@dharr you are act like a god of thus topic , i  like your work a lot... thank you so much more than you think 

@dharr  how we can plot is have any relation with trigonometric or hyperbolic  how we can plot this function  really i am intresting if we can plot this what is the shape of plotting?, i will do for  other  case,  if i got problem i will  update

@janhardo construct and chose thus function which make the ode to zero and make table ..

@dharr  i  am agree , i think if i put this picture it will give you notion of it 

this is link of paper is open access you can watch

 link-paper

C47 := {f[2] = 4/3*(Q^2 - (3*P)*R), f[3] = (4*Q)/27*(-2*Q^2 + (9*P)*R)};
          /                4  2         4    /           2\\ 
  C47 := { f[2] = -4 P R + - Q , f[3] = -- Q \9 P R - 2 Q / }
          \                3            27                 / 

CC := {P = P, Q = Q, R = R};
                  CC := {P = P, Q = Q, R = R}

ode46 := simplify(expand(subs(CC, ode)));
                               /         
ode46 := Typesetting:-mcomplete|F[ξ], 
                               \         

                    /[ d        ]\\            4          2    
  Typesetting:-_Hold|[---- F(xi)]||^2 = P F(xi)  + Q F(xi)  + R
                    \[ dxi      ]//                            


S46 := F(xi) = subs(C47, WeierstrassP(1/P*xi*f[2]*f[3]));
odetest(S46, ode46);
Error, (in WeierstrassP) expecting 3 argum

@janhardo  we have to apply long wave limit to eq(4)  and after tht we have to get eq(14) eq(17) eq(18)  beside this for each part of Exp(A[ij}) we have to apply long wave limit to for each B[ij]

@dharr  is been a year i am busy with this but still not any outcome, really i am not sure how to do it and it is making me angry a lot

which i can't find this while i undrestand whole topic but i can't see what missing in here once more i will explain it and i will put papers information if need i will send you i have a lot of them but if one of them come out the other is same, please help me for finding the  idea behind this i try to use Ai and he give me a lot idea but still not working, so lets begin explanations

before a year you and john hardo found this series Eq(4)

but we can't use long wave limit on it to get the function of eq(10), eq(13), eq(14)  eq(17), eq(18),...

i try to use all thus function by hand one by one and  i am not sure if  iam missign but i didn't found any error in using them , the problem is here when i found  for example in eq(5) we have exp(A[ij]) when i apply long wave limit on it it will be change and i used in interaction function of thus equation i mentioned  in top  instead of each B[ij] so exp(A[ij])=B[ij] after long wave limit, but how apply that i am not sure is some paper i did like they did by this method i already use but is not give me answer so i am confusing in here 

in this paper i mention in here i use eq(14) and i used long wave limit  is give me zero and don't have problem also when i use your trick numerically is not give me zero but is so near to zero, but when i used eq(17)  numerically is small variable -0.000034 and i didn't wait a long for not numerically solving by same way i did  , also i did for eq(18) again numerically is small number 0.00230291489 so please tell me which i am write about that or not also if i am true why my outcome is diferent from paper outcome specialy about that B[ij], becuase he mention something is so diferent from my outcome, i will put my file with picture here please watch carfully i will put in here one by one  first eq(40 by 3 maple file also for eq(17) and eq(18)

 eq14.mw

eq14-function.mw

eq14-function-Bij.mw

eq17.mw

eq17-function.mw

eq18.mw

eq18-function.mw

eq18-function-Bij.mw

also i have a diferent file for that series in case needed that without aplying long wave limit

N-series-Furmula.mw

@janhardo i did a 4 papers all of them have the same problem i did outcome and i am not sure about that limit becuase in some papers are different i will more and i will post in different way maybe i will post 4 papers for showing the different of outcome by applying that limit, but is strange why not give zero even i use the same as paper did

@janhardo  you have to use Eq17 in my file like author did also thus B[ij] come from (expA34) after apply long wave limit which you did in last time and theta[i] and eta[i] is clear in theta[i] just remove thus term have k[i] when i=1,2 but in eta[i] alll remain we have to replace thus part and then must pde be zero as i did  and i did a lot of other paper at there have same problem  i don't know why this issue is happen maybe we are do something wrong in long wave limit part 

@sand15  i change the name and that is the result as you mention, but our outcome is so different from him specially when we change a[ij] to Lambda[ij]  in eq(17), i am intrested a lot to find thus but have a trick which i can't see it

@janhardo  i have to find for all B[ij] becuase i have other function too, this is uncompleted we have to find B[12],B[13], and B[23] one by one and in other function sometime we have B[15] and so on   hence we need generalized this for finding all of them

restart;

# Kortere versie
e_ij := ((3*c1*ki*(ki - kj)*lj - c3)*li^2 - 3*(c1*kj*(ki - kj)*lj - (2/3)*c3)*lj*li - c3*lj^2) /
        ((3*c1*ki*(ki + kj)*lj - c3)*li^2 + 3*(c1*kj*(ki + kj)*lj + (2/3)*c3)*lj*li - c3*lj^2);

# Bereken direct B13 met k1 -> 0 limiet
B13_direct := limit((subs(ki=k1, kj=k3, li=l1, lj=l3, e_ij) - 1)/(k1*k3), k1=0);

# Vereenvoudig
B13_result := simplify(B13_direct);

"B_{13} = ";
B13_result;

in this part  if you can use limit for finding B12 and B23 and B13 it will be great can you i think in B12  K[1] and K[2]-->0 and in B13 K[1]-->0 and in B23 K[2]-->0

@dharr  the problem for thus ways is you can do a lot thing which not done before the idea is remains and open you can do anything and use any series combined with another ode which have a lot of solution and constract the ode and pde solution, i  combined the G'/G method with solution of herota bilinear method  and it give me answer of pdes so there is a lot idea is still open  and really if intrested in such topic i can put my 100 on this work with you in future and i am sure with your idea we can construct a lot new methods .  i will put a lot question which really are intresting after new year and really i have to construct a new method  and the graph of solution must be different, i hope  or i wish have a meeting with you and negotation about that topic

@sand15 thanks for your help and explaining that points, thanks a lot.

@dharr  your idea about hundling thus equation are emazing i liked a lot, as i mention before is a little bit long for explanation but i will try, the procces of solving pdes is like that i have transform the pde for two part real and imaginary then i take real part of pdes as my ode and i will use a methods for solving the ode then this solution of my ode will be solution of my pdes automatically becuase of that i use thus analyticall methods and there is a lot of them which i can't do it yet and becuase construct paper i don't have much time, there is some other way for solvng pdes which i don't have idea but the visualization is great i don't know where to start , like lax pair, lie symmetry  and darboux transformation thus are my future work and keep continuos on analytical methods  when something new come up i will try to solve, and really sorry for that if the questions are a little bit similar, in my mind still i am looking for long wave limit too, thank you a lot and any time have any question i will response and is my honor 

@janhardo don't have link yet