Critical behavior for scalar nonlinear waves
Abstract
In the long wave regime, nonlinear waves may undergo a phase transition from a smooth behavior to a fast oscillatory behavior. In this study, we consider this phenomenon, which is commonly known as dispersive shock, in the light of Dubrovin's universality conjecture (Dubrovin, 2006; Dubrovin and Elaeva, 2012) and we argue that the transition can be described by a special solution of a model universal partial differential equation. This universal solution is constructed using the string equation. We provide a classification of universality classes and an explicit description of the transition with special functions, thereby extending Dubrovin's universality conjecture to a wider class of equations. In particular, we show that the BenjaminOno equation belongs to a novel universality class with respect to those known previously, and we compute its string equation exactly. We describe our results using the language of statistical mechanics, where we show that dispersive shocks share many of the features of the tricritical point in statistical systems, and we also build a dictionary of the relations between nonlinear waves and statistical mechanics.
 Publication:

Physica D Nonlinear Phenomena
 Pub Date:
 February 2015
 DOI:
 10.1016/j.physd.2014.09.007
 arXiv:
 arXiv:1312.3880
 Bibcode:
 2015PhyD..292....1M
 Keywords:

 BenjaminOno equation;
 Dispersive shock;
 Kortewegde Vries equation;
 Phase transition;
 String equation;
 Tricritical point;
 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 10 pages, 7 figures, major modifications, published version