In this article, a model for the propagation of long waves over an uneven bottom is considered. We provide both theoretical and numerical results for this model. We also discuss the changes which occur in a solitary wave solution of the BBM equation as it travels through a channel of decreasing depth. | Turkish Journal of Mathematics Research Article Turk J Math (2013) 37: 652 – 664 ¨ ITAK ˙ c TUB doi: BBM equation with non-constant coefficients Amutha SENTHILKUMAR∗ Department of Mathematics, University of Bergen, Postbox 7803, 5008 Bergen, Norway Received: • • Accepted: Published Online: • Printed: Abstract: In this article, a model for the propagation of long waves over an uneven bottom is considered. We provide both theoretical and numerical results for this model. We also discuss the changes which occur in a solitary wave solution of the BBM equation as it travels through a channel of decreasing depth. Key words: Solitary waves, BBM equation, uneven bottom 1. Introduction Attention is given to the propagation of water waves for uneven bottoms with an incompressible, inviscid, irrotational fluid in a constant gravitational field. Our study investigates the motion of the free surface of fluid under the assumption of small amplitude shallow water. The first study was conducted by Russell in 1834, and he discovered solitary waves, which are stable and do not disperse with time [7, 13, 17]. Russell’s observations and experiments gave a motivation to find a mathematical formulation of such waves. Scientists such as Airy, Stokes, Boussinesq and Rayleigh investigated such waves in an attempt to understand this phenomenon. Boussinesq and Rayleigh separately got approximate descriptions of the solitary wave. Boussinesq derived non-linear onedimensional evolution equation (Boussinesq approximation). Finally Korteweg and de Vries derived a non-linear evolution equation in dimensional form u t + c0 u x + 3 c0 1 uux + c0 h20 uxxx = 0, 2 h0 6 () which approximates the propagation of unidirectional, small amplitude long waves in non-linear dispersive √ media. Here h0 denotes the undisturbed water depth for flat bottom, c0 = gh0 is the long wave speed, g is the .
