On the well-posedness of Benney's interaction equation of short and long waves

2x u = ↵|u| 2 u + nu, t, x 2 R, @ t n = @ x (|u| 2 ), u(x, 0) = u 0 (x), n(x, 0) = n 0 (x).(1.1)To describe the interaction between long and short water waves, Benney proposed two systems of dispersive equations ([1], [2]).One of the systems iswhere S and L denote the short wave envelope and the long wave profile, respectively; c g , c l , ↵, and are real constants.When = 0, this equation is a decoupled system and the first equation in (1.2) is the well-known cubic nonlinear Schrödinger equation which has been studied by many authors where both inverse scattering methods and nonlinear evolution equation techniques have been used.Further, N. Yajima-Oikawa used inverse scattering to obtain results for ↵ = 0 and > 0 ([26]) (also see Ma [16] for the case c g = c l ).From the point of view of evolution equations, M. Tsutsumi-Hatano in [21] and [22] investigated the well-posedness, existence, uniqueness, persistence, and continuous dependence upon initial data of the Cauchy problem for Benney's equation (1.2) where the main concern is on the initial data S 0 .They first established local well-posedness in fractional Sobolev spaces under the resonance condition c l = c g .When ↵ = 0 the initial-value problem is locally well-posed for H 1/2 (R); when ↵ 2 R it is locally wellposed in higher spaces, H j+(1/2) where j = 1, 2, 3, . . . .They improved these results to the nonresonance case, c g 6 = c l , and also obtained global well-posedness when ↵ = 0 in the same spaces as above.In both of these results, the largest function space for the initial data S 0 is H 1/2 (R).This is the largest space in which the nonlinear term