Stochastic Korteweg-de Vries Equation with and without Damping

Introducing the Korteweg-de Vries equation with a damping term and an external force, the behaviors of solitons under the Gaussian white noise is studied. In the non-damping case, a height and a width of the soliton are proportional to t -3/2 and t 3/2 respectively, for large t . The asymptotic form of the multi-soliton is also discussed. In the damping case, it is found that a width of the soliton is much narrow while a height is much small as compared with the non-damping case. A new idea for the analysis of inhomogeneous soliton equation is suggested.