Soliton evolution and radiation loss for the nonlinear Schrödinger equation

Modulation analysis of boundary-induced motion of optical solitary waves in a nematic liquid crystal

“…g is the amplitude of the radiation bed [31]. The first term in this trial function represents a varying nematicon-like beam, while the second term represents the diffractive radiation of low wavenumber which accumulates under the evolving nematicon [27,31].…”

“…The first term in this trial function represents a varying nematicon-like beam, while the second term represents the diffractive radiation of low wavenumber which accumulates under the evolving nematicon [27,31]. This radiation cannot remain flat, so it is assumed that g is non-zero in the interval −ℓ/2 ≤ y ≤ ℓ/2 [27,31]. Substituting the optical field form (4) into the director equation, the second of (2), and using the one space dimensional boundary condition θ = 0 at the cell walls y = ±L gives the solution…”

“…Applying the modulation method [28,30,31], based on the trial function (4) and the director solution (5), then gives the modulation equations governing the evolution of the nematicon. These equations and a short discussion of them are given in the Appendix.…”

Soliton Aided Propagation and Routing of Vortex Beams in Nonlocal Media

“…Since the radiation cannot remain indefinitely flat, g u is non-zero in the disc located symmetrically about the peak of the vortex [52]. The flat radiation shelves under the beams then match to shed radiation propagating away from them, enabling the beams to reach a steady state [33,51,53]. The expressions for R u and R v =w v can be found in references [50] and [52], respectively.…”

“…Here (r,ϕ) are polar coordinates centred at (ξ v ,0) These functions consist of two parts: the first terms in each are a varying nematicon and a varying vortex, respectively; the second terms result from the low-wavenumber diffractive radiation which accumulates under the wavepackets as they evolve due to its low group velocity [51]. The parameters g u and g v measure the height of this shelf and do not depend on x and y.…”