Numerical solutions of some stochastic hyperbolic wave equations including sine-Gordon equation

Abstract

Many wave equations, including Klein–Gordon, Liouville’s and the sine-Gordon equation, with added space–time white noise can be transformed to second order partial differential equations with mixed derivatives of the form Yxt=F(Y)+σWxt. Such equations are related to what Zimmerman (1972) called a diffusion equation. For such equations an explicit numerical scheme is employed in both deterministic and stochastic examples which is checked for accuracy against known exact analytical deterministic solutions. The accuracy is further tested in the stochastic case with F=0 by comparing statistics of solutions with those for the Brownian sheet. Generally the boundary conditions are chosen to be values of Y(x,0) and Y(0,t) on boundaries of the first quadrant or subsets thereof, existence and uniqueness of solutions having been established for such systems. For the linear case, solutions are compared at various grid sizes and wave-like solutions were found, with and without noise, for non-zero initial and boundary conditions. Surprisingly, putative wave-like structures seemed to emerge with zero initial and boundary conditions and purely noise source terms. Equations considered with nonlinear F included quadratic and cubic together with the sine-Gordon equation. For the latter, wave-like structures were apparent with σ≤0.25 but they tended to be shattered at larger values of σ. Previous work on stochastic sine-Gordon equations is briefly reviewed.