Analytic method for solving the modified nonlinear Schrödinger equation describing soliton propagation along optical fibers

Abstract

We give a direct method for obtaining exact solutions of the modified nonlinear Schrodinger equation iu, +u», "+2p~u~u+2iq (~u~'u) =0 describing the propagation of light pulses in optical fibers. By using a suggestive particlelike description, we classify all the obtained analytical solutions into one of the following categories: the "algebraic" soliton, the one-soliton solution, the bright solitary waves, and the regular periodic solutions which are very important from the physical point of view.PACS number(s): 42.65.Vh, 42.50.Rh, 03.65.Ge

Helmholtz algebraic solitons

“…, cos 1 2 where P is given by equation (12). That is, || > 0 decreases the power of the input beam relative to its unperturbed value (i.e., relative to the power of the exact solution with the same parameters).…”

“…Deep water waves and ion-acoustic waves in plasmas can be described by algebraic solitons of the derivative-nonlinear Schrödinger (NLS) [8] and the Kadomtsev-Petviashvili equations [2,3,9]. In photonics, algebraic solitons occur in such contexts as Raman scattering [10], self-induced transparency [11] (Maxwell-Bloch-type systems), pulse propagation in dispersive fibres [12] (derivative-NLS), electromagnetic modes of planar waveguides [13] (dual power-law NLS), and solitary-wave polaritons [14] (Boussinesq equation). Coupled modes and periodic systems can also support KdV-and NLS-type algebraic "gap solitons," respectively [15].…”

The Prolongation Structure of the Modified Nonlinear Schrödinger Equation and Its Initial-Boundary Value Problem on the Half Line via the Riemann-Hilbert Approach

“…Recently, many properties of this equation have been studied, such as the Hamiltonian structure [53], the Darboux transformation [54], the numerical solutions [55,56] and so on [57,58]. Actually, it can become the derivative NLS equation by certain gauge transformation [59]. In this paper, we mainly discuss the mNLS equation on the half line.…”

Riemann-Hilbert approach to the modified nonlinear Schröinger equation with non-vanishing asymptotic boundary conditions

“…The modified NLS equation (1.3) has been discussed extensively, for example, various solutions such as analytical solutions, soliton solutions, rational and multi-rogue wave so-lutions were found by analytical method, Hirota bilinear method and Darboux transformation respectively [21][22][23][24]. The Hamiltonian structure for the equation (1.3) was given [25].…”