A Virial-Morawetz approach to scattering for the non-radial inhomogeneous NLS

TL;DR

This paper develops a new approach combining Virial-Morawetz estimates to prove scattering for the inhomogeneous nonlinear Schrödinger equation in higher dimensions without assuming radial symmetry, broadening previous results.

Contribution

It introduces a Virial-Morawetz method to establish scattering for non-radial solutions in the inhomogeneous NLS, extending parameter ranges and simplifying proofs.

Findings

Proves scattering for a wider class of parameters p and b.

Removes the radial symmetry assumption in scattering results.

Provides a simpler proof avoiding the Kenig-Merle approach.

Abstract

Consider the focusing inhomogeneous nonlinear Schr\"odinger equation in $H^1(\mathbb{R}^N)$, $$iu_t + \Delta u + |x|^{-b}|u|^{p-1}u=0,$$ when $b > 0$ and $N \geq 3$ in the intercritical case $0 < s_c <1$. In previous works, the second author, as well as Farah, Guzm\'an and Murphy, applied the concentration-compactness approach to prove scattering below the mass-energy threshold for radial and non-radial data. Recently, the first author adapted the Dodson-Murphy approach for radial data, followed by Murphy, who proved scattering for non-radial solutions in the 3d cubic case, for $b<1/2$. This work generalizes the recent result of Murphy, allowing a broader range of values for the parameters $p$ and $b$, as well as allowing any dimension $N \geq 3$. It also gives a simpler proof for scattering nonradial, avoiding the Kenig-Merle road map. We exploit the decay of the nonlinearity, which, together with Virial-Morawetz-type estimates, allows us to drop the radial assumption.