Numerical implementation of the asymptotic boundary conditions for steadily propagating 2D solitons of Boussinesq type equations

In the present paper, a difference scheme on a non-uniform grid is constructed for the stationary propagating localized waves of the 2D Boussinesq equation in an infinite region. Using an argument stemming form a perturbation expansion for small wave phase speeds, the asymptotic decay of the wave profile is identified as second-order algebraic. For algebraically decaying solution a new kind of nonlocal boundary condition is derived, which allows to rigorously project the asymptotic boundary condition at the boundary of a finite-size computational box. The difference approximation of this condition together with the bifurcation condition complete the algorithm. Numerous numerical validations are performed and it is shown that the results comply with the second-order estimate for the truncation error even at the boundary lines of the grid. Results are obtained for different values of the so-called ‘rotational inertia’ and for different subcritical phase speeds. It is found that the limits of existence of the 2D solution roughly correspond to the similar limits on the phase speed that ensure the existence of subcritical 1D stationary propagating waves of the Boussinesq equation.