@salim-barzani You're wecome. I think you can get the extra functions for the HB (and other equations) by continuing solving the equations in turn, recognizing the resonant point ones will simplify to zero, after substituting in the earlier solutions (which shows the compatibility condition is satisfied, if I understand it correctly).

Sort was improved in 2025, but nothing much relevant to differential equations, so there is probably not much need for you to upgrade to 2025.

@dharr 

Here it is, so I think I am done!

AllStepsKdV.mw

AllStepsHB.mw

@salim-barzani I updated the solution to include step 3. It works for the Burgers equation, and it is clear that you have to think about the resonance points, and won't be able to just solve the whole system.

@salim-barzani I have modified it to work in Maple 2024.

@salim-barzani I think I fixed the step 1 part. I'm not sure what coeffcients to find and what to solve for. I collected and found coefficients of powers of psi(x,y,z,t) only (not its derivatives), stll that makes 14 equations. Then I'm solving the equations only for w[i](x,y,z,t), i=2..6 (somehow w[1] has disappeared) - in Jet notation these are w[2][],..w[6][], which I renamed W[2]..W[6]. But perhaps I am supposed to solve for the derivatives as well (but there are more tham 14)? 

GOM-Dr.D-test-1-2b.mw

Steps_1_and_2_DAH2.mw

@salim-barzani I updated Step1_DAH.mw - please check the statement in bold is correct. I will look at step 2.

@salim-barzani The errors in pdes-s-1-2.mw and step1-2.mw are because coeff can only handle integer powers, not symbolic ones. The findexp routine can handle symbolic powers.

@salim-barzani Thanks for the detailed explanation. I have finished inplementing step 1 for this case, but I still don't know how "higher nonlinearity" is defined in the general case. I will ok at the next steps later.

Step1_DAH.mw

@salim-barzani I did some more, but I still don't know how to complete step 1.

new-p_DAH.mw

@salim-barzani You say "is clear what they did". Would you please explain the steps to find alpha[1]. I do not understand that (and extra papers just seem to say the same thing). 

I don't understand what is happening, but maybe this is helpful as a start.

p1_DAH.mw

@michele @Carl Love's commands only work for 1-D input. I'm guessing you are using 2-D input, so the equivalent commands are

M[.., -[$1..n]]

M[-[$1..n]]

It can be quite difficult to get such complicated expressions into exactly the form you want in a paper. My usual procedure is the take the Maple expression A, copy it and modify it by hand to the way you want it, set it to B, and then simplify(A-B) to check that you haven't made a mistake.

@michele 
In U[.., p], the ".." means select all the rows (short for 1..4 in this case), and p is a list of columns to select, so for p=[4,3,2,1] the 4th, then 3rd then 2nd then first columns are selected.

See the help page ?MVselect for more details.