Discrete Nonlinear Schrödinger Equations with Time-Dependent Coefficients (Management of Lattice Solitons)

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The general topic of the book into which this chapter is incorporated is the discrete nonlinear Schrodinger (DNLS) equation as a fundamental model of nonlinear lattice dynamics. The DNLS equation helps to study many generic features of nonintegrable dynamics in discrete media [1]. Besides being a profoundly important model in its own right, this equation has very important direct physical realizations, in terms of arrays of nonlinear optical waveguides (as was predicted long ago [2] and demonstrated in detail more recently, see [3, 4] and references therein), and arrays of droplets in Bose–Einstein condensates (BECs) trapped in a very deep optical lattice (OL), see details in the original works [5–10] and the review [11]. In all these contexts, discrete solitons are fundamental localized excitations supported by the DNLS equation. As explained in great detail in the rest of the book, the dynamics of standing solitons, which are pinned by the underlying lattice, is understood quite well, by means of numerical methods and analytical approximations (the most general approximation is based on the variational method [12, 13]). A more complex issue is posed by moving discrete solitons [14–17]. While, strictly speaking, exact solutions for moving solitons cannot exist in nonintegrable lattice models because of the radiation loss, which accompanies their motion across the lattice, direct simulations indicate that a soliton may move freely if its norm (“mass”) does not exceed a certain critical value [17]. In the quasi-continuum approximation, the moving soliton may be considered, in the lowest (adiabatic) approximation, as a classical mechanical particle which moves across the effective Peierls–Nabarro (PN) potential induced by the lattice [18–21]. In this limit, the radiation loss is a very weak nonadiabatic effect, which attests to the deviation of the true soliton dynamics from that of the point-like particle. In the case of the DNLS equation describing arrays of nearly isolated droplets of a BEC trapped in a deep OL, an interesting possibility is to apply the Feshbach