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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.

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# Perturbation expansion when solving Kdv equation

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