Dynamics of complex singularities

 Pole dynamics and the exact solution of the Flierl - Petviashvili equation

Part of the elementary instinct of a theoretician consists of manifesting reserve against the raw formulation of a new problem. We usually receive a new problem expressed in terms that are strongly influenced by observations or experiment and this may not be an adequate framework for theoretical investigation. Almost always, it is better to expand the framework, imagining that the system has actually been projected on our plane of observation, and extensions are needed if we want to capture its true internal organization. We extend by adding complex variables or new dimensions or non-Abelian algebraic content.

Or, we look to an alternative formulation, not changing yet the physical picture. This happens when we make Fourier transformation (moving to the spectral space). Or, when we leave the real-space dynamics to look for the motion of the singularities of the solution in the complex plane of the time variable.

We apply this approach to the search for an exact solution of the Flierl Petviashvili equation. This equation is born in the physics of the atmosphere but occurs in two-dimansional plasma as well. It is interesting that in plasma it is derived under general assumption and, rather unexpectedly, provides a geometry of flow that looks like the zonal flows with alternating directions.

Here is a text written in 2003.

Pole dynamics for the Flierl - Petviashvili equation and zonal flow

The work is on arxiv.org

It has been published in Phys Rev Letters.

 Pole dynamics and stability of the tokamak zonal flows

We investigate the possible consequences of the exact solution of the Flierl Petviashvili equation, which we derived by re-construction from complex poles. The solution appears to offer general characteristics that are observed in experiments. Here we study the limit of stability of the structure of alternating poloidal flows.

Notes on a possible phenomenology of internal transport barriers in tokamak

The work is on arxiv.org

The numerical aspect should still be expanded.

 Pole dynamics applied to expanding fronts

We have applied known technical methods based on the dynamics of the complex singularities to problems of cloud expansion in the physics of the atmosphere.

This is in the Section Physics of the Atmosphere