Abstract
The time evolution of the Benjamin-Feir unstable mode in two dimensions is studied analytically when it resonates with a periodic soliton. The condition for resonance is obtained from an exact solution to the hyperbolic Davey-Stewartson equation. It is shown that a growing-and-decaying mode exists only in the backward (or forward) region of propagation of the periodic soliton if the resonant condition is exactly satisfied. Under a quasiresonant condition, the mode grows at first on one side from the periodic soliton, but decays with time. The wave field shifts to an intermediate state, where only a periodic soliton in a resonant state appears. This intermediate state persists over a comparatively long time interval. Subsequently, the mode begins to grow on the other side from the periodic soliton.
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