Mathematics and Computers in Simulation 65 (2004) 535–545 Exact solutions and invariants of motion for general types of regularized long wave equations S. Hamdi a,∗ , W.H. Enright a , W.E Schiesser b , J.J. Gottlieb c a Department of Computer Science, University of Toronto, 10 King’s College Road, Toronto, Canada M5S 3G4 b Mathematics and Engineering, Lehigh University, Bethlehem, PA 18015, USA c Institute for Aerospace Studies, University of Toronto, 4925 Dufferin Street, Toronto, Canada M3H 5T6 Abstract New exact solitary wave solutions are derived for general types of the regularized long wave (RLW) equation and its simpler alternative, the generalized equal width wave (EW) equation, which are evolutionary partial differential equations for the simulation of one-dimensional wave propagation in nonlinear media with dispersion processes. New exact solitary wave solutions are also derived for the generalized EW-Burgers equation, which models the propagation of nonlinear and dispersive waves with certain dissipative effects. The analytical solutions for these model equations are obtained for any order of the nonlinear terms and for any given value of the coefficients of the nonlinear, dispersive and dissipative terms. Analytical expressions for three invariants of motion for solitary wave solutions of the generalized EW equation are also devised. © 2004 IMACS. Published by Elsevier B.V. All rights reserved. Keywords: Equal width wave equation; Nonlinearity; Dispersion; Exact solutions; Solitary waves; Invariants of motion 1. Introduction The well-known Korteweg-de Vries (KdV) equation, u t + uu x + u xxx = 0, is a nonlinear partial differential equation (PDE) that models the time-dependent wave motion in one space dimension in media with nonlinear wave steepening and dispersion, such as shallow water waves and ion acoustic plasma waves. The pioneering study by Korteweg and de Vries [8] in 1895 showed that when nonlinear wave steepening, from the term uu x , is balanced by wave dispersion, owing to the term u xxx , their equation predicts a unidirectional solitary wave, that is a pulse which moves in one direction with a permanent shape and constant speed. Benjamin, Bona and Mahoney [2] advocated that the PDE u t + uu x + u x - μu xxt = 0 modeled the same physical phenomena equally well as the KdV equation, given the same assumptions and approximations that were originally used by Korteweg and de Vries [8]. This PDE of Benjamin et al. ∗ Corresponding author. Present address: Hydraulic Engineering, Civil and Marine Infrastructures, Canadian Coast Guard, Fisheries and Oceans Canada, 101 Champlain Blvd., Que., Canada G1K 7Y7. E-mail address: samir.hamdi@utoronto.ca (S. Hamdi). 0378-4754/$30.00 © 2004 IMACS. Published by Elsevier B.V. All rights reserved. doi:10.1016/j.matcom.2004.01.015