# On nonparaxial nonlinear Schr\"odinger-type equations

## TL;DR

This paper investigates the mathematical properties and numerical methods for the one-dimensional nonparaxial nonlinear Schrödinger equation, an alternative model to the classical equation when paraxial assumptions are invalid.

## Contribution

It provides theoretical analysis of well-posedness, Hamiltonian, and multi-symplectic structures, and develops structure-preserving numerical schemes for the nonparaxial nonlinear Schrödinger equation.

## Key findings

- Established linear well-posedness of the equation
- Derived Hamiltonian and multi-symplectic formulations
- Developed numerical methods that preserve these structures

## Abstract

In this paper the one-dimensional nonparaxial nonlinear Schr\"odinger equation is considered. This was proposed as an alternative to the classical nonlinear Schr\"odinger equation in those situations where the assumption of paraxiality may fail. The paper contributes to the mathematical properties of the equation in a two-fold way. First, some theoretical results on linear well-posedness, Hamiltonian and multi-symplectic formulations are derived. Then we propose to take into account these properties in order to deal with the numerical approximation. In this sense, different numerical procedures that preserve the Hamiltonian and multi-symplectic structures are discussed and illustrated with numerical experiments.