Long-Time Asymptotics and Modified Scattering

Determine the precise asymptotic behavior as time tends to infinity of large-data global solutions to the defocusing cubic nonlinear Schrödinger equation on the three-dimensional Friedlander model domain, including whether solutions scatter to a free linear evolution and whether resonant or near-resonant boundary modes produce nontrivial phase corrections or modified scattering.

Background

The paper proves global well-posedness for the defocusing cubic nonlinear Schrödinger equation on the three-dimensional Friedlander domain by combining energy conservation, a boundary-adapted Bourgain–Strichartz framework, Airy-mode multilinear estimates, and a high–low frequency decomposition. It does not analyze the behavior of these global solutions as time tends to infinity. In particular, the boundary’s glancing dynamics and the discrete normal spectral modes may generate long-range effects that are absent in the boundaryless Euclidean problem.

Resolving the problem would require estimates substantially finer than those used for global existence, including long-time dispersive bounds adapted to the glancing region, normal-form transformations, and a description of the radiative component. The unresolved issue is whether global solutions approach a free linear evolution or instead exhibit resonant phase corrections and modified scattering caused by the interaction between continuous tangential frequencies and discrete boundary modes.

References

Although the present work establishes global well-posedness by combining the uniform energy bound with the boundary-adapted nonlinear iteration, the precise asymptotic behavior of large-data solutions as $t\to\infty$ remains open. In the absence of a boundary, dispersive decay can provide a mechanism for scattering toward a free linear evolution. In the present setting, however, the interaction between boundary propagation, glancing dynamics, and the discrete normal spectral structure may lead to a more intricate long-time behavior.

— Large-Data Global Well-Posedness for the Defocusing Cubic Schrödinger Equation in 3D Convex Domains  (2609.30920 - Meas, 25 Sep 2026) in Section 7, Subsection 7.1, “Asymptotic Behavior and Modified Scattering for Large-Data Global Solutions”