Exactly Solvable Model for Nonlinear Pulse Propagation in Optical Fibers

Abstract

The nonlinear Schrödinger (NLS) equation is a fundamental model for the nonlinear propagation of light pulses in optical fibers. We consider an integrable generalization of the NLS equation, which was first derived by means of bi-Hamiltonian methods in [1]. The purpose of the present paper is threefold: (a) We show how this generalized NLS equation arises as a model for nonlinear pulse propagation in monomode optical fibers when certain higher-order nonlinear effects are taken into account; (b) We show that the equation is equivalent, up to a simple change of variables, to the first negative member of the integrable hierarchy associated with the derivative NLS equation; (c) We analyze traveling-wave solutions.

Baseband modulation instability as the origin of rogue waves

“…The FLE is a partial differential equation that has been derived as a generalization of the NLSE [26,27]. In the context of optics, the FLE models the propagation of ultrashort nonlinear light pulses in monomode optical fibers [27].…”

New Exact Solutions of the New Hamiltonian Amplitude-Equation and Fokas Lenells Equation

“…This equation is related to the nonlinear Schrödinger (NLS) equation in the same way that the Camassa-Holm equation is associated with the KdV equation [34].The Fokas-Lenells equation is a completely integrable equation which has been derived as an integrable generalization of the NLS equation using bi-Hamiltonian methods [35]. In optics, the FL equation models the propagation of nonlinear light pulses in monomode optical fibers when certain higher order nonlinear effects are taken into consideration [36]. The complete integrability of the FL equation has been exhibited by using the inverse scattering transform (IST) method [37].…”

Algebro-Geometric Solutions and Their Reductions for the Fokas-Lenells Hierarchy

“…Related results can also be directly applied to (1.1) and (1.3) since the existence of these simple transformations among them. It is shown that (1.4)/(1.5) is a completely integrable nonlinear partial differential equation possessing Lax pair, bi-Hamiltonian structure, and soliton solutions [5,[16][17][18]. One of the most remarkable feature of the FL equation is that it possesses various kinds of exact solutions such as solitons, breathers, etc..…”

“…An important feature of equation (1.3) is that it describes the firs negative fl w of the integrable hierarchy associated with the derivative nonlinear Schrödinger (DNLS) equation [5,12,13,16]. In this paper, we consider the following complexifie version of FL system q xt − q xx + iqq x r − 2iq x + q = 0, r xt − r xx − iqrr x + 2ir x + r = 0, (1.4) which is exactly reduced to the FL equation (α = β = 1 in (1.2)) q xt − q xx ∓ i|q| 2 q x − 2iq x + q = 0, (1.5) for r = ±q.…”

“…In ref. [5], Fokas proposed an integrable generalization of the NLS equation, iu t − νu tx + γu xx + σ |u| 2 (u + iνu x ) = 0, σ = ±1, x, ∈ R,t > 0 ν, γ, ρ ≡ constant ∈ R, (1 In the context of nonlinear optics, the FL equation models the propagation of nonlinear light pulses in monomode optical fiber when certain higher-oder nonlinear effects are taken into account [16].…”