Long-time evolution of an unstable water-wave train

The long-time evolution of an unstable wave train, consisting of a carrier wave and two 'side-band’ components, is investigated analytically. Mathematical expressions, involving Jacobian elliptic functions, for the wave envelope characteristics are derived. The solution yields the dependence of the long-time evolution on the initial disturbance. Of special interest is the simple formula for the modulation-demodulation recurrence period. The latter is shown to yield results in good agreement with those obtained from numerical solutions of the nonlinear Schrödinger equation.