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March 15, 2026Scientia Sinica Mathematica0 citations

Almost periodic solutions of one dimensional nonlinear Schrödinger equation with the constant potential

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SXShuaishuai Xue

Key Points

  • To prove the existence of a family of almost periodic solutions for the nonlinear Schrödinger equation under constant potential.
  • Analyzed frequency drift and strong Diophantine conditions.
  • Utilized an improved infinite-dimensional KAM theory.
  • Considered periodic boundary conditions in one dimension.
  • Established the existence of small amplitude, time quasi-periodic solutions.
  • The solutions lie primarily in a sparse admissible set rather than the maximum torus.

Abstract

在周期边界条件下, 考虑一维含常数位势 m 的非线性薛定谔方程 i uₜ - Δu + mu + f (|u|²) u = 0, t∈R, x∈T, f 是原点某邻域内的一个实解析函数, 满足 f (0) =0, f' (0) ≠0. 本文基于对频率漂移的分析, 强丢番图条件以及一种改进的无限维 KAM 理论, 证明对于任何给定的常数位势 m, 该方程存在一族 Whitney 光滑的小振幅、时间概周期解. 需要强调的是, 我们的解并非位于最大环面, 而是主要位于一个稀疏容许集上.

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Cite This Study

Shuaishuai Xue (2026) studied this question.

synapsesocial.com/papers/69b5ff4f83145bc643d1ba01https://doi.org/10.1360/ssm-2026-0014
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