@salim-barzani I really don't understand what you are asking, but this is my best guess

Dr.D.mw

@salim-barzani I'm don't understand what you are trying to do. See attached for the immediate solution to the error.

A.mw

@salim-barzani You seem to have gone back to using all monomials as before. The paper just says "after arduous calculations". So I don't know what to do here.

@Christopher2222 Yes, the double underscore is just for appearance; I guess you could say "one off". I don't think there is a convention for packages to use one or the other scheme.

@salim-barzani You can convert the answer to explicit forms using allvalues even if solve does not do it.

At first I thought it was a challenge to factor the two numbers given, but the first ends in a 5, so that is definitely too easy. I reread it and I still do not know what exactly you are asking. Would you please clarify.

@salim-barzani Once you have run the loop you know which answers are good, and can do latex or whatever on them later. I don't know why solving only for the a[i] doesn't work. In general you want the same number of variables as equations and if you don't as here then you may be lucky or not. I also don't know why explicit isn't working as it should.

@Christopher2222 Sometimes you want to use the subscript for a purpose, e.g., table index, so you need M[index], but sometimes you don't want the subscript to have any significance and just be part of the name, then M__index is appropriate. For integer indices the subscript is upright, e.g., for M[21], but italic for M__21 so you can see the difference.

@salim-barzani OK, I think you now have acceptable answers.

@janhardo I'm not sure what the problem is; it seems to work as well with f and g?

Using op(0,F) to find the function name seems not to make much sense for arbitrary expressions, but perhaps you only want to use it for functions. I think you should use declare rather than alias if you are using PDEtools. I'm not following what you actually are doing, but maybe this version works as expected?

Download ND.mw

@salim-barzani Isn't putting F1 into H to find u supposed to give a solution to the pde (or perhaps to the linear part?). If so you should check that before moving on to the a[i] part of the problem.

@salim-barzani I haven't been following the bilinear/Hirota part of the argument, so I can't help with that.

You didn't say which version you have. In Maple 2025 it works with the default settings. (The plot command is in the document but doesn't show in Mapleprimes.)

Download plotatzero.mw

@C_R @acer Yes, I noticed a significant slowing in the startup phase on Windows 11 64 bit, which I just attributed to loading the new ribbon interface. Like many other software packages, adding new things slows things down, but then you buy a faster computer and don't notice...

Anyway the startup times in my maple.ini are 7860 ms for 2024 and 21631 ms for 2025. (I didn't try to load a standard file in both cases.) 
Next time I ran 2025 it was 6161 ms, and I manually timed it as about 13 s.

This case is too complicated for solve. I think you will have to put in numerical values of your parameters (which you were going to do anyway for your plot) before solving.