On a class of homogeneous nonlinear Schrödinger equations

Abstract

A class of homogeneous, norm conserving, nonlinear wave equations of the Schrödinger type is studied. It is shown that those equations which derive from a Lagrangian can be linearized, but have no regular confined solutions, whereas the equations which cannot be obtained from a local Lagrangian do admit such confined solutions. The latter however are unstable against small perturbations of the initial data.

Information measures for inferring quantum mechanics

“…(This example highlights the point that a nonlinear Schrodinger equation cannot immediately, and in all generality, be declared pathological). However for the remaining values of λ one obtains after the change of variables [13] a linear diffusion equation. It will be assumed for the rest of this paper that the classical metric is used also in the information measure when symmetries are preserved.…”

“…If one allows in the measure a metricḡ ij which is still diagonal but different from the classical metric g ij then a nonlinear Schrodinger equation apparently ensues. However the nonlinearity can be removed by a change of variables (a nonlinear gauge transformation) [12,13] with the result that for a range of values of the Lagrange multiplier λ one actually recovers the usual linear Schrodinger equation. (This example highlights the point that a nonlinear Schrodinger equation cannot immediately, and in all generality, be declared pathological).…”

Резонансы Солитонов Огибающих И Немаделунговы Жидкости Типа Броера - Каупа

“…Линейное урав-нение Шредингера с таким членом было изучено Ауберсоном и Сабатьером [14]. НУШ с таким членом было выведено для гравитации Джакива-Тейтельбаума в ра-ботах [6], [15], [16] и в физике плазмы в работе [17].…”

Envelope soliton resonances and broer-kaup-type non-madelung fluids

“…Auberson and Sabatier studied the linear Schrödinger equation with such a term [14]. The NLS equation with such a term was derived for the Jackiw-Teitelboim gravity in [6], [15], [16] and in plasma physics in [17].…”