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In this paper, we derive and study two versions of the short pulse equation(SPE) in $(2+1)-$dimensions. Using Maxwell's equations as a starting point, andsuitable Kramers-Kronig formulas for the permittivity and permeability of themedium, which are relevant, e.g., to left-handed metamaterials and dielectricslab waveguides, we employ a multiple scales technique to obtain the relevantmodels. General properties of the resulting $(2+1)$-dimensional SPEs, includingfundamental conservation laws, as well as the Lagrangian and Hamiltonianstructure and numerical simulations for one- and two-dimensional initial data,are presented. Ultrashort 1D breathers appear to be fairly robust, while rathergeneral two-dimensional localized initial conditions are transformed intoquasi-one-dimensional dispersing waveforms.

Ultrashort pulses and short-pulse equations in $(2+1)-$dimensions