Split-Step Method for Generalized Non-Linear Equations: A Three Operator Story

TL;DR

This paper introduces an advanced exponential Fourier split-step method capable of efficiently solving a broader class of nonlinear partial differential equations, including complex models in optics, while maintaining high accuracy in multiple dimensions.

Contribution

The paper develops a generalized split-step method that handles additional nonlinear terms with third-order accuracy across all spatial dimensions and time, expanding modeling capabilities.

Findings

Maintains third-order error with complex nonlinear terms

Applicable to a wider class of PDEs in physics and engineering

Demonstrates implementation in a split-step framework

Abstract

This paper describes an updated exponential Fourier based split-step method that can be applied to a greater class of partial differential equations than previous methods would allow. These equations arise in physics and engineering, a notable example being the generalized derivative non-linear Schr\"odinger equation that arises in non-linear optics with self-steepening terms. These differential equations feature terms that were previously inaccessible to model accurately with low computational resources. The new method maintains a 3rd order error even with these additional terms and models the equation in all three spatial dimensions and time. The class of non-linear differential equations that this method applies to is shown. The method is fully derived and implementation of the method in the split-step architecture is shown. This paper lays the mathematical ground work for an upcoming paper employing this method in white-light generation simulations in bulk material.