Quadratic forms for the 1-D semilinear Schrödinger equation

Abstract

This paper is concerned with 1-D quadratic semilinear Schrödinger equations. We study local well posedness in classical Sobolev space H s H^s of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of s s which guarantees the desired local well posedness result. We prove that at least for the quadratic cases these values are negative and depend on the structure of the nonlinearity considered.