Generalized KdV equation for fluid dynamics and quantum algebras

We generalize the nonlinear 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 KdV equation together with its single-soliton solution.This article is dedicated to Max Jammer on the occasion of his 80th birthday. He is a clear friend, who was a visiting professor at the Johann Wolfgang Goethe-University in 1985. His brilliant lectures on Philosophische Probleme der modernen Physik, delivered in perfect German language, will long be remembered. Above all, Max Jammer's contributions to the philosophy of science have enriched our understanding forever.