Relationship between Benjamin–Feir instability and recurrence in the nonlinear Schrödinger equation

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Abstract
A simple relationship between the Benjamin–Feir instability associated with uniform solutions of the nonlinear Schrödinger equation and the long time evolution of the unstable solution is reported. The number of modes which actively participate in the energy sharing process associated with the instability is governed by the number of harmonics of the initial disturbance which lie within the unstable region as predicted by the Benjamin–Feir analysis. Generalization of this observation implies that equations which possess high wavenumber cutoffs in the instability characteristics should not thermalize in the conventional sense when undergoing such an instability, since active modes are confined to a finite range of wavenumbers.