The modified reductive perturbation method as applied to Boussinesq equation: strongly dispersive case Hilmi Demiray Faculty of Arts and Sciences, Department of Mathematics, Isik University, Bu ¨yu ¨kdere Caddesi, 34398 Maslak, Istanbul, Turkey Abstract In this work, we extended the application of ‘‘the modified reductive perturbation method’’ to Boussinesq equation for strongly dispersive case and tried to obtain the con- tribution of higher order terms in the perturbation expansion. It is shown that the first order term in the perturbation expansion is governed by the non-linear Schro ¨dinger equation and the second order term is governed by the linearized Schro ¨dinger equation with a non-homogeneous term. In the long-wave limit, a travelling wave type of solution to these equations is also given. Ó 2004 Published by Elsevier Inc. 1. Introduction In collisionless cold plasma, in fluid-filled elastic tubes and in shallow-water waves, due to nonlinearity of the governing equations, for weakly dispersive case one obtains the Korteweg–de Vries (KdV) equation for the lowest order term in the perturbation expansion, the solution of which may be described 0096-3003/$ - see front matter Ó 2004 Published by Elsevier Inc. doi:10.1016/j.amc.2004.06.076 E-mail address: demiray@isikun.edu.tr Applied Mathematics and Computation 164 (2005) 1–9 www.elsevier.com/locate/amc