White-Noise-Driven KdV-Type Boussinesq System

The white-noise-driven KdV-type Boussinesq system is a class of stochastic partial differential equations (SPDEs) that describe nonlinear wave propagation under the influence of random noise—specifically white noise—and generalize features from both the Korteweg–de Vries (KdV) and Boussinesq equations. We consider a Cauchy problem for two stochastic systems based on the KdV-type Boussinesq equations. For these systems, we determine sufficient conditions to ensure that this problem is locally and globally well posed for initial data in Sobolev spaces by the linear and bilinear estimates and their modification together with the Banach fixed point.