The Method of Moments for Nonlinear Schrödinger Equations: Theory and Applications

Abstract

Abstract. The method of moments in the context of Nonlinear Schrödinger Equations relies on defining a set of integral quantities, which characterize the solution of this partial differential equation and whose evolution can be obtained from a set of ordinary differential equations. In this paper we find all cases in which the method of moments leads to closed evolution equations, thus extending and unifying previous works in the field of applications. For some cases in which the method fails to provide rigorous information we also develop approximate methods based on it, which allow to get some approximate information on the dynamics of the solutions of the Nonlinear Schrödinger equation.

Nonlinear waves in Bose–Einstein condensates: physical relevance and mathematical techniques

“…This restriction is well-known to potentially lead to invalid results [195]; it is worthwhile to note, however, that there are efforts underway to systematically compute corrections to the variational approximation [196], thereby increasing the accuracy of the method. Another very useful tool for analyzing BEC dynamics is the so-called moment method [197], whereby appropriate moments of the wavefunction ψ = √ ρ exp(iφ) (where ρ = |ψ| 2 and φ are the BEC density and phase, respectively) are defined such as N = ρ dr (the number of atoms), X i = x i ρ dr (the center of mass location),…”

Evolution and Collapse of a Lorentz Beam in Kerr Medium

“…The collapse dynamics of elliptical beams have been extensively studied [22,24]. These studies pointed out significant differences between quantitative predictions of the aberrationless approximation and actual results obtained from NLS equation simulations [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40][41]. From our numerical calculations, we find a simple empirical expression for the critical power of asymmetrical Lorentz beam by fitting the results of the numerical calculation…”

“…In order to investigate further the effect of Kerr nonlinearity [23][24][25][26][27][28][29][30][31][32][33][34][35][36] on the Lorentz beam, we need to solve the NLS equation numerically. Numerical simulations were done using the parameters of wavelength λ = 0.53 µm, n 0 = 1, n 2 = 0.5 × 10 −4 cm 2 /GW, w 0x = 10 µm and z 0 = kw 2 0x /2 = 0.6 mm, respectively.…”

“…S 1 (z) and S 2 (z) are the beam normalized inverse of curvature. Note that a quadratic phase approximation (QPA) [30][31][32][33][34] has been introduced here. Substitution of the trial function Eq.…”

“…We apply the variational approach [27][28][29][30][31] to obtain important information about the nonlinear dynamics on the spatial distributions of the Lorentz beam in the Kerr medium. We obtain the critical power [32][33][34][35][36][37][38][39][40][41][42] of the Lorentz beam with uniform wavefront, which is required to collapse the beams, as a function of different beam parameters, particularly the waists w 0x and w 0y .…”