Explosive Development of the Quantum Kelvin-Helmholtz Instability on the He-II Free Surface

TL;DR

This paper investigates the nonlinear quantum Kelvin-Helmholtz instability on the He-II free surface, revealing explosive growth behavior and the destabilizing role of nonlinearity, with implications for quantum fluid dynamics.

Contribution

It introduces a nonlinear model describing the quantum Kelvin-Helmholtz instability using the Klein-Gordon equation and identifies conditions for explosive surface perturbation growth.

Findings

Nonlinear evolution governed by the $||^4$ Klein-Gordon equation.

Nonlinearity accelerates the instability regardless of density ratios.

Conditions for explosive growth of surface perturbations are established.

Abstract

We analyze nonlinear dynamics of the Kelvin-Helmholtz quantum instability of the He-II free surface, which evolves during counterpropagation of the normal and superfluid components of liquid helium. It is shown that in the vicinity of the linear stability threshold, the evolution of the boundary is described by the $|\phi|^4$ Klein-Gordon equation for the complex amplitude of the excited wave with cubic nonlinearity. It is important that for any ratio of the densities of the helium component, the nonlinearity plays a destabilizing role, accelerating the linear instability evolution of the boundary. The conditions for explosive growth of perturbations of the free surface are formulated using the integral inequality approach. Analogy between the Kelvin-Helmholtz quantum instability and electrohydrodynamic instability of the free surface of liquid helium charged by electrons is considered.