<p>In this work, a numerical simulation of the two-stream instability (TSI) in electrostatic plasmas is presented using the particle-in-cell (PIC) method. The TSI is a fundamental phenomenon in plasma physics, where two streams of charged particles with different velocities interact, generating electrostatic waves. These waves can give rise to non-linear structures, such as solitary waves, which are of great interest in applications ranging from nuclear fusion to astrophysics. Different perturbation amplitudes (<InlineEquation ID="IEq1"> <EquationSource Format="TEX">$A=0.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.0</mn> </mrow> </math></EquationSource> </InlineEquation>, 0.1 and 0.5) were explored to analyse how they affect the formation and propagation of electrostatic waves. The results show that in the absence of external perturbations (<InlineEquation ID="IEq2"> <EquationSource Format="TEX">$A=0.0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.0</mn> </mrow> </math></EquationSource> </InlineEquation>) the dynamic interaction between the particle flows generates Langmuir waves, which propagate without significant dissipation. With moderate perturbation (<InlineEquation ID="IEq3"> <EquationSource Format="TEX">$A=0.1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </math></EquationSource> </InlineEquation>), the formation of vortices in phase space is observed, which eventually merge to give rise to electrostatic solitary waves (ESW). For larger perturbation amplitudes (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$A=0.5$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </math></EquationSource> </InlineEquation>), the system enters a non-linear regime, where interactions between particles and fields give rise to non-dispersive solitary waves, such as Bernstein–Greene–Kruskal (BGK) waves. The PIC method has proven to be an effective tool for capturing both linear and non-linear effects of the TSI. These findings not only contribute to a better understanding of plasma dynamics, but also highlight the importance of the PIC method for studying non-linear phenomena in complex systems.</p>

Simulation of two-stream instability in electrostatic plasmas using the particle-in-cell method

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发表日期：

2026年1月20日

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M C González,
H Martínez,
R G Avila-Bonilla,
N Enriquez-Sánchez

摘要

In this work, a numerical simulation of the two-stream instability (TSI) in electrostatic plasmas is presented using the particle-in-cell (PIC) method. The TSI is a fundamental phenomenon in plasma physics, where two streams of charged particles with different velocities interact, generating electrostatic waves. These waves can give rise to non-linear structures, such as solitary waves, which are of great interest in applications ranging from nuclear fusion to astrophysics. Different perturbation amplitudes ( $A=0.0$ $A = 0.0$ , 0.1 and 0.5) were explored to analyse how they affect the formation and propagation of electrostatic waves. The results show that in the absence of external perturbations ( $A=0.0$ $A = 0.0$ ) the dynamic interaction between the particle flows generates Langmuir waves, which propagate without significant dissipation. With moderate perturbation ( $A=0.1$ $A = 0.1$ ), the formation of vortices in phase space is observed, which eventually merge to give rise to electrostatic solitary waves (ESW). For larger perturbation amplitudes ( $A=0.5$ $A = 0.5$ ), the system enters a non-linear regime, where interactions between particles and fields give rise to non-dispersive solitary waves, such as Bernstein–Greene–Kruskal (BGK) waves. The PIC method has proven to be an effective tool for capturing both linear and non-linear effects of the TSI. These findings not only contribute to a better understanding of plasma dynamics, but also highlight the importance of the PIC method for studying non-linear phenomena in complex systems.