A study of a new class of discrete nonlinear Schr dinger equations

A new class of 1D discrete nonlinear Schrödinger Hamiltonians with tunable nonlinearities is introduced, which includes the integrable Ablowitz–Ladik system as a limit. A new subset of equations, which are derived from these Hamiltonians using a generalized definition of Poisson brackets, and collectively referred to as the N-AL equation, is studied. The symmetry properties of the equation are discussed. These equations are shown to possess propagating localized solutions, having the continuous translational symmetry of the one-soliton solution of the Ablowitz–Ladik nonlinear Schrödinger equation. The N-AL systems are shown to be suitable for studying the combined effect of the dynamical imbalance of nonlinearity and dispersion and the Peierls–Nabarro potential, arising from the lattice discreteness, on the propagating solitary wave-like profiles. A perturbative analysis shows that the N-AL systems can have discrete breather solutions, due to the presence of saddle centre bifurcations in phase portraits. The unstaggered localized states are shown to have positive effective mass. On the other hand, large-width but small-amplitude staggered localized states have negative effective mass. The collision dynamics of two colliding solitary wave profiles is studied numerically. Notwithstanding colliding solitary wave profiles that are seen to exhibit nontrivial nonsolitonic interactions, certain universal features are observed in the collision dynamics. The future scope of this work and possible applications of the N-AL systems are discussed.