# Note on wavefront dislocation in surface water waves

## TL;DR

This paper investigates wavefront dislocation phenomena in one-dimensional surface water waves, linking phase singularities to the unboundedness of the Chu-Mei quotient, and demonstrates these phenomena in soliton solutions of the NLS equation.

## Contribution

It introduces the Chu-Mei quotient as a key indicator of wavefront dislocation and shows its unboundedness is necessary for phase singularities in linear wave fields.

## Key findings

- Unbounded Chu-Mei quotient correlates with wavefront dislocation.
- Phase singularities are generic at wave amplitude zeros.
- Soliton on finite background exhibits wavefront dislocation and unbounded quotient.

## Abstract

At singular points of a wave field, where the amplitude vanishes, the phase may become singular and wavefront dislocation may occur. In this Letter, we investigate for wave fields in one spatial dimension the appearance of these essentially linear phenomena. We introduce the Chu-Mei quotient as it is known to appear in the 'nonlinear dispersion relation' for wave groups as a consequence of the nonlinear transformation of the complex amplitude to real phase-amplitude variables. We show that unboundedness of this quotient at a singular point, related to unboundedness of the local wavenumber and frequency, is a generic property and that it is necessary for the occurrence of phase singularity and wavefront dislocation, while these phenomena are generic too. We also show that the 'soliton on finite background', an explicit solution of the NLS equation and a model for modulational instability leading to extreme waves, possesses wavefront dislocations and unboundedness of the Chu-Mei quotient.