Question about a scientific paper - Fluid Mechanics - Perturbation Theory
My question pertains to the following article: SpringerLink - Ocean Dynamics, Volume 47, Number 4
Gabriel Godin - Magnitude of Stokes' drift in coastal waters - Ocean Dynamics, 1995 - Springer
I have attached the relevant section to this post.
My question is whether Godin's assertion is correct or not - namely the sentence "Such a development ... additional terms" and the last sentence in the attachment. That is, that the only way to obtain terms higher than first-order in the Lagrangian velocity [tex] v(r_p,t) [\tex] is through numerical integration of eq. (1). He argues that an analytical solution via Taylor series expansions is not possible due to the strong non-linear nature of the problem due to the displacement [tex] r_p-r_0 [\tex] being unknown.
After speaking to several professors in my department they are surprised that this paper even passed the review process and claim that the analytical solution is indeed possible to obtain. However, I have attempted to obtain the analytical solution via a Taylor development (perturbation theory) and have been unsuccessful thus far. I have been successful in obtaining the numerical solution for the higher order terms, but since I am publishing a paper it would be nice to have the analytical solution as well or at least a confirmation that the numerical solution is the only way to go in this case.
I am not a mathematician and this subject is rather unfamiliar to me, I hope someone can let me know whether the author is correct in his assertion or not.