[Paper Review] The width-amplitude relation of three-dimensional Bernstein-Greene-Kruskal electron solitary waves

Li‐Jen Chen, D. J. Thouless+1 more|arXiv (Cornell University)|Mar 5, 2003

Dust and Plasma Wave Phenomena3 citations

TL;DR

This paper derives a width-amplitude inequality relation for three-dimensional Bernstein-Greene-Kruskal (BGK) electron solitary waves in magnetized plasmas with field-aligned electrons. The inequality shows that for a given amplitude, a broad range of widths is allowed, distinguishing BGK solitons from Korteweg-de Vries solitons and enabling easier excitation; this holds even when finite electron cyclotron radius effects are included, provided the radius is small compared to the structure's spatial scale.

ABSTRACT

Three-dimensional Bernstein-Greene-Kruskal (BGK) electron solitary wave solutions are constructed for magnetized plasmas in which electrons are exactly magnetic field-aligned. The form of the solitary potential is not tightly constrained. If one takes a Gaussian form, the relation between the solitary potential amplitude and its parallel and perpendicular widths is constrained by an inequality. The inequality width-amplitude relation allows us to distinguish, based on macroscopic observables, the BGK solitary waves from other solitons such as Korteweg-de Vries solitons whose widths and amplitudes are of a one-one mapping relation. The inequality relation allows BGK solitary waves to be excited easily, since for a fixed amplitude there exists a wide range of admitted widths. When finite cyclotron radius effect is taken into account, the properties obtained for the case of zero cyclotron radius are found to be still applicable under the condition that the electron cyclotron radius is much less than the distance over which the structure of the solitary wave varies.

Motivation & Objective

• To derive the width-amplitude relation for three-dimensional BGK electron solitary waves in magnetized plasmas with field-aligned electrons.
• To determine how the form of the solitary potential, particularly a Gaussian shape, constrains the relationship between amplitude and spatial widths.
• To establish a distinguishing criterion between BGK solitary waves and other solitons, such as Korteweg-de Vries solitons, based on macroscopic observables.
• To assess the robustness of the width-amplitude inequality when finite cyclotron radius effects are included in the model.

Proposed method

• Constructing three-dimensional BGK electron solitary wave solutions under the assumption that electrons are exactly magnetic field-aligned.
• Assuming a Gaussian functional form for the solitary potential to explore the constraints on its amplitude and parallel/perpendicular widths.
• Deriving an inequality that relates the amplitude of the potential to its parallel and perpendicular widths, rather than a one-to-one mapping.
• Analyzing the effect of finite electron cyclotron radius on the derived width-amplitude relation, assuming it is much smaller than the scale length of the solitary wave structure.
• Using the BGK formalism to ensure exact solutions under the given field alignment and electron dynamics constraints.
• Comparing the derived inequality with the one-to-one width-amplitude mapping characteristic of Korteweg-de Vries solitons to highlight the distinction.

Experimental results

Research questions

• RQ1 What is the functional relationship between the amplitude and the parallel and perpendicular widths of three-dimensional BGK electron solitary waves in magnetized plasmas?
• RQ2 How does assuming a Gaussian potential shape affect the constraints on the width-amplitude relation in BGK solitary waves?
• RQ3 Can the width-amplitude inequality distinguish BGK solitary waves from other solitons like Korteweg-de Vries solitons based on observable macroscopic parameters?
• RQ4 To what extent are the results for zero cyclotron radius valid when finite cyclotron radius effects are included in the model?

Key findings

• The width-amplitude relation for 3D BGK solitary waves is governed by an inequality rather than a one-to-one mapping, allowing a broad range of widths for a fixed amplitude.
• This inequality enables easier excitation of BGK solitary waves, as multiple width configurations are compatible with a given amplitude.
• The Gaussian potential assumption leads to a specific form of the width-amplitude inequality that constrains the possible combinations of amplitude and spatial widths.
• The inequality relation clearly distinguishes BGK solitary waves from Korteweg-de Vries solitons, which exhibit a one-to-one width-amplitude dependence.
• When finite electron cyclotron radius effects are included, the derived width-amplitude inequality remains valid under the condition that the cyclotron radius is much smaller than the spatial scale of the solitary wave structure.
• The robustness of the inequality under finite cyclotron radius effects confirms its relevance for realistic plasma conditions where electron motion is not perfectly free.