High-Order Davies' Approximation for a Solitary Wave Solution in Packham's Complex Plane

This paper considers a progressive solitary wave of permanent form in an ideal fluid of constant depth and explores Davies' approximation [Proc. R. Soc. Lond. A, 208 (1951), pp. 475--486] with high-order corrections to Levi-Civita's surface condition for the logarithmic hodograph variable. Using a complex plane that was originally introduced by Packham [Proc. R. Soc. Lond. A, 213 (1952), pp. 234--249], it is shown that a singularity at infinity can be regularized. Therefore, the solutions in Packham's complex plane under high-order Davies' approximation maintain two critical properties of a solitary wave, the correct exponential decay in the outskirt of wave and the harmonic property of a solution, that are often violated in classical long wave approximations. After introducing an accurate numerical method to compute solitary wave solutions in Packham's complex plane, we compare high-order Davies' approximate solutions with fully nonlinear solutions as well as long wave approximate solutions. The results demonstrate that high-order Davies' approximation produces rapidly converging series solutions even for relatively large amplitude waves and that Davies' approximate solutions compare much better with the fully nonlinear solutions than the long wave approximate solutions.