Excitations and management of the nonlinear localized gap modes

Abstract

We discuss about the theory of nonlinear localized excitations, such as soliton and compactons in the gap of the linear spectrum of the nonlinear systems. We show how the gap originates in the linear spectrum using examples of a few systems, such as nonlinear lattices, Bose–Einstein condensates in optical lattice and systems represented by coupled nonlinear evolution equations. We then analytically show the excitation of solitons and compacton-like solutions in the gap of the linear spectrum of a system of coupled Korteweg–de Vries (KdV) equations with linear and nonlinear dispersions. Finally, we discuss about the theory of Feshbach resonance management and dispersion management of the soliton solutions.