Journal of Applied Mathematics
Volume 2011 (2011), Article ID 863161, 11 pages
http://dx.doi.org/10.1155/2011/863161
Research Article

Stability of the NLS Equation with Viscosity Effect

N. Karjanto1 and K. M. Tiong2

1Department of Mathematics, University College, Sungkyunkwan University, Natural Science Campus, Jangan-gu, Suwon, Gyeonggi-do 440-746, Republic of Korea
2Nottingham University Business School, The University of Nottingham Malaysia Campus, Jalan Broga, Semenyih 43500, Selangor, Malaysia

Received 16 November 2010; Accepted 11 January 2011

Academic Editor: Ramon Codina

Copyright © 2011 N. Karjanto and K. M. Tiong. 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.

Abstract

A nonlinear Schrödinger (NLS) equation with an effect of viscosity is derived from a Korteweg-de Vries (KdV) equation modified with viscosity using the method of multiple time scale. It is well known that the plane-wave solution of the NLS equation exhibits modulational instability phenomenon. In this paper, the modulational instability of the plane-wave solution of the NLS equation modified with viscosity is investigated. The corresponding modulational dispersion relation is expressed as a quadratic equation with complex-valued coefficients. By restricting the modulational wavenumber into the case of narrow-banded spectra, it is observed that a type of dissipation, in this case the effect of viscosity, stabilizes the modulational instability, as confirmed by earlier findings.