The nonlinear Schrödinger limit and the initial layer of the Zakharov equations (Q5905555)

It is considered the formation of the convergence of solutions of the Zakharov equations, \[ (1) \quad i(\partial E/\partial t)+\Delta E=nE, \qquad (2) \quad \lambda^{-2}(\partial^ 2n/\partial^ 2t)-\Delta n=\Delta| E|^ 2, \] where \(E\) and \(n\) are functions on the time- space \(R\times R^ N\) with values in \(C^ N\) and \(R\), respectively, in the case when the ion sound speed \(\lambda\) goes to infinity. Analyzing the precise rate of convergence of the solutions as \(\lambda\to\infty\), the authors describe how the solutions of equations (1), (2) tend to the corresponding solutions for the nonlinear Schrödinger equation, the latter is known to have unique solutions, and it is exactly integrable by the inverse scattering transform.