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I am trying to simulate the propagation of light in material using the non-linear schrödinger equation (NLSE): $$\partial_zE=\frac{i}{2k_0}\nabla^2_\perp E+\frac{ik_0n_2}{n_0}\vert E\vert^2E-0.5\beta^{(2)}\vert E\vert^2E-\frac{\sigma}{2}(1+i\omega_0\tau_c)\varrho_eE$$ together with a plasma-generating term $$\partial_t\varrho_e=\frac{\beta^{(2)}\vert E\vert^4}{2\hbar\omega_0}$$ In the NLSE the last term corresponds to the interaction of the electric field with the free carriers generated in the second equation.

I would like to verify my implementation to check if all parts have been implemented correctly. I have an analytical solution for the diffraction term (first term in the NLSE) and an analytical solution for the second term (non-linear absorption) when assuming a collimated beam at the beginning. For the self-focusing process I can calculate the collapse distance, again for using a collimated beam at the input.

For the last term, though, I was not able to find a possible way for verifying if I implemented it correctly yet. What are my options here, and how can I proceed for verification (and future unit tests)? I currently see two options here:

  • Separate both equations, and assume that they are independent from each other. I then can find an analytical solution for the NLSE with a constant value of $\varrho_e$ and an analytical solution for the generation equation with a constant value of $E$.
  • Approximate the equations as close as possible and find an estimated solution, similar to the collapse distance of the self-focusing effect

The former approach would be the simpler approach, but would have the drawback that I can not test the interaction between both equations. The latter approach would be closer to my real implementation, but I am not sure if that approach is possible at all. Thus, are there other, simpler options?

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Use the Method of Manufactured Solutions to create whatever solution you like, crank it through the equations to give a forcing function and boundary conditions, and then put those into your solver and compare its answers to the ones you started with. This works best on nice domains.

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