Interaction of two quasi-monochromatic waves in shallow water

We study the nonlinear interaction of waves propagating in the same direction in shallow water characterized by a double-peaked power spectrum. The starting point is the prototypical equation for weakly nonlinear unidirectional waves in shallow water, i.e., the Korteweg–de Vries equation. In the framework of envelope equations, using a multiple-scale technique and under the hypothesis of narrow-banded spectra, a system of two coupled nonlinear Schrödinger equations is derived. The validity of the resulting model and the stability of their plane wave solutions is discussed. We show that when retaining higher order dispersive terms in the system, plane wave solutions become modulationally unstable.