I am submitting a paper with my brother as a co-author. We share the same initials and surname, and neither of us has a middle name.
How should we deal with being required to use the format Initial. Surname?
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The computer scientist brothers Marcin and Michał Pilipczuk have written many papers together. I think I've seen them referred to as "Ma." and "Mi.", though I can't find an example in writing. This is similar to an older style of initials for non-Roman alphabets which would, e.g., initialize Yuri as "Yu." and Phokion as "Ph."
However, journals seem to handle them inconsistently. For example, Algorithmica says we should cite one of their papers as "Cygan, M., Marcin Pilipczuk & Michał Pilipczuk". However, the same authors have a paper in the SIAM Journal on Computing which consistently cites the brothers as "M. Pilipczuk, M. Pilipczuk". There are also papers on ArXiv where they use initials in citations and just refer to themselves as "M. Pilipczuk", which suggests they don't feel a strong need to disambiguate. (I don't know them personally so this is pure speculation on my part.)
So, if it's possible, you could try using the second letters of your first names to make your initials unique but, if you do this, there's a good chance that the journals (and other authors!) will mess it up and you'll both be "J. Doe". And this doesn't work very well if you're James and Jason. An alternative would be to invent middle initials. There's no requirement to publish under your legal name, and there are famous cases of people having middle initials that don't stand for anything (e.g., Harry S. Truman and Ulysses S. Grant). However, the downside of this is that whoever chooses the initial earlier in the alphabet will always be listed first if your discipline orders authors alphabetically and, again, people will mess it up and not realize that they need to use both initials, especially for papers where only one of you is an author.
Another, rather off-the-wall suggestion is that some English names have abbreviations that were often used in company names up to, I'd guess, about a hundred years ago. So you'll often see "Wm" for William, "Thos" for Thomas, "Jas." for James, "Geo." for George and so on. But that only works for certain, traditional, English names, it's archaic and I doubt the journals would go with it.