An other Idea is:

MBE = {
     D[a[t, x], t] == -a[t, x] + d1[t, x] c0[t, x] - 
     d0[t, x] c1[t, x],
    D[b[t, x], t] == a[t, x] - b[t, x] - 
     d1[t, x] c0[t, x] + d0[t, x] c1[t, x],
    D[c0[t, x], t] == 
    c0[t, x] + (a[t, x] - b[t, x]) d1[t, x],
    D[c1[t, x], t] == 
   c1[t, x] + (a[t, x] - b[t, x]) d0[t, x],
    D[d0[t, x], x] == - c1[t, x],
    D[d1[t, x], x] ==  c0[t, x],
   a[t, 0] == 0, b[t, 0] == 0, c0[t, 0] == 0, c1[t, 0] == 0, 
   d0[t, 0] == 10^1, d1[t, 0] == 0,
   a[0, x] == 0, b[0, x] == 0, c0[0, x] == 0, c1[0, x] == 0, 
   d0[0, x] == 10^1, d1[0, x] == 0};
solutionMBE = 
 NDSolve[MBE, {a, b, c0, d0, c1, d1}, {t, 0, 
   3}, {x, 0, 20}]

with c=c0+I c1 and d=d0+I d0. Now the equations are real.

In this system of equations I get the Error massages:

NDSolve::pdord: Some of the functions have zero differential order, so the equations will be solved as a system of differential-algebraic equations.

and

NDSolve::ndcf: Repeated convergence test failure at t == 0.`; unable to continue.

What is now the problem of this system?

It is clear to me is I'd have to be more familiar with maxwell-bloch and I have not reached that in my studies yet. I understand fermi free electron theory and 3D shrod. but A99 requires more than introductory knowledge.

I tried eliminating "d" from eqs by taking D[,x] on eq. and replacing d with c in them. It became a sytem of pde and unsolvable using NDSolve. Maxwell-bloch is not a pde system (wikipedia shows an ODE system in d/dt) but the authors of A99 said 48-51 can fall in m-b form (how I cannot say). I'm unsure Frank Cappas comment but am not senior enough to say ignore his words.

Each function and symbol in 48-51 is a result of many of equated results as you know. I read A99 is not one of presenting experiment results but of finding a more agile closed form for describing such experiments as the previous used were "insufficient". And 48-51 are not "complete" however (they neglect some rules in the author said, for simplicity, so need to be applied with care).

It seems to me just ask who worked on the paper for guidance. The paper was made in 2018 2019 at least a few worked on it - so they are likely available.

I am left with question if your intermediate problem is not temporary (if you have not already abandoned your approach). It is an interesting problem and paper though.

An interesting problem. Is it from a book?

I'd like to know if the * in the equations refer to convolution (as they are used in some places but not everywhere).

Input[]:  system = {
  D[a[t, x], t] == a[t, x] + I Conjugate[d[t, x]] Conjugate[c[t, x]], 
  D[b[t, x], t] == b[t, x] + a[t, x] + I c[t, x]*d[t, x], 
  D[c[t, x], t] == c[t, x] + I (a[t, x] - b[t, x])*Conjugate[d[t, x]],
   D[d[t, x], x] == I c[t, x]}

solution = NDSolve[system, {a, b, c, d}, {t, 0, 10}, {x, 0, 10}]

Output[]

NDSolve::femibcnd: No DirichletCondition or Robin-type NeumannValue was specified for {a,b,c,d}; the result may not be unique.

{{a->InterpolatingFunction, b->InterpolatingFunction, c->ditto, d->ditto}}

Does that mean the conditions may be preventing a solution?

Is there a book solution C.H. would like to mention?

In this case, "ie" is "i.e." but without the expected punctuation. As one might expect, such under-punctuated posts pose issues for the moderator team. This is under discussion here.

I retract the question. A method I believed I saw was available, looking to check that it is, is not available. I cannot give an example of selecting an alternate method I know will work.

Ok. But I can't play with the system of differential equations. But why it works if everything is real:

system = {D[a[t, x], t] == a[t, x] + d[t, x]*  c[t, x],
   D[b[t, x], t] == b[t, x] + a[t, x] + c[t, x]*d[t, x],
   D[c[t, x], t] == c[t, x] + (a[t, x] - b[t, x])*d[t, x],
   D[d[t, x], x] == c[t, x],
   a[t, 0] == 1, b[t, 0] == 1, c[t, 0] == 0, d[t, 0] == 0,
   a[0, x] == 1, b[0, x] == 1, c[0, x] == 0, d[0, x] == 0};
solution = NDSolve[system, {a, b, c, d}, {t, 0, 10}, {x, 0, 10}]
Plot3D[{a[t, x] /. solution}, {t, 0, 2}, {x, 0, 2}]

Attachments:

test.jpg

You don't have derivatives of the functions with respect to both variables. That may be the problem.