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In his 1983 paper [3], R. F. Gundy introduced a new functional related to the Littlewood-Paley theory, called the$density of the area integral$. In this paper, we prove that this functional (although highly non-linear) can be expressed as the principal value of an explicit singular integral. This result provides us with a new and precise connection between the density of the area integral and the theory of Calderón-Zygmund operators. It does not seem to be a consequence of the standard Calderón-Zygmund-Cotlar theory, because the$sign$of a harmonic function in the half-space fails to have, in some appropriate sense, boundary limits.

Solitons of the power-energy saturation (PES) equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by solutions of the classical nonlinear Schrödinger (NLS) equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulse amplitude(s), velocity(ies), position(s), and phase(s) using integral relations derived from the PES equation. The results from the integral relations are shown to agree with multi-scale perturbation theory. It is shown that the single soliton case exhibits mode-locking behavior for a wide range of parameters, while the higher states form effective bound states. Using the fact that there is weak overlap between tails of interacting solitons, evolution equations are derived for the relative amplitudes, velocities, positions, and phase differences. Comparisons of interacting soliton behavior between the PES equation and the classical NLS equation are also exhibited.

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