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In the first part of this paper, Noether's theorem and the scale invariance of the nonlinear Schrodinger (NLS) equation are used to derive a modified conservation law for the NLS equation. When used with a soliton solution and applied to the perturbed NLS equation, this modified conservation law is shown to lead naturally to the perturbation equation for a soliton's phase (in a manner similar to the way in which the other, more familiar, conservation laws lead to perturbation equations for a soliton's amplitude, velocity, and position.) Thus, all four of the perturbation equations for the soliton parameters can be obtained directly from conservation laws. These perturbation equations then are applied to an example for which determining phase behavior is important: the parametric amplification of solitons in an optical fiber. The perturbation equations are shown to provide a good approximation to the full perturbed soliton dynamics.

A modified conservation law for the phase of the nonlinear Schrödinger soliton