Traveling Solitary Waves in the Periodic Nonlinear Schr\"odinger Equation with Finite Band Potentials

The paper studies asymptotics of moving gap solitons in nonlinear periodic structures of finite contrast ("deep grating") within the one dimensional periodic nonlinear Schr\"odinger equation (PNLS). Periodic structures described by a finite band potential feature transversal crossings of band functions in the linear band structure and a periodic perturbation of the potential yields new small gaps. Novel gap solitons with O(1) velocity despite the deep grating are presented in these gaps. An approximation of gap solitons is given by slowly varying envelopes which satisfy a system of generalized Coupled Mode Equations (gCME) and by Bloch waves at the crossing point. The eigenspace at the crossing point is two dimensional and it is necessary to select Bloch waves belonging to the two band functions. This is achieved by an optimization algorithm. Traveling solitary wave solutions of the gCME then result in nearly solitary wave solutions of PNLS moving at an O(1) velocity across the periodic structure. A number of numerical tests are performed to confirm the asymptotics.