everyone, I have a set of coupled differential equations as is shown below,
and these equations actually describe the interaction of a two-level atom (amplitude is ce(t)) and single-photon (photon with wavevector k and amplitude c(k,t)) in a one-dimensional waveguide. We can easily find the analytical solution by bringing the second formula into the first formula and changing the order of integration, then we can obtain the following equations
But I want to directly solve the equations numerically via Mathematica, I know it's difficult to do since we need solve an infinite number of differential equations in principle. I have searched the solution on stack exchange and find the following solutions,
and I write the coupled differential equations in Mathematica as follows
\[Kappa] = 1;
NDSolve[{ce'[t] == -I*(Sqrt[\[Kappa]]*Integrate[ck[k, t], {k, -3, 3}]),
D[ck[k, t], t] == -I*(k*ck[k, t] + Sqrt[\[Kappa]] ce[t]),
ck[k, 0] == 1/(k + 3*I/2), ce[0] == 0}, {ce[t], ck[k, t]}, {t, 0,
10}, {k, -3, 3}]
and the error happens,
so how can I write down the coupled differential equations with parameters that have integrals?