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Summary form only given. A coupled resonator optical waveguide (CROW) is a recently-introduced family of waveguides that structurally comprises a periodic array of isolated elements (e.g., high-Q resonators such as defects in photonic crystals) weakly coupled to one another. Waveguiding is fundamentally a consequence not of total internal reflection or Bragg reflection from a periodic structure, but instead of the overlap between the individual resonator modes of the structural elements. In direct correspondence with the description of electrons in a weak periodic potential in solid state physics, a CROW can be described using the tight binding approximation; this has been verified by recent experiments in the microwave and optical regimes. We have derived the fundamental representation of pulse propagation in a CROW that represents the effects of a nonlinear electric polarization generated by optical fields, as the starting point for the analysis of all nonlinear optical phenomena in the coupled mode framework.