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I want to solve the system of Schrodinger-Poisson equations numerically:

\begin{align} \chi_1''(r) + \frac{2}{r}\chi_1'(r)&=2U(r)\chi_1(r) \\ \chi_2''(r) + \frac{2}{r}\chi_2'(r)&=2\left(\frac{m_2}{m_1}\right)^2U(r)\chi_2(r) \\ U''(r)+ \frac{2}{r}U'(r) &= \chi_1^2(r) + \left(\frac{m_2}{m_1}\right)^2 \chi_2^2(r) \label{eq:sys4} \end{align} where $\chi_1(r)$ and $\chi_2(r)$ are scalar fields, U(r) is the gravitaional potential (shifted by some constant). This problem is a BVP, where $\chi_1(r)$ and $\chi_2(r)$ approach zero as r tends to infinity, and U(r) approaches some constant, but I can rewrite the problem as an IVP where the initial conditions are (for $\delta \ll 1$):

\begin{align} \chi_1(\delta) &=1 + \frac{U_0}{3}\delta^2\\ \chi_1'(\delta) &=2\frac{U_0}{3}\delta + \frac{1}{15} \left( 1 + 2U_0^2 + \left(\frac{m_2}{m_1}\right)^2 B_0^2 \right) \delta^3\\ \chi_2(\delta) &= B_0 \left(1 + \frac{1}{3}(U_0 + \Delta \gamma) \left(\frac{m_2}{m_1}\right)^2 \delta^2\right)\\ \chi_2'(\delta)&= \frac{B_0}{3}\left(\frac{m_2}{m_1}\right)^2 \left( 2(U_0 +\Delta \gamma ) \delta + \frac{1}{5}\left(1 + \left(\frac{m_2}{m_1}\right)^2(2(U_0 + \Delta \gamma)^2 + B_0^2)\right)\delta^3 \right) \\ U(\delta) &= U_0 + \frac{1}{6}\left(1 + \left(\frac{m_2}{m_1}\right)^2 B_0^2 \right) \delta^2 \\ U'(\delta) &= \frac{1}{3}\left(1 + \left(\frac{m_2}{m_1}\right)^2 B_0^2 \right)\delta + \frac{2}{15}\left(U_0 + \left(\frac{m_2}{m_1}\right)^4(U_0 + \Delta \gamma)B_0^2\right)\delta^3 \label{eq:sys5} \end{align}

I want to find the solution without any nodes (the ground state solution). When calculating, I can fix $B_0$ and $\frac{m_2}{m_1}$ to be some constant, and now I have to vary two parameters, $\Delta \gamma$ and $U_0$ in order to obtain the nodeless solution.

Can someone suggest me an algorithm that could solve this system of equations and obtain the values of $\Delta \gamma$ and $U_0$ for which the solution is nodeless?

The version with only one field $\chi$ can be solved using the shooting method where the parameter that is varied is $U_0$ (this is the only parameter in that case). In the case of the two fields, the shooting method doesn't look feasible.

Jovan Markov
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  • The problem is nonlinear so it would have to be some variation of the Newton method. – Maxim Umansky Mar 21 '21 at 04:11
  • But maybe a simple iteration process will converge, just like this: make an assumption for the initial potential U(r), then solve for the ground state eigenvectors by standard linear algebra software, then update the potential U(r) by solving the Poisson equation by standard linear algebra software etc. – Maxim Umansky Mar 21 '21 at 06:15
  • @MaximUmansky it would help me a lot if you can give me a link to some example of what you are talking about, I am completely new to this business, I don't even know where to start from to see what you suggested in the comment. – Jovan Markov Mar 22 '21 at 19:38
  • To be more specific: what part of the solution (per my suggestion above) do you not know how to do? Or what part do you know how to do? – Maxim Umansky Mar 23 '21 at 02:21
  • @MaximUmansky by ground state eigenvectors you mean the functions $\chi_1$ and $\chi_2$? – Jovan Markov Mar 25 '21 at 08:04
  • Yes, but specifically the ground state solution is solution without any nodes. – Maxim Umansky Mar 25 '21 at 14:54

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