On a Class of Solutions to the Generalized Derivative Schrödinger Equations

Abstract

In this work we shall consider the initial value problem associated to the generalized derivative Schrödinger equations ∂tu = i∂ 2x u + µ |u| α ∂xu, x, t ∈ R, 0 < α ≤ 1 and |µ| = 1, andFollowing the argument introduced by Cazenave and Naumkin [3] we shall establish the local well-posedness for a class of small data in an appropriate weighted Sobolev space. The other main tools in the proof include the homogeneous and inhomogeneous versions of the Kato smoothing effect for the linear Schrödinger equation established by Kenig-Ponce-Vega in [21].

On a class of solutions to the generalized derivative Schrödinger equations II

“…We observe that the local smoothing effect (1.13) obtained in Theorem 1.1 is slightly weaker than that obtained in [28] which was…”

“…In [28] we proved, roughly, Theorem 1.1 with M = 2 assuming that ν ≤ ǫ for some ǫ = ǫ(α; λ) > 0 small enough. One of the main tool used in [28] was the homogeneous and inhomogeneous versions of the so called Kato smoothing effect [20] for the free Schrödinger group {e it∂ 2…”

“…The last two terms can be handled using the argument presented in detail in [28]. Thus it remains to estimate Φ(v) in the last three norms in (3.8).…”

“…This is extended to solutions of the (1.3) with data u 0 satisfying (1.5) for m = m(α) and u 0 ∈ H s (R N ) with s sufficiently large. Stimulated by the ideas in [2] in our previous paper [28] we established a local well-posedness result for the IVP (1.1) for small data in a Sobolev weighted space with data satisfying (1.5). Our main objetive here is to remove the size restriction on the data in our previous paper [28].…”

On a class of solutions to the generalized KdV type equation

“…In [8], the arguments introduced in [2] were modified to study the IVP associated to the generalized derivative Schrödinger equation…”