Nonautonomous “rogons” in the inhomogeneous nonlinear Schrödinger equation with variable coefficients

Abstract

The analytical nonautonomous rogons are reported for the inhomogeneous nonlinear Schrödinger equation with variable coefficients in terms of rational-like functions by using the similarity transformation and direct ansatz. These obtained solutions can be used to describe the possible formation mechanisms for optical, oceanic, and matter rogue wave phenomenon in optical fibres, the deep ocean, and Bose-Einstein condensates, respectively. Moreover, the snake propagation traces and the fascinating interactions of two nonautonomous rogons are generated for the chosen different parameters. The obtained nonautonomous rogons may excite the possibility of relative experiments and potential applications for the rogue wave phenomenon in the field of nonlinear science.The nonlinear Schrödinger (NLS) equation is a foundational model describing numerous nonlinear physical phenomenon in the field of nonlinear science such as optical solitons in optical fibres [1,2,3], solitons in the mean-field theory of Bose-Einstein condensates [4,5,6], and the rogue waves (RWs) (also known as freak waves, monster waves, killer waves, giant waves or extreme waves) in the nonlinear oceanography [7,8,9,10]. RWs are single waves generated in the ocean with amplitudes much higher than the average wave crests around them [11,12]. Recently, the RWs have attracted more and more attention from the point views of both theoretical analysis [21,22,23,24,25,28,29,34] and experimental realization [13,14,30,31,32,33]. The oceanic RWs can be, under the nonlinear theories of ocean waves, modelled by the dimensionless NLS equation [7,8] i ∂ψ ∂t + 1 2which describes the two-dimensional quasi-periodic deep-water trains in the lowest order in wave steepness and spectral width. In addition, it has been shown that the RWs can be generated in nonlinear optical systems and the term "optical rogue waves" was coined by observing optical pulse propagation in the generalized NLS equaiton [13,14]. Some types of exact solutions of Eq. (1) have been presented to describe the possible formation mechanisms for the RW phenomenon such as the algebraic breathers (Peregrine solitons) [15], the time periodic

Manipulating matter rogue waves and breathers in Bose-Einstein condensates

“…Therefore, it is of high significance to study the dynamics of RWs and breather profiles of the GP equation (1). However only few attempts have been made to identify and analyze the RWs and breather solutions of (1) [35][36][37][38][39][40]. To the best of our knowledge neither higher-order RW solutions (with certain free parameters) nor higher order breather solutions of (1) have been taken up for study.…”

Nonautonomous Solitons: Applications from Nonlinear Optics to BEC and Hydrodynamics

“…Somewhat surprisingly, however, this figure indicates a sharp compression and strong amplification of the nonautonomous Peregrine soliton under the action of hyperbolic gain which, in particular, in the open ocean can be associated with "hyperbolic hurricane wind". It should be stressed that since the nonautonomous NLSE model is applied in many other physical systems such as plasmas and Bose-Einstein condensates (BEC), the results obtained in this Section can stimulate new research directions in many novel fields (see, for example, (Bludov et al, 2009;Yan, 2010)). …”

“…The law of soliton adaptation to an external potential has come as a surprise and this law is being today the object of much concentrated attention in the field. The interested reader can find many important results and citations, for example, in the papers published recently by Zhao et al Luo et al, 2009;Zhao et al, 2009;2008), Shin (Shin, 2008) and (Kharif et al, 2009;Porsezian et al, 2007;Yan, 2010). How can we determine whether a given nonlinear evolution equation is integrable or not?…”

Rogue waves in Lugiato-Lefever equation with variable coefficients

“…Literature includes rational solutions and their interactions, pulse splitting induced by higher-order modulation instability, and wave turbulence [25,26]. Exact rogue waves solution have also been obtained analytically for some physical models such as the Kadomtsev-Petviashvili equation in shallow water [27] and the NLSE with variable coefficients [28], higher orders [29], or higher dimensions [30,31]. However, to the best of our knowledge, the study of rogue waves in the Lugiato-Lefever equation (LLE) with variable coefficients remains unaddressed.…”