实直线上Szegö方程解的显式公式及其应用
Explicit formula for the solution of the Szegö equation on the real line and applications
摘要：
考虑上半平面Hardy空间中的三次Szegö方程i u_t=Pi（|u| ^2 u），其中Pi是正频率上的Szegö投影.它是一个完全非色散发展方程的模型，并且是完全可积的，因为它允许一个Lax对。我们找到了Szegö方程解的一个显式公式。作为应用，我们证明了孤子的分辨率在H^s的所有s>0，一般的数据。对于非一般数据，我们构造了一个例子，其中孤子分辨率仅在H^s，0<s<1/2中成立，而高Sobolev范数随时间增长到无穷大，即\lim_{t\to\pm\infty}| u（吨）|_{H^s}= \infty，如果s>1/2。作为第二个应用，我们通过求解出现在Lax对中的Hankel算子H_u的反问题，构造了显式广义作用角坐标。特别地，我们证明了具有一般数据的Szegö方程的轨迹是围绕拉格朗日环形柱T^N \times R^N的螺旋线。
英文摘要：
We consider the cubic Szegö equation i u_t=Pi(|u|^2u) in the Hardy space on the upper half-plane, where Pi is the Szegö projector on positive frequencies. It is a model for totally non-dispersive evolution equations and is completely integrable in the sense that it admits a Lax pair. We find an explicit formula for solutions of the Szegö equation. As an application, we prove soliton resolution in H^s for all s>0, for generic data. As for non-generic data, we construct an example for which soliton resolution holds only in H^s, 0<s<1/2, while the high Sobolev norms grow to infinity over time, i.e. \lim_{t\to\pm\infty}|u(t)|_{H^s}=\infty if s>1/2. As a second application, we construct explicit generalized action-angle coordinates by solving the inverse problem for the Hankel operator H_u appearing in the Lax pair. In particular, we show that the trajectories of the Szegö equation with generic data are spirals around Lagrangian toroidal cylinders T^N \times R^N.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
