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International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 4, Pages 593-629

A discrete variational approach for investigation of stationary localized states in a discrete nonlinear Schrödinger equation, named IN-DNLS

Institute of Physics, Bhubaneswar 751005, Orissa, India

Received 9 June 2004

Copyright © 2005 K. Kundu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


IN-DNLS considered here is a countable infinite set of coupled one-dimensional nonlinear ordinary differential difference equations with a tunable nonintegrability parameter. When this parameter vanishes, IN-DNLS reduces to the famous integrable Ablowitz-Ladik (AL) equation. The formation of unstaggered and staggered stationary localized states (SLSs) in IN-DNLS is studied here using a discrete variational method. The functional form of stationary soliton of AL equation is used as the ansatz for SLSs. Derivation of the appropriate functional and its equivalence to the effective Lagrangian are presented. Formation of on-site peaked and intersite peaked unstaggered SLSs and their dependence on the nonintegrability parameter are investigated. On-site peaked states are found to be energetically stable. Results are explained using the effective mass picture. Also, the properties of staggered SLSs of Sievers-Takeno- (ST-) like mode and Page- (P-) like mode are investigated and explained using the same effective mass picture. It is further shown here that an unstable SLS which is found in the truncated analysis of the problem does not survive in the exact calculation. For large-width and small-amplitude SLSs, the known asymptotic result for the amplitude is obtained. Further scope and possible extensions of this work are discussed.