# New variational characterization of periodic waves in the fractional Korteweg-de Vries equation

## TL;DR

This paper introduces a new variational approach to characterize periodic waves in the fractional Korteweg-de Vries equation, enabling a detailed analysis of their existence and spectral stability based on energy minimization principles.

## Contribution

It presents a novel variational characterization of periodic waves as constrained minimizers of the quadratic energy form, expanding understanding of their stability and existence regions.

## Key findings

- Unfolds the existence region of periodic waves.
- Provides a sharp spectral stability criterion.
- Establishes a monotonicity condition for stability analysis.

## Abstract

Periodic waves in the fractional Korteweg-de Vries equation have been previously characterized as constrained minimizers of energy subject to fixed momentum and mass. Here we characterize these periodic waves as constrained minimizers of the quadratic form of energy subject to fixed cubic part of energy and the zero mean. This new variational characterization allows us to unfold the existence region of travelling periodic waves and to give a sharp criterion for spectral stability of periodic waves with respect to perturbations of the same period. The sharp stability criterion is given by the monotonicity of the map from the wave speed to the wave momentum similarly to the stability criterion for solitary waves.