The initial value problem for the 1-D semilinear Schrödinger equation in Besov spaces

Abstract

We define a class of Besov type spaces which is a generalization of that defined by Kenig-Ponce-Vega ([{4}], [{5}]) in their study on KdV equation and non-linear Schrödinger equation. Using these spaces, we prove the following results. the 1-dimendional semilinear Schrödinger equation with the nonlinear term c₁u²+c₂overline{u}² has a unique local-in-time solution for the initial data∈ B_{2, 1}^-3/4, and that with cuoverline{u} has a unique local-in-time solution for the initial data∈ B_{2, 1}^{-1/4, \#}. Note that B_{2, 1}^{-1/4, \#}(\bm{R})⊃ B_{2\bm{, }1}^-1/4(\bm{R})⊃ H^s(\bm{R}) for any s>-1/4.