@dharr "and about step one is not about appearing twice i think, is about non linear term which we chose which a-N is bigger and non linear term is about a+b-N which N is begger and have same a then we can find b like the non couple equation ."

Let me see if I understand this. Say some terms give a-3, a-4, a-5 then we take the minimum a-5. Now suppose we have also a+b-3, a+b-4, a+b-5 so we take a+b-5. So now we set them equal and find b=0. Then the value a-5 and a+0-5 appears twice (since we set them equal). If we have different N, then we get different b (for the same a), e.g.,

a-6 = a+b-5 leads to b=-1, and then the value a-6 = a-1-5 appears twice.

Is there a problem with this?

@salim-barzani You say "intrested about the u[0],v[0] i think they are not the same which i am not sure which is corect  i don't have issue with yours one of g[x] is extra if you watch you will see". I don't see an extra g[x]; please specify exactly where this is (there are some g's there because I am not not calculating u[0] directly; do I have them incorrect?).

" the other point is about that condition how he got that condition in eq((16),  and when N=2 we have to find the u[1],u[2],..u[6] and v[1],v[2],...v[6] ". I don't see N=2 in eq 16. Please specify exactly where you mean.

You have now added the other paper here which I didn't deal with. That paper has p =1 and not p=-1 so that seems wrong to me. 

Basically I am not understanding; please spend a bit more time specifying some details of what you think is wrong or needs to be done. 

@salim-barzani To get the resonances and other solutions correct you have to evaluate with the earlier solutions (see how I did the KdV example), and the resonant ones set to zero (not sure why, but otherwise their deriatives don't disappear). 

all-steps-Dr.D.mw

I am working to understand the coupled case.

@salim-barzani I have updated the files here to reflect my improved understanding of the step 1 procedure. In particular see the plot in AllStepsHBnew.mw for how to interpret the lowest exponents rule. I'll work on the coupled case next.

@salim-barzani This paper doi.org/10.2991/jnmp.2006.13.1.8 seems clearer about how to do step 1, so I'll work through it; it also should help with the case of multiple equations as in the non-linear Schroedinger equations.

@acer's routine (and my general practice) expects the pde to be an expression, but you have added =0.

schrodinger-test.mw

@Alfred_F So in general cos/sin/exp of an algebraic number is transendental. The only thing I know about the exceptions is that Maple sometimes finds roots of a polynomial in trig form, when they might (or might not?) be also be expressible in radical form. The case of cyclotomic polynomials where these are roots of unity is the easiest to find.

Download trig.mw

@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!

AllStepsKdVnew2.mw

AllStepsHBnew.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.