Let lambda is an element of R and let H = -1/2 partial derivative(2)(x) + q delta(0) be the one-dimensional Schr & ouml;dinger equation with a repulsive delta potential. We study the Cauchy problem for the nonlinear equation {i partial derivative(t)u(t,x)=Hu(t,x)+lambda divided by u(t,x)divided by(2)u(t,x),(t,x) is an element of R & times; R, u(0,x) = u(0)(x), in L-p-based spaces. Using the boundedness of wave operators, a characterization of Besov space adapted to H, and the cancellation property of the trilinear form T(v(1)(tau),v(2)(tau),v(3)(tau))=U(-tau)(U(-tau)v(1)(tau)U(tau)v2(tau)U(tau)v3(tau)) with U(tau)=e-itH, we demonstrate that under the linear transformation v(t)=U(-t)u(t), the problem is locally well-posed in L-p(R) for 1
