论文标题
部分可观测时空混沌系统的无模型预测
Rigidity of smooth finite-time blow-up for equivariant self-dual Chern-Simons-Schrödinger equation
论文作者
论文摘要
我们考虑了在模棱两可的对称性中的自偶联chern-schrödinger方程（CSS）的长时间动力学。 （CSS）是一个自dual $ l^{2} $ - 具有伪形式不变性和孤子的关键方程。在本文中，我们表明，任何$ M $ equivariant，$ m \ geq1 $，$ h^{3} $有限的时间爆破解决方案（CSS）都是伪合成的爆炸解决方案。更准确地说，这种解决方案分解为一个以伪符号速率$λ（t）\ sim t-t $和辐射的调制的孤子总和。相反，我们还获得了$ m $ equivariant，$ m \ geq1 $，$ h^{3,3} $ solutions的精制孤独的分辨率定理：任何此类解决方案在伪独立式策略中爆炸（到$ 0 $），或与某些定期级别的阶段相互竞争。据我们所知，这是对任意平滑和空间衰减解决方案的动态的完整分类的第一个结果，包括规模和相位的动力学，在非线性Schrödinger方程类别中，这些方程式不完全集成。 我们的分析不仅基于以前的作品，尤其是作者Kwon和OH的Soliton决议定理，而且还完善了通常在爆破动力学前进构建中通常采用的论点的所有步骤。证明的关键特征是，我们可以识别任何$ h^{3} $有限的爆破解决方案的奇异和规则部分，因此，单数部分的演变受通用调制动力学的控制，而常规零件则保持$ h^{3} $，甚至限制在爆炸时间。作为副产品，我们还观察到渐近谱具有通用的奇异结构。
We consider the long time dynamics for the self-dual Chern-Simons-Schrödinger equation (CSS) within equivariant symmetry. (CSS) is a self-dual $L^{2}$-critical equation having pseudoconformal invariance and solitons. In this paper, we show that any $m$-equivariant, $m\geq1$, $H^{3}$ finite-time blow-up solution to (CSS) is a pseudoconformal blow-up solution. More precisely, such a solution decomposes into the sum of one modulated soliton that contracts at the pseudoconformal rate $λ(t)\sim T-t$, and a radiation. Applying the pseudoconformal transform in reverse, we also obtain a refined soliton resolution theorem for $m$-equivariant, $m\geq1$, $H^{3,3}$ solutions: any such solutions blow up in the pseudoconformal regime, scatter (to $0$), or scatter to a modulated soliton with some fixed scale and phase. To our knowledge, this is the first result on the full classification of the dynamics of arbitrary smooth and spatially decaying solutions, including the dynamics of scale and phase, in the class of nonlinear Schrödinger equations which are not known to be completely integrable. Our analysis not only builds upon the previous works, especially the soliton resolution theorem by the author, Kwon, and Oh, but also refines all steps of the arguments typically employed in the forward construction of blow-up dynamics. The key feature of the proof is that we can identify the singular and regular parts of any $H^{3}$ finite-time blow-up solutions, such that the evolution of the singular part is governed by a universal modulation dynamics while the regular part is kept $H^{3}$-bounded even up to the blow-up time. As a byproduct, we also observe that the asymptotic profile has a universal singular structure.
