The Boussinesq equation for the surface waves reduces to a 4th-order elliptic equation for steady moving waves. This is investigated for bifurcation in 2D by devising a finite-difference scheme and an iterative algorithm. We prove that the truncation error of the scheme is second-order in spac. Next, we develop a perturbation series with respect to the small parameter (square of the phase speed of the wave). Within 2nd order of the small parameter, we derive a hierarchy of 1D equations that are 4th-order in the radial variable and solve the ODEs. We create special approximations to handle the...