nonlinear evolution, and B is the M ฯซ N dimensional noise term, which is a functional of , and multiplies an A robust semi-implicit central partial difference algorithm for the numerical solution of coupled stochastic parabolic partial differen-N dimensional real or complex Gaussian-distributed stotial equations (PDEs) is described. This can be used for calculating chastic field . All terms in these equations are assumed correlation functions of systems of interacting stochastic fields. to be evaluated at the same point in the independent vari-Such field equations can arise in the description of Hamiltonian able t. The functional notation [] denotes an arbitrary and open systems in the physics of nonlinear processes, and may functional dependence on the fields [], not necessarily include multiplicative noise sources. The algorithm can be used
for studying the properties of nonlinear quantum or classical field evaluated at the same location x (even for field theories theories. The general approach is outlined and applied to a specific derived from local interactions), generically denoted as x. example, namely the quantum statistical fluctuations of ultra-short The stochastic fields (t, x) are generally delta-correlated optical pulses in (2) parametric waveguides. This example uses a in t, although not always in x, so that non-diagonal coherent state representation, and correctly predicts the sub-shot noise level spectral fluctuations observed in homodyne detection measurements. It is expected that the methods used will
be applicable for higher-order correlation functions and other physical problems as well. A stochastic differencing technique for reduc-
ing sampling errors is also introduced. This involves solving nonlinear stochastic parabolic PDEs in combination with a reference process, which uses the Wigner representation in the example pre-One of the most well-known examples of this type of sented here. A computer implementation on MIMD parallel architecstochastic partial differential equation is the time-depentures is discussed. แฎ 1997 Academic Press dent Ginsburg-Landau [4] equation with a stochastic source. This is commonly used to describe systems with critical-point phase-transitions, like super-fluids, lasers, 312