AI 中文总结
本文综述一维非线性薛定谔方程的两类适定性方法，将可处理全范围幂次及组合非线性项的方法应用于含有限组合非线性项的NLS，解决物理应用中的相关难题。
AI 中文摘要
我们研究一维非线性薛定谔方程，其非线性项为|u|^α u，其中幂次α>0，综述两类求解该方程的适定性方法：一是初值属于L^2、H^1或H^1加权子空间的局部适定性方法，该方法基于Strichartz估计，通常要求非线性项幂次α≥1时H^1适定性成立；二是直接应用与导数可交换的加权估计及初值的特定下确界条件，该方法可处理0<α<∞全范围的非线性项，还能应对不同非线性项的组合。最后，我们将第二类方法应用于含有限个组合非线性项的NLS，该问题对物理应用（如激光光学）具有重要意义，由于缺乏标度不变性，用第一种方法获取局部适定性极具挑战性，甚至可能无法实现。
英文摘要
We consider the nonlinear Schrödinger equation in one dimension with nonlinearities of type $|u|^αu$ for any power $α>0$ and review two different methods for obtaining solutions, namely, local well-posedness, with initial data either in $L^2$ or $H^1$, or in the weighted subspace of $H^1$. One approach is based on the Strichartz estimates, and thus, $H^1$ well-posedness typically holds for nonlinearities with power $α\geq 1$. The other one is a direct application of weighted estimates commuting with derivatives and a certain infimum condition on the initial data, and thus, can treat nonlinearities for the whole range $0 < α< \infty$; furthermore, it can handle a sum of different nonlinearities. We then conclude with an application of the second approach to the NLS with {\it finitely} many combined nonlinearities, important for physical applications (e.g., in laser optics), as it is more challenging, if at all possible, to obtain local well-posedness with the first method due to the lack of scaling invariance.
Commentsforthcoming in Involve
