论文标题
重建的KDV方程中的浆果阶段
Berry Phases in the Reconstructed KdV Equation
论文作者
论文摘要
我们考虑了一个圆上的KDV方程及其偏离的重建，这让人联想到流体颗粒的运动方程。对于周期性波，频道重建的运动受到迭代地图的控制，其庞加莱旋转数量产生漂移速度。我们表明该数字具有几何来源：它是动态阶段，浆果相和“异常相”的总和。最后两个数量是通用的：它们仅是由于基本的Virasoro组结构。尤其是浆果相，以前在[Arxiv：1703.06142]中用于二维形式的保形场理论，并从传播波产生的绝热变形中进行了描述。我们用cNoidal波说明了这些一般结果，由于我们得出的统一图，可以以封闭形式评估所有阶段。一路走来，当波变得不可均匀时，我们遇到“轨道分叉”：存在共鸣楔，在cnoidal参数空间中，其中粒子运动锁定在波浪中，而没有这样的锁定在楔形外。
We consider the KdV equation on a circle and its Lie-Poisson reconstruction, which is reminiscent of an equation of motion for fluid particles. For periodic waves, the stroboscopic reconstructed motion is governed by an iterated map whose Poincaré rotation number yields the drift velocity. We show that this number has a geometric origin: it is the sum of a dynamical phase, a Berry phase, and an "anomalous phase". The last two quantities are universal: they are solely due to the underlying Virasoro group structure. The Berry phase, in particular, was previously described in [arXiv:1703.06142] for two-dimensional conformal field theories, and follows from adiabatic deformations produced by the propagating wave. We illustrate these general results with cnoidal waves, for which all phases can be evaluated in closed form thanks to a uniformizing map that we derive. Along the way, we encounter "orbital bifurcations" occurring when a wave becomes non-uniformizable: there exists a resonance wedge, in the cnoidal parameter space, where particle motion is locked to the wave, while no such locking occurs outside of the wedge.
