# Split-step-method for coupled equations

I have implemented a split-step-method for an equation of the shape $$\partial_z E = i\partial_x^2E+ic|E|^2E$$ resulting in a split into the linear part $$L=\partial_x^2$$ and the nonlinear part $$N=c|E|^2E$$ giving me $$E(x, z+\Delta z)=\exp(i\Delta z(N+L))E(x, t)$$ But now my $c$-variable is non-constant, but rather $$\partial_tc=|E|^4$$ How do I couple that to my equation above in the split-step approach (either using an FFT or a DHT)? I can not include it as nonlinear term, after it does not depend on $E$, only on the absolute value.

• Why not use a method whereby you first compute $E$ based on the previous $c$, and then compute $c$ based on the just-computed $E$? – Wolfgang Bangerth Nov 8 '17 at 0:44
• @WolfgangBangerth: I calculate $c$ after the initial linear step (I am using a $LNL$-method) using $c = |E|^4\cdot dt + c_0$, and use this value for calculating the nonlinear step. Is there a possible estimation about the error of that approach? – arc_lupus Nov 8 '17 at 9:14
• You are using a first-order operator splitting method for the coupled system. There are plenty of books about how to estimate the error. – Wolfgang Bangerth Nov 8 '17 at 23:21
• I use a modified version of the code above, resulting in an error of $\Delta z^2$, but I was wondering how the error changes when I calculate $c$ in the middle of the half-step. – arc_lupus Nov 9 '17 at 8:15