环面上散焦三次半波方程的无穷级联

中文摘要 AI 辅助

我们构造了周期散焦三次半波方程的全局解，对于每个 $1/2<s<3/2$，其 $H^s$ 范数随着 $t\to+\infty$ 呈指数增长。这样的解存在于每个正质量上。构造从线性摄动 Szego 方程的显式集中轨迹出发，并求解全半波流（包括其负频率分量）的无穷时间终值问题。我们获得了精确的增长率和守恒量。有限 Blaschke 乘积使我们能够指定有限多个集中点及其相对宽度，并计算极限临界动能测度。我们还确定了所构造数据的端点傅里叶衰减，并证明了在共振质量切片上和一般标量参数下的尖锐对数正则性阈值。特别地，据我们所知，这提供了环面上散焦分数阶非线性薛定谔方程真正无穷级联的第一个例子。

英文摘要

We construct global solutions of the periodic defocusing cubic half-wave equation whose $H^s$ norms grow exponentially as $t\to+\infty$ for every $1/2<s<3/2$. Such solutions exist at every positive mass. The construction starts from explicit concentrating trajectories of a linearly perturbed Szego equation and solves an infinite-time final-value problem for the full half-wave flow, including its negative-frequency component. We obtain the exact growth rate and conserved quantities. Finite Blaschke products allow us to prescribe finitely many concentration points and their relative widths, and we compute the limiting critical kinetic-energy measures. We also identify the endpoint Fourier decay of the constructed data and prove the sharp logarithmic regularity threshold on the resonant mass slice and for generic scalar parameters. In particular, to the best of our knowledge, this provides the first example of a genuine infinite cascade for a defocusing fractional nonlinear Schrödinger equation on the torus.

发表机构

• University of Basel（巴塞尔大学）

机构由 AI 辅助整理，请以论文原文为准。