# On symmetry of traveling solitary waves for dispersion generalized NLS

## TL;DR

This paper proves symmetry properties of traveling solitary waves in a broad class of dispersion generalized nonlinear Schrödinger equations, using Fourier space rearrangements, applicable to various higher-order and fractional NLS models.

## Contribution

It establishes symmetry results for solitary waves in dispersion generalized NLS equations with integer powers, extending to many NLS variants using Fourier rearrangement techniques.

## Key findings

- Symmetry of traveling solitary waves proven for integer power nonlinearities.
- Results apply to fourth-order, fractional, and half-wave NLS equations.
- Uses Fourier space Steiner rearrangements for proofs.

## Abstract

We consider dispersion generalized nonlinear Schr\"odinger equations (NLS) of the form $i \partial_t u = P(D) u - |u|^{2 \sigma} u$, where $P(D)$ denotes a (pseudo)-differential operator of arbitrary order. As a main result, we prove symmetry results for traveling solitary waves in the case of powers $\sigma \in \mathbb{N}$. The arguments are based on Steiner type rearrangements in Fourier space. Our results apply to a broad class of NLS-type equations such as fourth-order (biharmonic) NLS, fractional NLS, square-root Klein-Gordon and half-wave equations.