@dharr i send you in mail i hope you got the mean, if you undrestand the problem it will be solved automatically

@dharr is just about aranging function with letter all thus function i used are u and when we use that laplace we have just one function which is u[0] when we substitute we get u[1] and for u[2] we use u[1] for get u[3] we have to use u[2] but in adomian polynomial i can't use conjugate(u) or diff(u,x) is not work like that becuase of that i use v for conjugate and z for derivative of u . 

@janhardo  the code are too comlicate , if you watch the accer answer is so short and  mine is true too but i add equal sign sometime code not work with = sign .

@dharr  let me give you the more detail i know each steps but coding for me is prblem and i did my trail, thus B[0] and B1[0] and T[o] they non linear term in my pdf which i changed to P[i] and Q[i] and R[i] as paper did that  and when i add three term of non linear part it give  A[0]=P[0]+Q[0]+R[0] and so on for other A[i], 
the second part i have to define thus nonlinear term by adomian polynomial i have two definition i try to apply more easier but i fail so at end i did somehow which make my term true  but  in function susbtitution i got problem  becuase in definition of adomian polynomial i change each cojugate(u(x,t)) to v(x,t)  and for diff(u(x,t),x) i try use z[i,x] which this i is change and x is derivative and for diff(conjugate(u(x,t)),x) i use v[i,x] but all of thus z and v are u function, really is make a lot hard for me to arange them , and my invistagation is stuck becuase of that there is 10 equation i am stoped here for this step 

i will update the figure here also adomian definition for nonlinear term  for more undrestanding i hope you find out really i needed that

if you watch my step are not wrong and outcome are completly true  
if you need other information please just mention it 

note: |u|^2=u*conjugate(u)  by complex property

and this is adomian polynomial  for non linear term

@janhardo  don't be angry i didn't mean that, is perfect for some one who know how work with it and i am know you are expert with it and 100 year i work on that ode which you give me the idea i could not found that idea is depend on the question sometime give us answer sometime not 

@acer Ai is trash in 100 time just one time work , when i am stuck i will post here and my work is rare i cant find it any where i have to do step by step,  many thanks for you

@acer Dr. david and maccdara write thus code each one write one of them i try to used but i am fail , i have a lot question they easy but i  don't know how apply code when i apply not work 

@janhardo  i have to change my topic, i will go to another topic and i will ask a lot question in future

@janhardo  i put condition on it but didn't work it and i don't know how he reach this goal.

@janhardo  but again i didn't get result

ode1 := diff(G(xi), xi)^2 = A^2 + 2*A*B*G(xi) + 2*A*C*G(xi)^2 + 2*B*C*G(xi)^3 + C^2*G(xi)^4;
                              /         
ode1 := Typesetting:-mcomplete|G[ξ], 
                              \         

                    /[ d        ]\\      2
  Typesetting:-_Hold|[---- G(xi)]||^2 = A 
                    \[ dxi      ]//       

   + 2 A B Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)]))

   + 2 A C 

                                                        2
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

   + 2 B C 

                                                        3
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

      2                                                       4
   + C  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 


W := B = -2*A*C;
                        W := B = -2 A C

ode2 := subs(W, ode1);
                              /         
ode2 := Typesetting:-mcomplete|G[ξ], 
                              \         

                    /[ d        ]\\      2
  Typesetting:-_Hold|[---- G(xi)]||^2 = A 
                    \[ dxi      ]//       

        2                                                         
   - 4 A  C Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)]))

   + 2 A C 

                                                        2
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

          2 
   - 4 A C  

                                                        3
  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 

      2                                                       4
   + C  Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) 


;
G25 := G(xi) = -1/(2*C)*(sqrt(-2*A*C) + sqrt(-6*A*C)*tanh(1/2*sqrt(-6*A*C)*xi));
G25 := Typesetting:-mcomplete(G, Typesetting:-_Hold([G(xi)])) = - 

          (1/2)           (1/2)     /1         (1/2)   \
  (-2 A C)      + (-6 A C)      tanh|- (-6 A C)      xi|
                                    \2                 /
  ------------------------------------------------------
                           2 C                          


(simplify(odetest(G25, ode2)) assuming (A < 0, 0 < C));

@janhardo  we have to change the ode when we change the ode we have to replace A,B,C changed to thus in eq4 and 5 but how satisfy i don't have idia

@janhardo  after equation 24 the reds one is not satisfy the equation i have to use equation 4-5 condition but i don't know how they use and satisfy the ode 

@janhardo  in equation 4 to 5 there is some substitution maybe they use that can you arrange the condition

@dharr  i got the problem with you explanation i didn't see that when i copy and paste this xi[0] from a mw file to another i don't know why this is happen and when i rewrite again this xi[0] by the same xi[0] it will fix it , but in my function i didn't saw that what  you mentioned thank you so much dear dctor

@janhardo i try this i did this before for another ode , thank you so much john

G15 := G(xi) = -(1/(xi__0 + xi) + B/2)/C