# Energy preserving methods for nonlinear Schr\"odinger equations

## TL;DR

This paper analyzes energy-preserving numerical methods for nonlinear Schrödinger equations, proving the order of the relaxation method and introducing a generalized version for various nonlinearities, supported by numerical simulations.

## Contribution

It provides a rigorous proof of the relaxation method's order and proposes a generalized energy-preserving method for different nonlinearities.

## Key findings

- Relaxation method has order 2 for cubic nonlinear Schrödinger equations.
- Generalized relaxation method effectively handles various power law nonlinearities.
- Numerical simulations demonstrate the efficiency of the proposed methods.

## Abstract

This paper is concerned with the numerical integration in time of nonlinear Schr\"odinger equations using different methods preserving the energy or a discrete analog of it. The Crank-Nicolson method is a well known method of order 2 but is fully implicit and one may prefer a linearly implicit method like the relaxation method introduced in [10] for the cubic nonlinear Schr{\"o}dinger equation. This method is also an energy preserving method and numerical simulations have shown that its order is 2. In this paper we give a rigorous proof of the order of this relaxation method and propose a generalized version that allows to deal with general power law nonlinearites. Numerical simulations for different physical models show the efficiency of these methods.