Marginal State of Modulational Instability —Note on Benjamin-Feir Instability—

Abstract

This paper deals with the marginal state of modulational instability of a nonlinear wave which is governed by a cubic nonlinear Schrödinger (shortly NS) equation far from the marginal state. Considering the gravity water wave as a typical example for which the coefficient of the nonlinear term of the NS equation does vanish at the marginal state, we reduce a new governing equation near the marginal state instead of the NS equation. Based on this new equation, the Benjamin-Feir instability of the Stokes wave is found to be enhanced by the higher order nonlinearity.