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 so-called "behavioral"
conditions at the point of singularity. Comparison of
the results obtained with the two different
techniques is in excellent agreement and validated.
We discover that the shape of the moving soliton
decays as inverse-square of the radial distance from
the center of the base, while the profile of the
standing soliton decays exponentially. This means
that the asymptotic behavior of the solution is not
robust, a novel result. Our results are of importance
both for the mathematical theory of Boussinesq
solitons in multi-dimension, and for their physical
applications.