Classical and quantum mechanical studies of nonlinear lattices.

Abstract

A class of nonlinear Hamiltonian lattice models that includes both the nonintegrable discrete nonlinear Schrodinger and the integrable Ablowitz-Ladik models is investigated classically and quantum mechanically. In general, the model under consideration is nonintegrable and its Hamiltonian structure is derived from a nonstandard Poisson bracket. It is shown that solutions of the classical model can, under appropriate and well-defined conditions, become infinite in finite time (blowup). Under suitable restrictions, it is demonstrated that an associated quantum lattice does not exhibit this singular behavior. In this sense, quantum mechanics can regularize a singular classical phenomenon. A fully nonlinear modulation theory for plane wave solutions of the classical lattice is developed. For cases of the model exhibiting blowup, numerical evidence is presented that suggests the existence of both stable and unstable modulated wavetrains. At the present time, it is unclear the extent to which one may relate the onset of instability to blowup. The Hartree approximation is applied to a generalized discrete self-trapping equation (GDST), with the result that the effective Hartree dynamics are described by a rescaled version of the GDST itself. In this manner, the Hartree approximation gives a direct connection between classical and quantum lattice models. Finally, Weyl's ordering prescription is shown to reproduce perturbative results for a weakly nonlinear oscillator. These results are extended to Hamiltonians that are nonpolynomial functions of the number operator. Extensions to the methodology that permit the treatment of other ordering prescriptions are given and compared with Weyl's rule.