# Nonlinear Wave Equation (Nonlinear Helmholtz)

• A
jgk5141
TL;DR Summary
Looking to solve a nonlinear wave equation using some sort of numerical method. Struggling with convergence issues.
I am trying to solve a PDE (which I believe can be approximated as an ODE). I have tried to solve it using 4th Order Runge-Kutta in MATLAB, but have struggled with convergence, even at an extremely high number of steps (N=100,000,000). The PDE is:
$$\frac{\partial^2 E(z)}{\partial z^2} + \frac{\omega^2}{c^2}\epsilon(z)E(z)=0$$
Where $\epsilon(z) = \epsilon[ 1 + i \gamma |E(z)|^2]$. This is the "nonlinear" part where the imaginary part of the permittivity ($\epsilon$) is dependent on the intensity of the field ($~ |E(z)|^2$). The coefficient $\gamma$ can be considered the Loss Coefficient. With low $\gamma$ the intensity dependent term is negligible and the RK4 approach will converge, but with high intensities or higher $\gamma$ then it blows up.

I am using the numerical outputs from another simulation to give the field and the value of it's derivative for initial conditions.

Any suggestions on how to solve this type of differential equation? Analytically or numerically? I do not have much experience in numerical methods.

#### Attachments

• NL-RK4_High_Gamma.png
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• NL-RK4_Low_Gamma.png
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