Burgers' equation

Burgers' equation is a fundamental partial differential equation occurring in various areas of applied mathematics, such as fluid mechanics,nonlinear acoustics,gas dynamics, traffic flow. It is named for Johannes Martinus Burgers (1895–1981).

For a given field and diffusion coefficient (or viscosity, as in the original fluid mechanical context) , the general form of Burgers' equation (also known as viscous Burgers' equation) in one space dimension is the dissipative system:

Added space-time noise forms a stochastic Burgers' equation

This stochastic PDE is equivalent to the Kardar-Parisi-Zhang equation in a field upon substituting . But whereas Burgers' equation only applies in one spatial dimension, the Kardar-Parisi-Zhang equation generalises to multiple dimensions.

When the diffusion term is absent (i.e. d=0), Burgers' equation becomes the inviscid Burgers' equation:

which is a prototype for conservation equations that can develop discontinuities (shock waves). The previous equation is the 'advection form' of the Burgers' equation. The 'conservation form' is