Global existence and modulation analysis for the stochastic Degasperis–Procesi equation with transport noise

Abstract We study the stochastic Degasperis–Procesi (DP) equation with transport noise. We first establish the local existence and uniqueness of the stochastic DP equation by the viscous truncated approximation equation. Then, we show two global existence results under different initial value conditions. One is the global existence under a non-negative condition of the initial value. The other is the global existence in large probability with the small initial value condition. Moreover, we investigate the influence of random perturbations to the smooth solitary wave of the DP equation. We prove that if the initial condition of the solution is a solitary wave solution of the unperturbed equation, the solution decomposes into the sum of a randomly modulated solitary wave and a small remainder. We derive an estimate on the time up to which the modulation procedure is available. We also show that the remainder converges to the solution of a stochastic linear equation as the noise intensity tends to zero.