Arrest of Langmuir wave collapse by quantum effects

TL;DR

This paper rigorously proves that quantum effects prevent the blow-up of Langmuir waves in plasma, demonstrating time-periodic solutions and discussing challenges in three-dimensional cases.

Contribution

It provides a rigorous mathematical proof that quantum effects arrest Langmuir wave collapse in multiple dimensions and explores solution behaviors near singularity thresholds.

Findings

Quantum effects prevent blow-up in Langmuir wave equations.

Time-periodic solutions exist near classical singularity thresholds.

Challenges in extending perturbative approaches to three dimensions.

Abstract

The arrest of Langmuir-wave collapse by quantum effects, first addressed by Haas and Shukla [Phys. Rev. E 79, 066402 (2009)] using a Rayleigh-Ritz trial-function method is revisited, using rigorous estimates and systematic asymptotic expansions. The absence of blow up for the so-called quantum Zakharov equations is proved in two and three dimensions, whatever the strength of the quantum effects. The time-periodic behavior of the solution for initial conditions slightly in excess of the singularity threshold for the classical problem is established for various settings in two space dimensions. The difficulty of developing a consistent perturbative approach in three dimensions is also discussed, and a semi-phenomenological model is suggested for this case.