In particular, for the one dimensional cubically nonlinear Schrödinger equation and for the cubically nonlinear wave equation we will study moving solitary waves (moving gap solitons) in narrow spectral gaps of perturbed finite band potentials. The amplitude of the perturbation, or equivalently the width of the gap, will play the role of a small asymptotic parameter. The choice of finite band potentials should guarantee the opening of narrow spectral gaps from points of transversal crossings of spectral functions in the band structure. This transversality is expected to lead to the existence of a whole family of gap solitons with an interval of velocities which are independent of the asymptotic parameter. This is in contrast with the standard case of structure with an arbitrary contrast, where only asymptotically slow gap solitons are known asymptotically close to the spectrum.
Firstly, an asymptotic envelope approximation of gap solitons via coupled mode equations will be given. The asymptotics will then be rigorously justified and the approximation error estimated for large but finite time intervals. Numerical computations will then corroborate the analysis, show examples of gap solitons and check the asymptotic convergence rates. The second, and more challenging, part will deal with the existence of exact moving generalized breathers of the two equations. These breathers are moving pulses, whose shape changes periodically in space and time. Based on related existing results only generalized breathers are expected, i.e. such pulses whose profiles are very close to localized ones on very large but finite spatial intervals. It is, once again, the aim to produce a family of breathers with an interval of velocities.