On scattering for generalized NLS on waveguide manifolds

Abstract

In this paper, we prove the large data scattering for fractional nonlinear Schrödinger equations (FNLS) on waveguide manifolds R d × T, d ≥ 3. This result can be regarded as the fractional analogue of [43,44] and the waveguide analogue of [16]. A key ingredient of the proof is a Morawetz-type estimate for the setting of this model. This result also extends the recent result [35] by proving the scattering behavior.

On well-posedness results for the cubic-quintic NLS on $\mathbb{T}^3$

“…For interested readers, we also refer to [7,8,14,15,16,17,29,30,32,34,35,36] for further well-posedness results for NLS (with single nonlinear potential) on tori or waveguide manifolds based on the atomic space theory. (See [24,31] for other dispersive equations on waveguides.) Despite that small data well-posedness results are satisfactory to certain extent, it is more interesting (and hence also more challenging) to deduce well-posedness results where the initial data are not necessarily small.…”

On scattering asymptotics for the 2D cubic resonant system

“…One may consider other related problems such as 'other dispersive equations on waveguides' and 'NLS on other manifolds/product spaces'. See [16,31,38] for examples (Klein-Gordon equations on waveguides, Fractional NLS on waveguides and Fourth order NLS on waveguides respectively). 7.2.…”