Coupled differential equations with convolution and correlation

[schulzy]

Homework Statement

I have two equations:
$$\frac{\partial}{\partial z}E_{L}\left(z,\omega \right) = i \frac{2 \mu d_{eff}(\omega+\omega_{0})^{2}}{k(\omega+\omega_{0})}\int E_{L}(z,\omega-\omega_{T})E_{T}(z,\omega_{T})d\omega_{T}$$
$$\frac{\partial}{\partial z}E_{T}\left(z,\omega_{T} \right) = i \frac{\mu d_{eff}\omega_{T}^{2}}{2k(\omega_{T})}\int E_{L}(z,\omega+\omega_{T})E_{T}^{*}(z,\omega)d\omega$$

Homework Equations

$$\mu$$ is the magnetic permeability
k is the real wave vectors at respective frequencies, which are determined by the dispersion relation
$$d_{eff}$$ is the effective nonlinear optical coefficient.
$$E_{L}(z,\omega)$$ optical frequency component $$E_{T}(z,\omega_{T})$$ THz frequency components.

The Attempt at a Solution

I tied solve this equations whit Runge- Kutta method, but I think, I ruined something. I attached my MATLAB solution.
Where I may have ruined it?

 

[NateD]

§§ COMUpdateI added more information:I have two equations:\frac{\partial}{\partial z}E_{L}\left(z,\omega \right) = i \frac{2 \mu d_{eff}(\omega+\omega_{0})^{2}}{k(\omega+\omega_{0})}\int E_{L}(z,\omega-\omega_{T})E_{T}(z,\omega_{T})d\omega_{T} \frac{\partial}{\partial z}E_{T}\left(z,\omega_{T} \right) = i \frac{\mu d_{eff}\omega_{T}^{2}}{2k(\omega_{T})}\int E_{L}(z,\omega+\omega_{T})E_{T}^{*}(z,\omega)d\omega The parameters are: \mu is the magnetic permeabilityk is the real wave vectors at respective frequencies, which are determined by the dispersion relationd_{eff} is the effective nonlinear optical coefficient.E_{L}(z,\omega) optical frequency component E_{T}(z,\omega_{T}) THz frequency components.My goal is to solve these two coupled equations with initial conditions and boundary conditions. I want to solve the equations for the two fields E_{L}(z,\omega) and E_{T}(z,\omega_{T}) as a function of the propagation distance z. I tried to solve this equations with Runge- Kutta method, but I think, I ruined something. I attached my MATLAB solution. Where I may have ruined it?