KINKS IN A STOCHASTIC PDE Grant Lythe Department of Applied Mathematics, University of Leeds, LS2 9JT, UK. ∗ kinkdens@stochastic.org.uk Salman Habib Theoretical Division, Los Alamos National Laboratory, NM87545, USA Abstract We consider a stochastic PDE with localized coherent structures known as kinks. The availability of exact results for the steady state and in- creasing computer power means that precise quantitative comparison between theory and numerics is possible. One of the quantities of in- terest is the steady-state density of kinks, maintained by a balance be- tween nucleation and annihilation of kink-antikink pairs. The density as measured from numerical solutions is sufficiently accurate to resolve the difference between analytical predictions based on the exact value of the correlation length, and those based on the WKB approximation to it. Keywords: Stochastic PDEs, kinks, convergence of algorithms. Many extended systems have localized coherent structures that main- tain their identity as they move about and are buffeted by local fluctua- tions [1, 2]. To study the dynamics, we must accurately solve nonlinear extended systems with noise and devise a sensible method for identi- fying and tracking the coherent structures. Until fairly recently, com- puter memory and performance restrictions were sufficiently severe that stochastic partial differential equations (stochastic PDEs) describing ex- tended systems with noise could only be studied at low resolution. These restrictions are being overcome, at least in one space dimension [3, 4, 5], and good progress has been made towards studying, understanding, and improving the accuracy of numerical solutions. Numerical solutions of stochastic PDEs are cannot be performed in the infinite-dimensional space in which the equation is defined [6, 7], but ∗ http://stochastic.org.uk 1