonlinear Schrödinger equations

“…(3) is a system of six equations for six unknowns (real and imaginary parts of f, g and h). 8 RWs are derived by taking the long wavelength limit of breather solutions of Eq. (3).…”

Manipulating localized matter waves in multicomponent Bose-Einstein condensates

“…The CNLSE also models water wave interactions [19] and wave propagation in fiber communication system [13]. The exact analytical RW solutions of the CNLSE (integrable Manakov case) [20,21] and its generalized version have been presented in [22]. Further, BECs have also been shown to constitute a good platform to explore the RWs which allow one to understand deeply the nature and dynamics of RWs under laboratory conditions.…”