TY - JOUR
T1 - Local Well-posedness of Interaction Equations for Short and Long Dispersive Waves
JO - Journal of Partial Differential Equations
VL - 2
SP - 137
EP - 151
PY - 2004
DA - 2004/05
SN - 17
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jpde/5382.html
KW - Short and long dispersive waves
KW - the Fourier restriction norm
KW - the Smoothing effects
AB -  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) ×C([0, T);H^{s-\frac{1}{2}}) for any initial data (u_0, v_0) ∈ H^s(\mathbb{R}) ×H^{s-\frac{1}{2}} (\mathbb{R})(s ≥ \frac{1}{4}) and the solution depends continuously on the initial data.