Solve nonlinear complex numerical differential equations

Questions
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1

Dear Wolfram community,

I have a nonlinear complexe system with twovariables.

   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],
   a[t, 0] == 1, b[t, 0] == 1, c[t, 0] == I, d[t, 0] == 0,
   a[0, x] == 1, b[0, x] == 1, c[0, x] == I, d[0, x] == 0};
solution = NDSolve[system, {a, b, c, d}, {t, 0, 10}, {x, 0, 10}]

and get as an error massage:

NDSolve:Some of the functions have zero differential order, so the equations will be solved as a system of differential-algebraic equations.
NDSolve: For the method NDSolve`IDA, only machine real code is available. Unable to continue with complex values or beyond floating-point exceptions.

What can I do to solve this system correctly.
Thank for your help.

2

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

3

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}]

test.jpg
test.jpg360×296 19.9 KB

4

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}]

test.jpg
test.jpg360×296 19.9 KB

5

The error message answers that question.

For the method NDSolve`IDA, only machine real code is available. Unable to continue with complex values or beyond floating-point exceptions.

6

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.

7

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

8

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.

9

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?

10

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?

11

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.

12

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.

13

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.

14

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?

15

“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?