高维含时势导数非线性Schrödinger系统的渐近行为

Asymptotic Behavior of Derivative Nonlinear Schrödinger Systems with Time-Dependent Potentials in High-Dimensional Space

摘要

考虑高维（d≥3）空间中含时势的二次导数型非线性项的Schrödinger系统的初值问题. 首先, 在质量共振条件下, 利用能量不等式、 嵌入定理等工具得到系统解的先验估计； 其次, 利用先验估计证明具有小初值的非线性Schrödinger系统解的整体存在性； 最后, 通过构造辅助函数给出质量共振条件下系统解是渐近自由的.

Abstract

We considered the initial problem of derivative Schrödinger systems with time-dependent potentials and quadratic nonlinearities in high-dimensional （d≥3）space. Firstly, under the condition of mass resonance, we obtained the priori estimates of solutions to the systems by using the tools such as energy inequalities and embedding theorems. Secondly, we proved the global existence of solutions for the nonlinear Schrödinger systems with small initial value by using priori estimates. Finally, by constructing auxiliary functions, we demonstrate that the solutions to the systems are asymptotically free under the condition of mass resonance.

关键词

导数非线性 / Schrödinger系统 / 含时势函数 / 质量共振关系 / 渐近行为

Key words

derivative nonlinear Schrödinger system / time-dependent potential / mass resonance relation / asymptotic behavior

参考文献

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基金资助

国家自然科学基金(12361051)

吉林省教育厅项目(JJKH20250396KJ)