Solving NLSE with Fourier Split Step in MATLAB

[n0_3sc]

I'm not sure if this is the right section to post this thread but here goes...

I am solving the NLSE (nonlinear schroedinger equation) using the Fourier Split Step Method in MATLAB. To avoid MATLAB's boundary reflection problems and phase discontinuity's you naturally have to make the time/freq windows very very large. As a consequence you need to considerably increase the number of points and hence simulation time...Except increasing the number of points also makes your pulse less visible.

My question is whether or not their is a way to either:
- avoid the Fourier transform boundary reflections OR
- some how create a code that takes more points where your actual 'pulse of light' is and less points in the region where nothing exists.

 

[Aaron Bex]

It sounds like you're looking for a way to reduce the number of points needed in your simulation. One way could be to use a method such as adaptive mesh refinement (AMR), where you would only sample more points in areas where it is necessary, and fewer points in areas where it is not. Another option could be to use polynomial extrapolation methods, which would allow you to extrapolate from the points that have already been sampled, instead of having to calculate them all. Finally, you could also try using a spectral method, which uses a combination of Fourier transforms and numerical integration to solve the equation more efficiently.

 

[oswaler]

I commend you for using the Fourier Split Step Method to solve the NLSE in MATLAB. This is a powerful and widely used technique in numerical simulations of nonlinear systems. However, as you have mentioned, there are some challenges with this method, particularly in dealing with boundary reflections and phase discontinuities.

One way to address the issue of boundary reflections is to use a technique called "padding". This involves adding zeros to the end of your data before performing the Fourier transform. This effectively extends the length of your signal and can reduce the impact of boundary reflections. You can then remove the zeros after the transform is complete.

Another approach is to use a different numerical method that is less sensitive to boundary reflections, such as the split-step Fourier method or the pseudospectral method. These methods have been shown to be more accurate and efficient for solving the NLSE.

In terms of optimizing the number of points in your simulation, there are techniques such as adaptive mesh refinement that can dynamically adjust the number of points in different regions of your simulation domain. This can help to reduce the simulation time while still maintaining accuracy in regions where the pulse is present.

In conclusion, while the Fourier Split Step Method is a powerful tool for solving the NLSE, it is important to consider the limitations and potential challenges associated with it. By using techniques such as padding and adaptive mesh refinement, you can improve the accuracy and efficiency of your simulations.