@acer you did a great job but

in your case why calculate seperatly not all our parameter toghether map must be [4]? and a[0] if zero is ok but each parameter of ODE equation and in series solution assumption a[1] must not be zero which multiply by G(xi) ! in paper assumption he get 15 and more  how he achived them he must use limit i think he used limit for auxiliary equation

@janhardo ..................................................

@nm  this is your last answer? How if we don't have data like that i want find more we can finded by changing the limit or not? Like you did if we say m->1 left say m->2 right what will happen?

@janhardo I don't know why they did that 

@nm I don't know why they delete it

@janhardo  your code not run and please upload the mw file
and how if we don't have data we must able to find more solution we must looking for finding them by assumption of limiting...i don't know how but i saw a lot they did like that,

this problem is hard one really and long pde equation is (1+1) dimensional (concatenation model used for polarization-preserving fibers)

@nm  How we can add more parameter  by limit or anything else becuase there is a lot method used this technique if i can find for example at lease 100 parameter whcih give me solution this is a big achievment, i get idea from your comment so we can replace the parameter by another parameter by get limitng the result will change or not?  for example in first row m->1 (left) we can say m->2 right, what will happen to solution it will change or it will be error, how we try this? we must find thus solution even if we don't have thus data and find more solution?

@nm i will add another one question of this kind  topic again with new idea please help me in next question, i will mention you.

@janhardo  nevermind i got idea i was wrong in third part

@janhardo  that is all i find all step in last file but i want make it easier and this equation is easy one i want update the code for the hard equation and there is a lot of method i wnat apply the code for all of them... and this part by @dharr  is so good and work for me i don't have issue but when we done this  i figure out is just for one case which is A,B not zero but if in begin our auxilary A at there is zero then our system of equation will change and out come of case will change  it almost done all part we just one collect them and get out come

@nm open please 

@janhardo this code is emazing and different   and determine all cases one by one but i am looking for U(xi) and list them for each case for example A=0 the U(xi)=@ my mean not G(xi) also if do testode for each case it will be emazing, i upload in top and i will upload here too if provide me this it will be emazing

all_case.mw

@nm this code is emazing  and determine all cases one by one but i am looking for U(xi) and list them for each case for example A=0 the U(xi)=@ my mean not G(xi) i upload in top and i will upload here too if provide me this it will be emazing

all_case.mw

@dharr first thanks for this code
there is one case missing which is A  and B not equal to zero !
also i am  looking for U(xi) with telling me when for example A=0  then U(xi)=@  with substitute case in U(xi), also i need to clarify the odetes, i know it is work but want be sure for substitute odetest must be zero,  i am sure you didn't see this file i upload later 

i hope undrestand my mean

all_case.mw

@Kitonum it seems you have a good expreince in changing variable always i have problem in physic how we can remove the absolute value automatically i do that by hand in equation where  q=U(x)exp(I*x) then |q|=U(x) and |q|^2=U(x)^2 how maple do that automatically and do the change variable all by programing something like that in below picture also i have esier   if needed i just want to know how  replace this absolute value and do that automatically without need hand