Approximation of the Korteweg–de Vries Equation by the Nonlinear Schrödinger Equation

In 1968 Zakharov derived the Nonlinear Schrödinger equation as an approximation equation for the water wave problem. It is still an open problem if there are really solutions of the water wave problem which can be approximated by the solutions of the Nonlinear Schrödinger equation. In answering this question we made some progress for a model problem—the approximation of the Korteweg–de Vries equation by the Nonlinear Schrödinger equation.