Perturbations of dark solitons

Abstract

A direct perturbation method for approximating dark soliton solutions of the nonlinear Schrödinger (NLS) equation under the influence of perturbations is presented. The problem is broken into an inner region, where the core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton that propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated, including linear and nonlinear damping type perturbations. Keywords: perturbation theory; solitons; opticsPerturbation theory as applied to solitons that decay at infinity, i.e. so-called bright solitons, has been developed over many years (cf. Karpman & Maslov 1977;Kodama & Ablowitz 1981;Herman 1990). The analytical work employs a diverse set of methods including perturbations of the inverse scattering transform (IST), multi-scale perturbation analysis, perturbations of conserved quantities, etc.; the analysis applies to a wide range of physical problems. In optics, a central equation that describes the envelope of a quasi-monochromatic wave train is the nonlinear Schrödinger (NLS) equation, which in normalized form readswhere D, n are constants. In this paper, we consider the NLS equation in a typical nonlinear optics context, where D corresponds to the group-velocity dispersion (GVD), n > 0 is related to the nonlinear index of refraction, z is the

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“…This is illustrated in Figure 4, where the greater amount of radiation for the evolving nematicon is clearly visible. The shelf of low wavenumber radiation under the dark nematicons can also be seen in this figure as the beam has been shifted off |u| = 0, in agreement with theory [28]. The interaction between the nematicon, shelf and radiation changes the nematicon waist in a periodic manner as waves are periodically shed from the edge of the shelf.…”

“…The existence of this radiation shelf has also been proven using perturbation theory [28]. Integrating from the edge of the shelf at ξ + /2 to ∞ gives the mass lost to radiation which propagates to the right of the soliton as…”

“…The trial function (5) does not contain a shelf term, as for an evolving bright NLS soliton [8]. The long wavelength radiation shed by an evolving soliton has low group velocity and so accumulates under the soliton, forming a shelf [8,28]. If such a shelf term were included in the trial function (5), at first order it would give zero contribution as all integrals involving the even shelf and the odd dark soliton integrate to zero in the averaged Lagrangian.…”

“…An alternative approach accounting for this radiative loss has been developed [8,9,21], with the radiative terms derived from perturbed inverse scattering [8]. The form of these terms has independently been obtained using standard soliton perturbation theory [26][27][28]. The NLS equation (2) has the Lagrangian…”

“…The shelf width [28] was chosen to be = w f as this gives the correct decay rate of the soliton waist onto the steady state value for a broad range of initial conditions. This value differs from that for bright NLS beam evolution, for which ≈ 3w f [8].…”