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The initial value problems for the nonlinear modulation of dispersive waves are investigated by virtue of the method developed by Zakharov and Shabat. It is studied in general how the modulated waves evolve to decay into solitons moving with their respective speeds or to form the bound state of solitons. The perturbation analysis is applied to investigate the condition for the bisymmetric decay of modulated waves into moving solitons. As a special example, the initial condition of a hyperbolic function type is considered in details. The numerically computed solutions are also shown.
Satsuma et al. (1974) studied this question.