The attractive nonlinear delta-function potential

Abstract

We solve the continuous one-dimensional Schrödinger equation for the case of an inverted nonlinear delta-function potential located at the origin, obtaining the bound state in closed form as a function of the nonlinear exponent. The bound state probability profile decays exponentially away from the origin, with a profile width that increases monotonically with the nonlinear exponent, becoming an almost completely extended state when this approaches two. At an exponent value of two, the bound state suffers a discontinuous change to a delta-like profile. Further increase of the exponent increases again the width of the probability profile, although the bound state is proven to be stable only for exponents below two. The transmission of plane waves across the nonlinear delta potential increases monotonically with the nonlinearity exponent and is insensitive to the sign of its opacity.

Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity I: Basic theory

“…where δ = δ(x) is the delta function supported at the origin, and ψ(x, t) is a complexvalued wave function for x ∈ R. The equation (1.1) can be interpreted as the free linear Schrödinger equation Appropriate function spaces in which to discuss solutions, and the corresponding meaning of (1.2) will be given below. Specifically important is the case p = 3, the 1D focusing NLS with cubic point nonlinearity (1.3) i∂ t ψ + ∂ 2 x ψ + δ|ψ| 2 ψ = 0 This equation could model the following two physical settings, as proposed by [MB01]. First, (1.3) could model an electron propagating in a 1D linear medium which contains a vibrational "impurity" at the origin that can couple strongly to the electron.…”

“…In the approximation where one considers the vibrations completely "enslaved" to the electron, (1.3) is obtained as the effective equation for the electron. [MB01] further remark that an important application of (1.3) is that of a wave propagating in a 1D linear medium which contains a narrow strip of nonlinear (general Kerr-type) material. This nonlinear strip is assumed to be much smaller than the typical wavelength.…”

Scattering for the 𝐿² supercritical point NLS

“…In this paper, we address a theoretical study on a model, proposed in [16], that describes a wave propagation in a 1D linear medium containing a narrow strip of nonlinear material, where the nonlinear strip is assumed to be much smaller than the typical wavelength. Considering such nonlinear strip may allow to model a wave propagation in nanodevices, in particular the authors in [13] consider some nonlinear quasi periodic super lattices and investigate an interplay between the nonlinearity and the quasi periodicity.…”

Construction of solutions for some localized nonlinear Schrödinger equations

“…To our knowledge, this situation has not been studied systematically and provides interesting and useful insights for more general cases, see e.g. [8,31]. By studying a one dimensional basic model we provide a construction of infinitely many bound states for any negative energy of the system (see Theorems 3.3 and 3.4).…”