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I'm trying to efficiently solve a system of complex differential equations. As an example (note that these are not the actual differential equations I'm using) let's consider the following set of equations for the complex-valued functions $y_1(t)$ and $y_2(t)$: \begin{eqnarray} \dot{y_1} &=& A(y_1+y_1^*)+y_2 \\ \dot{y_2} &=& B (y_1^*y_2-y_1y_2^*), \end{eqnarray}

where $A$ and $B$ are complex constants. In order to solve ODEs efficiently numerically, it is beneficial to provide the Jacobian to the solver. How would the proper Jacobian look like in this case? I'm a bit unsure because I don't know how to deal with the complex conjugates

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  • $\begingroup$ Are $y_1$ and $y_2$ supposed to be functions of $x_1$ and $x_2$? What variable is the dot differentiating with respect to? $\endgroup$
    – eyeballfrog
    Jun 20 at 16:25
  • $\begingroup$ sorry, updated the question. Dot means differentiation with respect to time $\endgroup$
    – wa4557
    Jun 20 at 16:28
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    $\begingroup$ One option is to convert to a system of real equations for the real and imaginary parts of the $y_i$. You get twice as many equations but you don't need to worry about conjugates anymore. Does that work? $\endgroup$
    – eyeballfrog
    Jun 20 at 18:40
  • $\begingroup$ yeah. In order to solve the system numerically I do that anyways. I was just wondering if there is a general way to compute the jacobian of such a complex valued set $\endgroup$
    – wa4557
    Jun 20 at 19:01

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