一维空间次临界长程非线性下散焦非线性薛定谔方程的大数据修正波算子
Large-data modified wave operators for the defocusing nonlinear Schrödinger equation in one space dimension with subcritical long-range nonlinearity
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中文总结 AI 辅助
针对一维散焦NLS次临界长程非线性，通过线性化与相位修正构造修正波算子，证明无大小限制下全局解散射到给定渐近轮廓。
中文摘要 AI 辅助
我们研究一维空间中具有幂非线性$|u|^{2\sigma}u$且在次临界长程区域$\frac{2}{\sqrt{7}}<\sigma<1$内的散焦非线性薛定谔方程（NLS）的终态问题解的长时行为。给定一个在加权$L^2$空间中预先指定的渐近轮廓，且无大小限制，该轮廓通过用非线性多项式相位修正修改自由解获得，我们构造了该NLS的唯一全局解，该解散射到此轮廓，从而证明了修正波算子的存在性。证明依赖于两个新要素。扩展我们先前关于三次情形的工作，我们通过围绕渐近轮廓线性化NLS，将非线性项的主要部分作为线性势并入线性部分，并证明线性化方程的全局修正能量估计。我们还利用了线性化产生的非线性项的特定结构，该结构在修正能量空间中估计非线性项时产生关键抵消，使我们能够在次临界情形下控制非线性相位修正的多项式增长。
英文摘要
We study long-time behavior of the solutions to the final state problem for the defocusing nonlinear Schrödinger equation (NLS) in one space dimension with the power nonlinearity $|u|^{2σ}u$ in the subcritical long-range regime $\frac{2}{\sqrt{7}}<σ<1$. Given a prescribed asymptotic profile in a weighted $L^2$-space, without size restriction, obtained by modifying the free solution with a nonlinear polynomial phase correction, we construct a unique global solution of the NLS that scatters to this profile, thereby proving the existence of modified wave operators. The proof relies on two new ingredients. Extending our previous work for the cubic case, we incorporate the leading part of the nonlinear term into the linear part as a linear potential by linearizing the NLS around the asymptotic profile and prove a global modified energy estimate for the linearized equation. We also exploit a specific structure of the nonlinearity arising from the linearization, which gives rise to a crucial cancellation when estimating the nonlinear terms in the modified energy space and enables us to control the polynomial growth of the nonlinear phase correction in the subcritical case.
发表机构
- Research Institute for Interdisciplinary Science, Okayama University（冈山大学跨学科科学研究所）
- Department of Mathematics, Graduate School of Science, The University of Osaka（大阪大学理学研究科数学系）
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