NLS with mass-subcritical combined nonlinearities: small mass $L^2$-scattering

TL;DR

This paper proves small data scattering for a mass-subcritical nonlinear Schrödinger equation with combined nonlinearities, using pseudo-conformal transformation and variational methods, with smallness only on initial mass.

Contribution

It establishes small mass scattering results for NLS with mixed focusing and defocusing nonlinearities, extending previous understanding in the mass-subcritical regime.

Findings

Proves small data scattering for the specified NLS model.

Uses pseudo-conformal transformation and variational techniques.

Smallness condition is only on initial mass, not full norm.

Abstract

We prove small data scattering in the mass-subcritical regime for the NLS equation with double nonlinearities, where a focusing leading term is perturbed by a lower order defocusing nonlinear term. Our proof relies on the pseudo-conformal transformation in conjunction with a general variational argument used to obtain the positivity of certain modified energies. Moreover, the smallness assumption is only on the mass of the initial data, and not on the whole $\Sigma$-norm.