Soliton dynamics for the stochastic modified Camassa-Holm equation

We study the soliton dynamics for the stochastic modified Camassa-Holm 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. Moreover, we derive the equations for the modulation parameters and show that the remainder converges to the solution of a stochastic linear equation as amplitude of the noise tends to zero.