Scattering in spacetime cones for the nonlinear Schrödinger equation
Abstract
We consider the nonlinear Schrödinger equation on ℝ3,1, with a class of `mixed' nonlinearities that admit soliton solutions. We are interested in the long-time behaviour of solutions with finite, but possibly large, mass and/or energy. By a spacetime cone, we mean an open set in ℝ3,1x,t that is invariant under dilation (x, t) ↦ (αx, αt) for all α> 0. Our main result is as follows: let $u(x, t)$ be the solution with initial condition in H2(ℝ3). Let ℳ, ℰ be the mass and energy of u, and let η= η(ℳ, ℰ) > 0 be sufficiently small. Let Cε be the complement of N ≥ 1 of sufficiently thin spacetime cones { (x, t) | |x−vjt| ≤ εt }, where v1, …, vN are fixed velocities and ε is sufficiently small. If the L∞ norm of u in Cε is ≤ η, we prove that u scatters in C2ε ⊂ Cε.
Related
- Global well-posedness and scattering for the defocusing nonlinear Schrödinger equation below energy space
- Threshold Scattering for the Three-Dimensional Focusing Energy-Critical Nonlinear Schrödinger Equation
- Global well-posedness and scattering of energy-critical defocusing nonlinear Schrödinger equation on high-dimensional hyperbolic spaces
- An inverse scattering problem for the nonlinear Schrödinger equation under partial harmonic confinement
- On 2D Scattering for a Critical Nonlinear Schrödinger Flow
BibTeX
@misc{hassell2026scattering,
title = {{Scattering in spacetime cones for the nonlinear Schrödinger equation}},
author = {Andrew Hassell and Qiuye Jia and Xiaoxu Wu},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.29265},
eprint = {2610.29265},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.29265},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Scattering in spacetime cones for the nonlinear Schrödinger equation
AU - Andrew Hassell
AU - Qiuye Jia
AU - Xiaoxu Wu
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.29265
UR - https://arxiv.org/abs/2610.29265
AB - We consider the nonlinear Schrödinger equation on ℝ^3,1, with a class of `mixed' nonlinearities that admit soliton solutions. We are interested in the long-time behaviour of solutions with finite, but possibly large, mass and/or energy. By a spacetime cone, we mean an open set in ℝ^3,1_x,t that is invariant under dilation (x, t) ↦ (αx, αt) for all α> 0. Our main result is as follows: let $u(x, t)$ be the solution with initial condition in H^2(ℝ^3). Let ℳ, ℰ be the mass and energy of u, and let η= η(ℳ, ℰ) > 0 be sufficiently small. Let C_ε be the complement of N ≥ 1 of sufficiently thin spacetime cones { (x, t) | |x−v_jt| ≤ εt }, where v_1, …, v_N are fixed velocities and ε is sufficiently small. If the L^∞ norm of u in C_ε is ≤ η, we prove that u scatters in C_2ε ⊂ C_ε.
ER -