Asynchronous development of the Benjamin-Feir unstable mode: solution of the Davey-Stewartson equation

The long time evolution of the Benjamin-Feir unstable mode in two dimension is described by the growing-and-decaying mode solution to the Davey-Stewartson equation. The solution of the hyperbolic Davey-Stewartson (the so-called Davey-Stewartson I) equation is analyzed to show that the resonance between line soliton and growing-and-decaying mode exists. If the resonant condition is exactly satisfied, the growing-and-decaying mode exists only in the forward region of propagation of soliton and the soliton is accelerated (or decelerated). Under the quasiresonant condition, the growing-and-decaying mode grows at first in the forward region, and after the sequence of the evolution has done in the forward region the mode starts to grow in the backward region of the soliton.