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The modulational stability of ground state solitary wave solutions of nonlinear Schrödinger equations relative to perturbations in the equation and initial data is studied. In the “subcritical case” ground states are shown by variational methods to be stable modulo time-dependent adjustments (modulations) of free parameters. These parameters satisfy the modulation equations, a coupled system of nonlinear ODE’S governing the amplitude, phase, position and speed of the dominant solitary wave part of the solution.
Michael I. Weinstein (Wed,) studied this question.