Anomalous hydrodynamics of two-dimensional vortex fluids.

A dense system of vortices can be treated as a fluid and itself could be described in terms of hydrodynamics. We develop the hydrodynamics of the vortex fluid. This hydrodynamics captures characteristics of fluid flows averaged over fast circulations in the intervortex space. The hydrodynamics of the vortex fluid features the anomalous stress absent in Euler's hydrodynamics. The anomalous stress yields a number of interesting effects. Some of them are a deflection of streamlines, a correction to the Bernoulli law, and an accumulation of vortices in regions with high curvature in the curved space. The origin of the anomalous stresses is a divergence of intervortex interactions at the microscale which manifest at the macroscale. We obtain the hydrodynamics of the vortex fluid from the Kirchhoff equations for dynamics of pointlike vortices.

Singularities of the Hele-Shaw flow and shock waves in dispersive media.

We show that singularities developed in the Hele-Shaw problem have a structure identical to shock waves in dissipativeless dispersive media. We propose an experimental setup where the cell is permeable to a nonviscous fluid and study continuation of the flow through singularities. We show that a singular flow in this nontraditional cell is described by the Whitham equations identical to Gurevich-Pitaevski solution for a regularization of shock waves in Korteveg-de Vriez equation. This solution describes regularization of singularities through creation of disconnected bubbles.

Quantum hydrodynamics, the quantum benjamin-ono equation, and the Calogero model.

Collective field theory for the Calogero model represents particles with fractional statistics in terms of hydrodynamic modes--density and velocity fields. We show that the quantum hydrodynamics of this model can be written as a single evolution equation on a real holomorphic Bose field--the quantum integrable Benjamin-Ono equation. It renders tools of integrable systems to studies of nonlinear dynamics of 1D quantum liquids.

Unstable fingering patterns of Hele-Shaw flows as a dispersionless limit of the Kortweg-de vries hierarchy.

We show that unstable fingering patterns of two-dimensional flows of viscous fluids with open boundary are described by a dispersionless limit of the Korteweg-de Vries hierarchy. In this framework, the fingering instability is linked to a known instability leading to regularized shock solutions for nonlinear waves, in dispersive media. The integrable structure of the flow suggests a dispersive regularization of the finite-time singularities.