Stochastic D-bifurcation for a damped sine-Gordon equation with noise

Abstract

We investigate the stochastic bifurcation of a damped sine-Gordon equation with Dirichlet boundary conditions under the influence of multiplicative Gaussian white noise. Introducing a slow time scale, we derive the amplitude equations near the trivial solution by multiscale analysis. And the stationary probability density functions are formulated analytically using the stochastic averaging of energy envelope. The numerical calculations show that the system undergoes a stochastic D-bifurcation of energy envelope from a delta measure to new stationary measures when the control parameter crosses a critical point. C 2015 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License. [http://dx.

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

“…Technical analysis of stochastic sine-Gordon equations and its generalizations has been performed. Recently, such endeavours have been pursued energetically: for example by Hairer and Shen (2015) for 2 space dimensions, Anton et al (2015) with two-parameter white noise on the unit square, Guarcello et al (2015) with a numerical study of breathers in long Josephson junctions and Huang et al (2015) with a study of stochastic bifurcation theory.…”