Integrability of an inhomogeneous nonlinear Schrödinger equation in Bose–Einstein condensates and fiber optics

Abstract

In this paper, we investigate the integrability of an inhomogeneous nonlinear Schrödinger equation, which has several applications in many branches of physics, as in Bose–Einstein condensates and fiber optics. The main issue deals with Painlevé property (PP) and Liouville integrability for a nonlinear Schrödinger-type equation. Solutions of the integrable equation are obtained by means of the Darboux transformation. Finally, some applications on fiber optics and Bose–Einstein condensates are proposed (including Bose–Einstein condensates in three-dimensional in cylindrical symmetry).

Matter-wave dark solitons in boxlike traps

“…is conserved within the GPE, as confirmed by our numerical simulations. Equation (II.2) have been investigated over the years in terms of the "complete integrability" (see [56] and reference therein). This property (even though still not univocally defined) regards the existence of infinite number of conservation laws and the possibility of relating the nonlinear PDE (partial differential equation) to a linear PDE by an explicit transformation.…”

“…The main feature is that Eq. (II.2) is not completely integrable except for the case V (x) = ax + b, with a and b constants [56]. For V (x) = 0, the GP equation (II.2) has the following exact dark soliton solution [1,9],…”

Blow-up results and soliton solutions for a generalized variable coefficient nonlinear Schrödinger equation

“…In modern nonlinear sciences some of the most important models are the variable coefficient nonlinear Schrödinger-type ones. Applications include long distance optical communications, optical fibers and plasma physics, see [4], [5], [8], [12], [15], [23], [24], [25], [30], [41], [48], [49], [51], [52], [53], [61], [63], [65] and references therein.…”

On integrability of nonautonomous nonlinear Schrödinger equations

“…Recently several nonautonomous (with time-dependent coefficients) and inhomogeneous (with space-dependent coefficients) nonlinear Schrödinger equations have been discussed as (possible) new integrable systems [7], [8], [14], [16], [20], [42], [48], [52], [53], [59], [66], [67], [76], [81], [82], [83], [84], [85], [87], [93], [96], [97], [98], [104] (see also [2], [3], [9], [18], [19], [21], [45], [55], [92] and references therein for earlier works). They arise in the theory of Bose-Einstein condensation [35], [75], fiber optics [6], [47], superconductivity and plasma physics [18], [19], [70], [71].…”