This chapter studies nonlinearity in electromagnetic media. Nonlinearity appears in a number of cases in which the dispersion relations or equivalently the coefficients of the constitutive relations change as a function of the field amplitudes. This behaviour is very common in dielectrics, where it has been experimentally verified, theoretically studied, and widely studied mathematically. Furthermore, the interplay between dispersion and nonlinearity in dielectrics has led to the observation of solitary waves, which has important applications to optical communications. This chapter begins by introducing a model for the study of nonlinear phenomena in complex media, and then provides some rigorous results concerning the solvability and well posedness of the nonlinear model using techniques from the theory of nonlinear PDEs.
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