On the cauchy problem for the coupled Klein Gordon Schrödinger system with rough data

Abstract

Aninteraction equations of the complex scalar nucleon field andreal scalar meson field is considered. we show that the Cauchy problem of the Klein-Gordon-Schrödinger system$ iu_{t}+u_{x x}=-uv, $$ v_{t t}-v_{x x}+v=|u|^2,$$u(0, x)= u_0(x), v(0, x)= v_0(x), v_t(0, x)= v_1(x)$ is locally well-posed for weak initial data $(u_0, v_0, v_1)\in H^s\times H^{s-1/2}\times H^{s-3/2}$ with $s\geq 0$. We use the analogous method for estimate the nonlinear couple terms developed by Bourgain and refined by Kenig, Ponce and Vega.