# Nonlinear Wave Equation (Nonlinear Helmholtz)

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• jgk5141
In summary, the conversation discusses the challenges in solving a partial differential equation (PDE) that can be approximated as an ordinary differential equation (ODE). The PDE involves the second derivative of a function E(z) and a nonlinear term, where the imaginary part of the permittivity is dependent on the intensity of the field. The use of 4th Order Runge-Kutta method in MATLAB for solving the equation has been unsuccessful due to convergence issues, particularly at high values of the Loss Coefficient. The individual seeking advice is considering both analytical and numerical approaches and has limited experience in numerical methods. The equation is relevant to non-linear optics and a suggested solution is to account for absorption caused by the complex index of refraction.
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.

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• NL-RK4_High_Gamma.png
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• NL-RK4_Low_Gamma.png
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Your d.e. describes non-linear optics. Please see https://en.wikipedia.org/wiki/Soliton_(optics) for the closed form derivation that results in a soliton. You will have to account for absorption due to your complex index of refraction.

jim mcnamara

## 1. What is a nonlinear wave equation?

A nonlinear wave equation is a mathematical equation that describes the behavior of nonlinear waves, which are waves that do not follow the principle of superposition. This means that the amplitude of the wave is not directly proportional to the applied force or disturbance.

## 2. What is the Nonlinear Helmholtz equation?

The Nonlinear Helmholtz equation is a specific type of nonlinear wave equation that describes the propagation of light in a nonlinear medium. It takes into account the effects of nonlinearities, such as self-focusing and self-phase modulation, on the behavior of light waves.

## 3. What are some applications of the Nonlinear Helmholtz equation?

The Nonlinear Helmholtz equation has many applications in fields such as optics, photonics, and quantum mechanics. It is used to study the behavior of light in nonlinear optical materials, as well as in the design and analysis of optical devices and systems.

## 4. How is the Nonlinear Helmholtz equation solved?

The Nonlinear Helmholtz equation is typically solved using numerical methods, such as finite difference methods or finite element methods. These methods involve discretizing the equation and solving it iteratively to obtain a numerical solution.

## 5. What are some challenges in solving the Nonlinear Helmholtz equation?

Solving the Nonlinear Helmholtz equation can be challenging due to the complexity of the equation and the nonlinear behavior of the waves. It also requires a good understanding of numerical methods and the ability to choose appropriate parameters for the problem at hand.

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