A Map Approach to Stationary Solutions of the DNLS Equation

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In this chapter we discuss the well-established map approach for obtaining stationary solutions to the one-dimensional (1D) discrete nonlinear Schrodinger (DNLS) equation. The method relies on casting the ensuing stationary problem in the form of a recurrence relationship that can in turn be cast into a two-dimensional (2D) map [1–5]. Within this description, any orbit for this 2D map will correspond to a steady state solution of the original DNLS equation. The map approach is extremely useful in finding localized solutions such as bright and dark solitons. As we will see in what follows, this method allows for a global understanding of the types of solutions that are present in the system and their respective bifurcations. This chapter is structured as follows. In Sect. 11.2 we introduce the map approach to describe steady states for general 1D nonlinear lattices with nearest-neighbor coupling. In Sect. 11.3 we present some of the basic properties of the 2D map generated by the 1D DNLS lattice and how these properties, in turn, translate into properties for the steady-state solutions to the DNLS. We also give an exhaustive account of the possible orbits that can be generated using the map approach. Specifically, we describe in detail the families of extended steady-state solutions (homogeneous, periodic, quasi-periodic, and spatially chaotic) as well as spatially localized steady states (bright and dark solitons and multibreather solutions). In Sect. 11.4 we study the limiting cases of small and large couplings. We briefly describe the bifurcation process that is responsible for the mutual annihilation of localized solutions through a series of bifurcations. For a more detailed account of the bifurcation scenaria for the DNLS using the map approach, see [3].