Generalized KdV Equation for Fluid Dynamics and Quantum Algebras

Abstract

We generalize the non-linear one-dimensional equation of a fluid layer for any depth and length as an infinite order differential equation for the steady waves. This equation can be written as a q-differential one, with its general solution written as a power series expansion with coefficients satisfying a nonlinear recurrence relation. In the limit of long and shallow water (shallow channels) we reobtain the well known Korteweg-de-Vries equation together with its single-soliton solution.Comment: 17 pages, Latex, PACS: 47.20.Ky, 43.25.Rq, 47.35.+i, 03.40.Kf, 43.25.Fe, 02.20.Tw, MSC: 16W30, 17B37, 81R50, 35Q51, 34B15, 34L30, 76E3