On integrable discretization of the inhomogeneous Ablowitz-Ladik model.
نویسنده
چکیده
On integrable discretization of the inhomogeneous Ablowitz-Ladik model. Abstract An integrable discretization of the inhomogeneous Ablowitz-Ladik model with a linear force is introduced. Conditions on parameters of the discretization which are necessary for reproducing Bloch oscillations are obtained. In particular, it is shown that the step of the discretization must be comensurable with the period of oscillations imposed by the inhomo-geneous force. By proper choice of the step of the discretization the period of oscillations of a soliton in the discrete model can be made equal to an integer number of periods of oscillations in the underline continuous-time lattice.
منابع مشابه
A note on the integrable discretization of the nonlinear Schrödinger equation
We revisit integrable discretizations for the nonlinear Schrödinger equation due to Ablowitz and Ladik. We demonstrate how their main drawback, the non-locality, can be overcome. Namely, we factorise the non-local difference scheme into the product of two local ones. This must improve the performance of the scheme in the numerical computations dramatically. Using the equivalence of the Ablowitz...
متن کاملAn Alternative Approach to Integrable Discrete Nonlinear Schrödinger Equations
In this article we develop the direct and inverse scattering theory of a discrete matrix Zakharov-Shabat system with solutions Un and W n. Contrary to the discretization scheme enacted by Ablowitz and Ladik, a central difference scheme is applied to the positional derivative term in the matrix Zakharov-Shabat system to arrive at a different discrete linear system. The major effect of the new di...
متن کاملIntegrable Discretization of the Coupled Nonlinear Schrödinger Equations
A discrete version of the inverse scattering method proposed by Ablowitz and Ladik is generalized to study an integrable full-discretization (discrete time and discrete space) of the coupled nonlinear Schrödinger equations. The generalization enables one to solve the initial-value problem. Soliton solutions and conserved quantities of the full-discrete system are constructed.
متن کاملA systematic method for constructing time discretizations of integrable lattice systems: local equations of motion
We propose a new method for discretizing the time variable in integrable lattice systems while maintaining the locality of the equations of motion. The method is based on the zero-curvature (Lax pair) representation and the lowest-order “conservation laws”. In contrast to the pioneering work of Ablowitz and Ladik, our method allows the auxiliary dependent variables appearing in the stage of tim...
متن کاملOn an integrable discretization of the modified Korteweg-de Vries equation
We find time discretizations for the two ”second flows” of the Ablowitz–Ladik hierachy. These discretizations are described by local equations of motion, as opposed to the previously known ones, due to Taha and Ablowitz. Certain superpositions of our maps allow a one–field reduction and serve therefore as valid space–time discretizations of the modified Korteweg-de Vries equation. We expect the...
متن کامل