Local Well-posedness of Interaction Equations for Short and Long Dispersive Waves

Abstract

" The well-posedness of the Cauchy problem for the system {i∂_tu + ∂²_xu = uv + |u|²u, t, x ∈ \\mathbb{R}, ∂_tv + ∂_xΗ∂_xv = ∂_x|u|², u(0, x) = u_0(x), v(0, x) = v_0(x), is considered. It is proved that there exists a unique local solution (u(x, t), v(x, t)) ∈ C([0, T);H^s) x\u0002C([0, T);H^{s-\\frac{1}{2}}) for any initial data (u_0, v_0) ∈ H^s(\\mathbb{R}) x\u0002H^{s-\\frac{1}{2}} (\\mathbb{R})(s ≥ \\frac{1}{4}) and the solution depends continuously on the initial data."