Solitons in dissipative systems subjected to random force within the Benjamin–Ono type equation

Solitary wave dynamics is investigated under the assumption of small dissipation and an external random force. Through a change of variables, the problem becomes homogeneous, allowing for the derivation of asymptotic algebraic soliton solutions. This change of variables makes the randomness manifest primarily on the soliton phases. Consequently, the averaged soliton field and the statistical moments can be computed analytically, assuming that the phase follows a uniform distribution. In the absence of Reynolds dissipation, we show that the soliton-averaged field tends to spread and dampen as the dispersion increases. In addition, in the presence of Reynolds dissipation, we demonstrate that algebraic solitons can transition between thick and thin soliton states. Moreover, when there is viscosity in the upper moving layer, the averaged soliton field exhibits a dynamic evolution from soliton to thick soliton to soliton, contingent upon the parameter settings.