We consider synchronization by noise for stochastic partial differential equations which support traveling pulse solutions, such as the FitzHugh–Nagumo equation. We show that any two pulse-like solutions which start from different positions but are forced by the same realization of a multiplicative noise, converge to each other in probability on a time scale σ^-2≪ t≪exp(σ^-2), where σ is the noise amplitude. The noise is assumed to be Gaussian, white in time, colored and periodic in space, and nondegenerate only in the lowest Fourier mode. The proof uses the method of phase reduction, which allows one to describe the dynamics of the stochastic pulse only in terms of its position. The position is shown to synchronize building upon existing results, and the validity of the phase reduction allows us to transfer the synchronization back to the full solution.

We study a parametrically forced nonlinear Schrödinger (PFNLS) equation, driven by multiplicative translation-invariant noise. We show that a solitary wave in the stochastic equation is orbitally stable on a timescale which is exponential in the inverse square of the noise strength. We give explicit expressions for the phase shift and fluctuations around the shifted wave which are accurate to second order in the noise strength. This is done by developing a new perspective on the phase-lag method introduced by Krüger and Stannat. Additionally, we show well-posedness of the equation in the fractional Bessel space H for any s∈ [0,0∞), demonstrating persistence of regularity.