Dispersive Blowup of Solutions of Generalized Korteweg-de Vries Equations

The strong effect of dispersion on short-wavelength disturbances featured by the Korteweg-de Vries equation and some of its generalizations is exploited to provide solutions of these equations that correspond to infinitely smooth initial data, which exhibit a specified loss of spatial smoothness at particular times. The points in space-time at which smoothness is lost may even comprise an arbitrary discrete subset of the upper half-plane ) { (x, t): x , t ≥ 0}. Our results are related to recent work on smoothing of solutions of such equations, some of which are sharpened here, and they show that in certain aspects these earlier results are not far from being optimal. The theory makes use of new results concerning well-posedness of such equations in weighted Sobolov spaces and some detailed analysis of the linear Korteweg-de Vries equation.