Solitons in parametrically driven discrete nonlinear Schrödinger systems with the exploding range of intersite interactions

Abstract

We present the sequence of parametrically driven discrete nonlinear Schrödinger systems with the progressively extending range of intersite couplings. In the case of time-independent coupling parameters the sequence is reduced to the Ablowitz–Ladik hierarchy, which is known to be integrable by the inverse scattering transform. However the models with the time-dependent intersite interactions are shown to be integrable too irrespective of a particular form of time dependencies of coupling parameters. Any of such parametrically driven systems might exhibit rather complex soliton dynamics and is described by the unconserved Hamiltonian function. We reveal an important subclass of parametrically driven systems demonstrating the parametrical localization of soliton dynamics on a confined domain of space. Meanwhile an appropriate choice of time dependencies in intersite interactions allow us to transform the original parametrically driven system into another one but subjected to the linear external potential. As a result the latter system can be readily integrated as well. In particular the peculiarities of Bloch oscillations in the systems with time-independent long range intersite interactions and linear external potential of constant strength are analyzed. In general, regulating the range of intersite couplings, the strengths and time dependencies of coupling parameters, we are able to model a number of physically important quasi-one-dimensional systems. We develop an alternative approach to solve the Marchenko equations permitting one to obtain the multisoliton solutions in the most simple and natural way. Finally, we point out how to reformulate any model in row in terms of corrected amplitudes with the standard Poisson brackets.