We describe an approach to nonlocal, nonlinear advection in one dimension that extends the usual pointwise concepts to account for nonlocal contributions to the flux. The spatially nonlocal operators we consider do not involve derivatives. Instead, the spatial operator involves an integral that, in a distributional sense, reduces to a conventional nonlinear advective operator. In particular, we...

It is pointed out that for the case of (compressible) magnetohydrodynamics (MHD) with the fields v y (y, t) and B x (y, t) one can have equations of the Burgers type which are integrable. We discuss the solutions. It turns out that the propagation of the non-linear effects is governed by the initial velocity (as in the Burgers case) as well as by the initial Alfvén velocity. Many results previo...

We investigate the non-perturbative results of multi-dimensional forced Burgers equation coupled to the continuity equation. In the inviscid limit, we derive the exact exponents of two-point density correlation functions in the universal region in arbitrary dimensions. We then find the universal generating function and the tails of the probability density function (PDF) for the longitudinal vel...

This article is a short introduction to the surprising appearance of Burgers equation in some basic probabilistic models.