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

Abstract

" The well-posedness of the Cauchy problem for the system {i\u2202_tu + \u2202\u00b2_xu = uv + |u|\u00b2u, t, x \u2208 \\mathbb{R}, \u2202_tv + \u2202_x\u0397\u2202_xv = \u2202_x|u|\u00b2, 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)) \u2208 C([0, T);H^s) \u00d7\u0002C([0, T);H^{s-\\frac{1}{2}}) for any initial data (u_0, v_0) \u2208 H^s(\\mathbb{R}) \u00d7\u0002H^{s-\\frac{1}{2}} (\\mathbb{R})(s \u2265 \\frac{1}{4}) and the solution depends continuously on the initial data."