"I don't understand this because 10^15 is only a number." Perhaps the factor makes the estimative term error grow and Mathematica "tosses out" the result. Best to focus on the simple solution before details.

maxwell-bloch not correct? wikipedia shows the equations all in d/dt but A99 clearly shows 3 in d/dt and the 4th d/dx. I will re-check that against other online sources.

Again: your sub-problem is quite different from the problem in the paper. What PDE method are you using to claim this sub-problem (connects or ads) to solve the original problem? Is it a Fourier of the original problem? Is it split from the original in some other way?

Later (today, tomorrow?) I'll look at this again. I wrote down your new MBE eq. and will look at the original problem again. I will at least post the original equation set.

Reproduce Figure 3. The 6 complex plots are graphed from eq's but each requires more equations applied. When they graphed you have to ask if they used experiment data or equation. (example: if was Pee was data (not unknown when graphed) it would make "solving" totally different). For fig 3 it says to use eq 36-39 (then later that 48-51 are mostly equivalent but "assuming the initial absence of fields and coherences, neither of them will appear later on according to Eqs. (50) and (51)". The author suggests ref. 8 and 12 (12 about plotting), they are non-free and not guaranteed to contain the equations sought.

Did you contact the original authors? Or are they your professors and you cannot ask them?

The author of the paper are the guys who where I apply for the PhD and I can start if I solve this problem and they programmed it in python. A solution of the system is given in: Design and Characteristics of a Population Inversion X-ray Laser Oscillator. And I want to reproduce Figure 3 of this paper

The Maxwell-Bloch-equation are a pde. I think the equations on Wikipedia are incorrect.

What I did is the same as you:

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

But I'm not sure that I can kick the Conjugation of d. I know that a and b are real functions and I think that the derivative dont act on complex argument.

And a other problem is that is doesn't work for large prefactors 10^15 a[t,x]. Then I get: NDSolve::icfail: Unable to find initial conditions that satisfy the residual function within specified tolerances. Try giving initial conditions for both values and derivatives of the functions. Why do I get this Error. I don't understand this because 10^15 is only a number.

No, the problem come not from a book.

The problem comes from the preparation for my PhD and thought that I break the problem down to a general one.

I thought I could make any conditions and after I understand Mathematica I include the correct conditions.

The system which I want to solve comes from the Maxwell-Bloch-equations and the paper Quantum theory of superfluorescence based on two-point correlation functions equations 48-51.

* is the normal product between two functions.

OK. But I don't understand how can I solve my problem?

Can I solve this with NDSolve, if I restrict the functions a and b to be real and between 0 and 1?

OK, I use Mathematica 12. No, never change the method. What is ie?

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