Existence and smoothing effect of solutions for the Zakharov equations (Q1327859)

The paper under review concerns the Zakharov equations \[ (1) \quad i {\partial E \over \partial t} + \Delta E = nE, \qquad (2) \quad {\partial^2n \over \partial t^2} - \Delta n = \Delta |E |^2,\;t > 0,\;x \in \mathbb{R}^N \] with initial data \[ E(0,x) = E_0 (x), \quad n(0,x) = n_0 (x), \quad {\partial \over \partial t} n(0,x) = n_1 (0,x). \tag{3} \] Here, \(E\) is a function from \(\mathbb{R}^+_t \times \mathbb{R}^N_x\) to \(\mathbb{C}^N\), \(n\) is a function from \(\mathbb{R}^+_t \times \mathbb{R}^N_x \) to \(\mathbb{R}\) and \(1 \leq N \leq 3\). The system of equations (1)--(3) describes the long wave Langmuir turbulence in a plasma. A certain limiting case of this system is related to the nonlinear Schrödinger equation \[ i {\partial E \over \partial t} + \Delta E = - |E |^2 E. \tag{4} \] It is well known that equation (4) possesses nice smoothing properties, and it is natural to conjecture whether some of these properties are also shared by the system (1)--(3). The authors study the local solvability and the smoothing effect of the (1)--(3). They establish local existence and uniqueness of strong solutions in certain Sobolev and weighted Sobolev spaces. Their results include the case of initial data \((E_0, n_0, n_1) \in H^2 \oplus H^1 \oplus L^2\). Concerning the study of the smoothing properties, the authors prove a nice result that clarifies the similarities and differences between the system (1)--(3) and the nonlinear Schrödinger equation (4).