Nonlinear Waves and Growing Interfaces: 1-D Burgers–KPZ Models
The present chapter studies behavior of two standard 1-D nonlinear dynamics models described by partial differential equations of order two and higher: the Burgers equation and the related KPZ model. We shall concentrate our attention on the theory of nonlinear fields of hydrodynamic type, where the basic features of the temporal evolution of nonlinear waves can be studied in the context of competition between the strengths of nonlinear and dissipative and/or dispersive effects. Apart from being model equations for specific physical phenomena, Burgers–KPZ equations are generic nonlinear equations that often serve as a testing ground for ideas for analysis of other nonlinear equations. They also produce a striking typically nonlinear phenomenon: shock formation.