Defocusing nonlinearity with nonnegative localized potential: global existence and absence of localization

Determine whether adding a defocusing nonlinear term to the Schrödinger equation with a nonnegative, localized, smooth potential ensures global existence and precludes the formation of spatially localized solutions beyond the small‑data regime.

Background

In the context of nonlinear inverse scattering, the author highlights a basic regime where no linear bound states are present but the dynamics include a defocusing nonlinearity. While small‑data scattering results suggest dispersion, a full large‑data theory ensuring global well‑posedness and absence of localization for such backgrounds is not yet established.

Resolving this question would clarify whether defocusing interactions generically rule out localized solutions when combined with nonnegative trapping‑free potentials, aligning nonlinear behavior with linear intuition.

References

Let us then consider a simpler situation where there are no bound states solutions to the in the equation. For example, it is expected (still open) that if a non-negative localized smooth potential is perturbed by a defocusing nonlinear term, there will be global existence and no localized solutions in many cases.

— A New Paradigm For Scattering Theory of Linear And Nonlinear Waves: Review And Open Problem (2408.14269 - Soffer, 26 Aug 2024) in Section: Open Problems — Inverse Scattering, Problem 2