Perturbation solution for the 2D Boussinesq equation

Boussinesq equation arises in shallow water flows and in elasticity of rods and shells. It contains nonlinearity and fourth-order dispersion and has been one of the main soliton models in 1D. To find its 2D solutions, a perturbation series with respect to the small parameter ɛ = c2 is developed in the present work, where c is the phase speed of the localized wave. Within the order O(ɛ2) = O(c4), a hierarchy is derived consisting of one-dimensional fourth-order equations. The Bessel operators involved are reformulated to facilitate the creation of difference schemes for the ODEs from the hierarchy. The numerical scheme uses a special approximation for the behavioral condition in the singularity point (the origin). The results of this work show that at infinity the stationary 2D wave shape decays algebraically, rather than exponentially as in the 1D cases. The new result can be instrumental for understanding the interaction of 2D Boussinesq solitons, and for creating more efficient numerical algorithms explicitly acknowledging the asymptotic behavior of the solution.