On the existence of gap solitons

The role of high harmonics in the gap soliton evolution is studied. The problem is considered from two viewpoints. First, it is shown that being treated as purely mathematical objects (i.e., localized in space and periodic in time solutions of a nonlinear wave equation), gap solitons do not exist due to the energy transfer to high harmonics, which are spatially delocalized. Second, in spite of this fact, by direct numerical solution of the nonlinear wave equation it is found that, subject to certain conditions (namely, when the CW frequency is sufficiently close to the definite stop band edge), gap solitons can be treated as existing in the physical sense, i.e., as long-living electromagnetic pulses. Moreover, they can be described well by the usual approximation of the lowest harmonics, i.e., within the framework of the parabolic approximation. It is shown that the decay of the gap solitons is accompanied by emission of linear waves and by formation of shock waves.