Quadratic forms for the 1-D semilinear Schrödinger equation

This paper is concerned with 1-D quadratic semilinear
Schrödinger equations. We study local well posedness in classical Sobolev space of the associated initial value problem and periodic boundary value problem. Our main interest is to obtain the lowest value of 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.