The Evolution of Longwave Solutions to the Nonlinear Schrödinger Equation

Abstract

In water of moderate depth, the behavior of small perturbations superimposed on Stokes wave trains is described by the nonlinear (cubic) Schrödinger equation. In the present study wave‐like solutions to this equation are examined, and it is shown that when these perturbations are neutrally stable and sufficiently long, solutions to the Schrödinger equation may be approximated by the well‐known Korteweg–deVries equation. As a result, sufficiently long perturbations to Stokes wave trains may be regarded as mathematically analogous to those imposed on a free surface separating two fluids of different densities. This result is established independently by singular perturbation techniques, numerical computation, and comparison of exact stationary wave solutions.