Modulation leading to frequency downshifting of water waves in the vicinity of the Benjamin-Feir transition

TL;DR

This paper demonstrates that water waves near the Benjamin-Feir transition can undergo a permanent frequency downshift through a heteroclinic connection, explained via Whitham modulation theory and confirmed by numerical simulations.

Contribution

It reveals a new mechanism for frequency downshifting in water waves without viscosity, using asymptotic phase dynamics and energetic analysis.

Findings

Heteroclinic connection links stable and unstable wave families.

Frequency downshift occurs without viscous effects.

Numerical simulations support the theoretical predictions.

Abstract

For Stokes waves in finite depth within the neighbourhood of the Benjamin-Feir stability transition, there are two families of periodic waves, one modulationally unstable and the other stable. In this paper we show that these two families can be joined by a heteroclinic connection, which manifests in the fluid as a travelling front. By shifting the analysis to the setting of Whitham modulation theory, this front is in wavenumber and frequency space. An implication of this jump is that a permanent frequency downshift of the Stokes wave can occur in the absence of viscous effects. This argument, which is built on a sequence of asymptotic expansions of the phase dynamics, is confirmed via energetic arguments, with additional corroboration obtained by numerical simulations of a reduced model based on the Benney-Roskes equation.