Analyticity and smoothing effect for the coupled system of equations of Korteweg-de Vries type with a single point singularity

Using Bourgain spaces and the generator of dilation P=3t ∂ _t +x ∂ _x, which almost commutes with the linear Korteweg-de Vries operator, we show that a solution of the initial value problem associated for the coupled system of equations of Korteweg-de Vries type which appears as a model to describe the strong interaction of weakly nonlinear long waves, has an analyticity in time and a smoothing effect up to real analyticity if the initial data only have a single point singularity at x=0.