Chaos near the gap soliton in a Kerr grating

Differences are investigated between solutions of the one-dimensional Helmholtz equation for monochromatic waves in a Kerr grating and its approximation, the (generalized) coupled mode (GCM) equations. First it is pointed out that use of the latter can be justified on the basis of averaging theory, and an upper bound is given for the error made this way. Second, the qualitative difference that arises because of the nonautonomous nature of the Helmholtz equation is investigated. The latter property causes that part of the trajectories to be chaotic, in contrast with the periodicity of the solutions of the (autonomous) GCM equations. In particular, standing waves near the gap soliton with (envelope) wavelength of the order of the inverse squared of the index contrast show irregular features. This is concluded from the observed scaling behavior of the dimensions of the chaotic region in the phase plane.