Abstract

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