Weakly damped KdV soliton dynamics with the random force

The soliton dynamics in the random field is studied in the framework of the Korteweg-de Vries-Burgers equation. Asymptotic solution of this equation with weak dissipation is found and the average wave field is analyzed. All formulas can be given explicitly for the uniform (table-top) distribution function of the random field. Weakly damped KdV soliton on large times transforms to the “thick” soliton or KdV-like soliton depending from the statistical properties of the force. New scenario of KdV soliton transformation into the thick soliton and then again in KdV-like soliton is predicted for certain conditions.