Some local well posedness results in weighted Sobolev space $H^{1/3}$ for the 3-KdV equation

Abstract

The paper analyses the local well posedness of the initial value problem for the nonlinear k-generalized Korteweg-de Vries equation for k = 3 with irregular initial data. k-generalized Korteweg-de Vries equations serve as a model of magnetoacoustic waves in plasma physics, of the nonlinear propagation of pulses in optical fibers. The solvability of many dispersive nonlinear equations has been studied in weighted Sobolev spaces in order to manage the decay at infinity of the solutions. We aim to extend these researches to the k-generalized KdV with k = 3. For initial data in classical Sobolev spaces there are many results in the literature for several nonlinear partial differential equations. However, our main interest is to investigate the situation for initial data in Sobolev weighted spaces, which is less understood. The low regularity Sobolev results for initial value problems for thisnonlinear dispersive equation was established in unweighted Sobolev spaces with s ≥ 1/12 and later further improved for s ≥ −1/6. The paper improves theseresults for 3-KdV equation with initial data from weighted Sobolev spaces.