Continuum limits of integrable systems

In the investigation of dispersive phenomena for nonlinear partial differential equations, there are specific systems (referred to as “completely integrable”) which provide a mathematically viable avenue to analyze and classify novel phenomena. In many physical media the predominant behavior is described by nonlinear hyperbolic partial differential equations, whose solutions exhibit shock formation (also called gradient catastrophe). However, in the real physical medium being modeled, this gradient catastrophe does not happen due to micro-scale processes not present in the hyperbolic approximation. Sometimes, a gradient catastrophe is regularized by diffusive processes; such situations have been carefully analyzed in both mathematics and physics communities, and the detailed nature of the diffusive mechanism of regularization is understood in quite general terms. Another mechanism, the dispersive regularization, is much less studied, but perhaps equally natural. The analysis of dispersive regularizations for nonlinear hyperbolic PDEs is important to understand and model physical systems as they transit through the critical point of the shock appearance. There are a host of explicit examples in which the PDE, after regularization, occurs to be an integrable system. Such regularizations are not only useful for describing the microstructure which actually stops the shock formation, but they also provide new methods of solution and analysis of the original nonlinear hyperbolic PDE. The first mathematical analysis of an integrable dispersive regularization has been carried out by P. Lax and C. D. Levermore in their investigation of the Burger's equation as the zero dispersion limit for the KdV equation. We shall focus in continuum (in space variables) generalization of this approach.