Abstract:
The paper analyses the local well posedness of the initial value problem for the 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 this dispersive equation was established in unweighted Sobolev spaces with
s ≥ 1/12 and later further improved for s ≥
−1
6
. The paper improves these results for 3-KdV
equation with initial data from weighted Sobolev spaces.
