Nonlinear development of modulated two-dimensional gravity wavetrains in deep water

Abstract

This paper gives an analytic description of the nonlinear development of the modulational instability with the two-dimensional nonlinear (cubic) Schrodinger equation governing the evolution of a weakly nonlinear, deep-water gravity wavetrain and tries to shed some light on the previous numerical and analytical results on this problem. Toward this objective, the author first develops a proper formulation of the method of multiple scales to describe the long-time behaviour of the linearly unstable modulation near the threshold for linear instability for the two-dimensional case and investigate whether the nonlinear development of this modulation indicates a quasi periodic motion in the latter case.