Optical Solitary Waves in the Higher Order Nonlinear Schrödinger Equation

Abstract

We study solitary wave solutions of the higher order nonlinear Schrödinger equation for the propagation of short light pulses in an optical fiber. Using a scaling transformation we reduce the equation to a two-parameter canonical form. Solitary wave (1-soliton) solutions always exist provided easily met inequality constraints on the parameters in the equation are satisfied. Conditions for the existence of N -soliton solutions (N ≥ 2) are determined; when these conditions are met the equation becomes the modified KdV equation. A proper subset of these conditions meet the Painlevé plausibility conditions for integrability.42.81. Dp, 02.30.Jr, 42.65.Tg, 42.79.Sz The propagation of nonlinear waves in dispersive media is of great interest since nonlinear dispersive systems are ubiquitous in nature. Propagation of ultrashort light pulses in optical fibers is of particular interest because of the common expectation that solitary waves may be of extensive use in telecommunication and even revolutionarize it. The existence of solitary wave solutions implies perfect balance between nonlinearity and dispersion which usually requires rather specific conditions and cannot be established in general. The objective of the present paper is to study the conditions under which the existence of solitary waves is guaranteed for ultrashort pulses.The propagation of light pulses in fibers is well described by the higher order nonlinear Schrödinger equation (HONSE) [1-4], a partial differential equation (PDE) whose right hand side includes the effects of group velocity dispersion, self-phase modulation, third order dispersion, self-steepening, and self-frequency shifting via stimulated Raman scattering, respectively:When the last three terms are omitted this propagation equation for the slowly varying envelope of the electric field, E, reduces to the nonlinear Schrödinger equations (NSE), which is integrable (meaning it not only admits N -solitary wave solutions, but that the evolution of any initial condition is known in principle) [4][5][6]. We call these N -solitary wave solutions N -solitons, and mean by this that the solitary waves scatter elastically and asymptotically preserve their shape upon undergoing collisions, just like true solitons. However, for short duration pulses the last three terms are non-negligible and should be retained. In general, the presence or absence of solitary wave solutions depends on the coefficients α appearing in Eq. (1), and therefore, on the specific nonlinear and dispersive features of the medium. Here, we reduce the HONSE to a two-parameter equation and derive a general solitary wave (1-soliton) solution. We determine conditions when N -soliton solutions exist and display the solutions. We also study the Painlevé plausibility conditions for integrability and show that these are only a proper subset of the conditions necessary for N -soliton solutions to exist. We begin by scaling the HONSE, letting E = b 1 A, z = b 2 ζ, and t = b 3 τ . Substituting into the HONSE we obtainChoosing b 1 = (α 3 1...

Rational W-shaped solitons on a continuous-wave background in the Sasa-Satsuma equation

“…(1) reduces to the standard NLS equation which has only the terms describing lowest order dispersion and self-phase modulation. The soliton solutions have been presented on the zero background in [18][19][20]. Here, we study rational solutions on a CW background,…”

Multisoliton solutions and integrability aspects of coupled higher-order nonlinear Schrödinger equations

“…With the inclusion of all these effects, Kodama and Hasegawa [4] have proposed that the dynamics of femtosecond pulse propagation be governed by a higher-order NLS (HNLS) equation. The HNLS equation allows soliton-type propagation only for certain choices of parameters [5].…”

Propagation of Ultrashort Pulses in a Nonlinear Medium

“…sn(ξ), dn(ξ) to zero leads to a set of algebraic equations for a j . By solving these equations, we obtain the final result for u in the form (29). We will apply this method below to find the soliton solutions in two cases: for the Eq.…”

“…Substituting (29) into (28) and equating the coefficients of all power of cn(ξ). sn(ξ), dn(ξ) to zero leads to a set of algebraic equations for a j .…”

“…For calculating the value of the envelope function by (29) we should know how the action of the linear and nonlinear operators on the envelope function is calculated. Because these operators contain the time partial derivatives one can calculate them just by Fourier Transform.…”