NLS with mass-subcritical combined nonlinearities: small mass $L^2$-scattering
Jacopo Bellazzini, Luigi Forcella, Vladimir Georgiev
TL;DR
The paper proves small data scattering in the mass-subcritical regime for a nonlinear Schrödinger equation with mixed nonlinearities: a focusing mass-subcritical term perturbed by a defocusing lower-order term. The authors combine a pseudo-conformal transformation with a variational energy framework, introducing a time-dependent modified energy $E_A(\tau,\varphi)$ whose evolution is controlled under a mass threshold, leading to $L^2$-scattering for data with small mass $\|\psi_0\|_{L^2}<\rho^{\star}$. A key part of the approach is the variational analysis of a defocusing–focusing energy $E^{\alpha,\beta,\gamma}$ and the associated ground-state threshold masses, establishing a dichotomy that underpins the scattering result. The work culminates in a precise mass threshold $\rho^{\star}$, established via a monotone limit as a parameter $A$ approaches $1$, and clarifies the role of standing waves in obstructing scattering by comparing $\rho^{\star}$ with $\rho_{0,SW}$ and $\rho_{0,E}$, thereby providing a robust framework for $L^2$-scattering in the short-range regime with minimal mass assumptions.
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 $Σ$-norm.