Hi all. I have actually asked this question in my another thread but seems not many people notice that. I am going to repeat the question here and hope to get some replies. The question is about assuming a perturbation series for the KdV equation. u_t + 6uu_x + u_xxx = 0 Why one would assume a pertubation series of u = eu_1 + e^2u_2 + e^3u_3 + ... but not u = u_0 + eu_1 + e^2u_2+ e^3u_3+... ? Can someone explain the subtleties inside this to me? The same thing is obsreved in this article: www-personal.engin.umich.edu/~jpboyd/op121_boydchennlskdv.pdf and a paper. (Please see the figure) https://www.physicsforums.com/attachment.php?attachmentid=9655&d=1175451667 Please kindly help.