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I'm new to Mathematica, and I am trying to solve a system of coupled equations with complex conjugates using DSolve before trying NDSolve.

So far my code does not work, I was wondering if maybe you could help me with this.

Element[{a, b, w}, Reals > 0]
DSolve[{f'[x] == I*a*g[x]*Conjugate[f[x]], 
  g'[x] == I*a*f[x]*f[x] + I*b*Conjugate[u[x]]*v[x], 
  u'[x] == I*b*Conjugate[g[x]]*v[x], v'[x] == I*b*g[x]*u[x], 
  f[0] == w, g[0] == 0, u[0] == w, v[0] == 0 }, {f[x], g[x], u[x], 
  v[x]}, x]

I'm not sure i'm using Conjugate properly, and there might be a lot of unseen mistakes, the answer I get is :

DSolve[{Derivative[1][f][x] == I a Conjugate[f[x]] g[x], 
  Derivative[1][g][x] == I a f[x]^2 + I b Conjugate[u[x]] v[x], 
  Derivative[1][u][x] == I b Conjugate[g[x]] v[x], 
  Derivative[1][v][x] == I b g[x] u[x], f[0] == w, g[0] == 0, 
  u[0] == w, v[0] == 0}, {f[x], g[x], u[x], v[x]}, x]

Thank you for the kind use of your precious time!

PS : For those who are interested, it's a coupled system of nonlinear Optics, with 2 cascaded interactions generating the so-called third-harmonic beam.

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  • 1
    $\begingroup$ I don't think there is anything wrong with the code (beyond the fact that Element[{a, b, w}, Reals > 0] does nothing and means nothing). It's just that the system has no analytic solution. Try NDSolve instead. $\endgroup$ – AccidentalFourierTransform Jul 3 at 12:29
  • $\begingroup$ $Assumptions = a>0 && b>0 && w>0; reference.wolfram.com/language/ref/$Assumptions.html After that you can verify it "works" by trying this Simplify[Sqrt[a^2]] $\endgroup$ – Bill Jul 3 at 15:38

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