| The Boussinesq equation for the surface waves over a shallow layer of inviscid liquid contains nonlinearity and fourth-order dispersion. For steady moving waves, it reduces to a fourth-order elliptic equation which must be investigated for bifurcation. For solving this higher-order boundary-value problem we use two techniques.;First, we devise a finite-difference scheme and an iterative algorithm for the direct numerical solution in which we consider the "false transient" with respect to an artificial time. We prove that the truncation error of the scheme is second-order in space. The second order rate of convergence is proved theoretically and demonstrated practically by conducting numerical experiments with different grid resolutions.;Second, we develop a perturbation series with respect to the small parameter epsilon = c2, where c is the phase speed of the wave. Within the order O(epsilon2) = O(c4), we derive a hierarchy of one-dimensional equations that are fourth-order in the radial variable. The ODEs are also solved by means of difference schemes, but on grids with much better resolution than the 2D case. To this end, we create special approximations which account automatically for the so-called "behavioral" conditions at the point of singularity.;The results obtained with the two different techniques are compared and shown to be in excellent agreement which validates both of them. We discover that the shape of the moving soliton decays at infinity as 1/r 2, with the polar coordinate r, while the profile of the standing soliton (c = 0) decays exponentially. This means that the asymptotic behavior of the solution is not robust, which is a novel result not available from the literature. Thus, the shapes of stationary Boussinesq solutions are obtained by means of two different mathematical techniques. Our results are of importance both for the mathematical theory of Boussinesq solitons in multi-dimension, and for their physical applications. |
