On Stability of Small Solitons of the 1-D NLS with a Trapping Delta Potential

Abstract

We consider a Nonlinear Schrödinger Equation with a very general non linear term and with a trapping δ potential on the line. We then discuss the asymptotic behavior of all its small solutions, generalizing a recent result by Masaki et al. [30]. We give also a result of dispersion in the case of defocusing equations with a non-trapping delta potential.

Failure of scattering to solitary waves for long-range nonlinear Schrödinger equations

“…Our Theorems 1.1 and 1.2 assert that while the 'dispersive' part v(t) may decay, it will not contain a scattering component in the range 0 < p ≤ 1. The appearance of the exponent p = 1 in both [6] and the present work appears to be coincidental. Theorems 1.1 and 1.2 provide an extension of the well-known results concerning non-existence of linear scattering for long-range nonlinearities to the more general setting of scattering to solitary waves, which is an ongoing area of active research interest.…”

“…As mentioned above, the generality of our assumptions on V for d = 1 permits us to include interesting cases such as the delta potential, which amounts to choosing V to be a point mass at the origin. In particular, we may compare our results to the recent results of [6], in which the authors treat the NLS with a delta potential and power-type nonlinearities |u| p u. They show that for small H 1 initial data and any p > 0, the solution admits a unique decomposition of the form…”

Decay of small odd solutions for long range Schrödinger and Hartree equations in one dimension

“…(1.10) 12) and Q c > 0 solving Q ′′ c − cQ c + Q p c = 0, Q c ∈ H 1 . Note that this solution is even in space and small in H 1 provided c ≪ 1.…”

“…See e.g. the recent paper by Cuccagna and Maeda [12], and the NLKG paper by Kowalczyk, Martel and Muñoz [30].…”

A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II

“…Remark 5.5. Many of the papers on asymptotic stability of standing waves, focus on small standing waves which bifurcate from eigenvalues of a Schrödinger operator, see [120,121,115], [128]- [131], [127,61,55,54,57,58,108,97,37]. In these papers the spectrum of the Schrödinger operator is rather simple.…”