@janhardo  i try use Ai but didn't give me anything usefull , when i test all of them give me error

@dharr i try to explain more in here step by step , i hope this time undrestand otherwise i have to send you a easy paper with solution , if you try use my first code which just u remain to put inside u[i] and do magic stuff like assume real or something like that maybe work , take your time i will be wait 

fully_explain.mw

@dharr is not just about u_xx all part is together when we say u[i+1] when i=0 it mean u[1]=invlapalce(1/s*laplace(linear part + nonlinear part)
which linear part we have u_xx we put u[0]  so we have diff(u[0]=beta*exp(I*x),xx) 
also non linear part we have A[0]  put u[0] inside each term which conclude u[0] diff(u[0],x) and conjugate(u[0]) at this part this conjugate again make problem becuase function is exponential is not evaluate true 

just one time slowly look at my supervisor code and watch the pdf you will get idea even my code is better but he take out all part and for laplace about that t you said  i just copy the code of supervisor so if it is wrong why worked at there 

@dharr  this time  all structure work pretty but this u[i] remain when substitute why not evaluate the conjugate like paper did, i did but this conjugate  i think make a problem the important part remain which calculating u[i+1] in my file how we can get that

u[0] := beta*exp(x*I);
                     u[0] := beta exp(I x)

for i from 0 to 4 do
    u[i + 1] := invlaplace(1/s*laplace(-diff(u[i], x $ 2)*I + A[i], t, s), s, t);
end do;

@dharr i am not sure is exactly the same or not becuase there is a lot of term like |u|^2 which is |u|^2=u*conjgate(u) and |u|^4=u^2*conjugate(u)^2 also i checked the p[1] is not same but if we devide by 2 is same but in my code is directly give it to me 

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@dharr i don't know how define tha B in adomian which make it be true and in end if i just have u[0] if i do substitute is calculated automatically but in here in my last file i used v and z which make it not work , 
i have to define B[0] in B[n] and B1[0] in B1[n] and T[0] in T[n] they have same structure  

i have to define like that , is give me wrong if you watch the paper you will see they are not the same 

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but when i use changing of variable to subscript is give the exact answer as paper did 

the true one is this  function with subscript make problem 

watch the paper is exactly like him 

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@dharr sorry for that this is same as paper which i have to put in adomian polynomial 

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@dharr this is just one term of nonlinear i have to  put this function in adomian polynomial for calculatin P[0],p[1],P[2] and also for Q and R then add them to get A[0],A[1] which A[n] is all term of nonlinearity, we don't want calculate  one by one of them we calculate all term together one time , 
how you define in this adomian polynomian 

if you watch the paper you will see the directly  i did the same 

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@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