On optimal exponential decay properties of solutions to the Korteweg-de Vries Equation

We consider the Cauchy problem associated to the Korteweg-de Vries equation (KdV) and study the preservation of exponential decay of order 3/2 on the right of the x-axis as time evolves. More precisely, for a solution of the equation which decays at t=0 as e−a0x3/2 for x>0, we find an optimal function a(t) with a(0)=a0 such that the solution decays as e−a(t)x3/2 for x,t>0.