Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions

Рудерман, М. С.; Talipova, Tatiana; Pelinovsky, Efim

doi:10.1017/s0022377808007150

Cited by 102 publications

(54 citation statements)

References 38 publications

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“…Similar derivations for particular cases can be found in [50,2,96,85]. The main purpose of our derivation is to identify the generic conditions under which the mKdVB model dynamics persist.…”

Section: Overview Of This Workmentioning

confidence: 68%

“…In that case, third-order terms become necessary in the expansions (2.4) to account for the higher-order nonlinear term in the resulting equation for v 1 . Then balance is achieved with new scaling parameters p = 1 and q = 3 in the transformation (2.5) (see, e.g., [96], [85]). Substituting the new multiple scales expansions (2.4), (2.5) into (2.1) and equating like powers of ε up to ε 4 , we obtain the mKdVB equation for the first order corrections,…”

Section: Mkdvb Approximationmentioning

confidence: 99%

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Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

El¹,

Hoefer²,

Shearer³

2017

SIAM Rev.

108

150

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Abstract. We consider two physically and mathematically distinct regularization mechanisms of scalar hyperbolic conservation laws. When the flux is convex, the combination of diffusion and dispersion are known to give rise to monotonic and oscillatory traveling waves that approximate shock waves. The zero-diffusion limits of these traveling waves are dynamically expanding dispersive shock waves (DSWs). A richer set of wave solutions can be found when the flux is non-convex. This review compares the structure of solutions of Riemann problems for a conservation law with non-convex, cubic flux regularized by two different mechanisms: 1) dispersion in the modified Korteweg-de Vries (mKdV) equation; and 2) a combination of diffusion and dispersion in the mKdV-Burgers equation. In the first case, the possible dynamics involve two qualitatively different types of DSWs, rarefaction waves (RWs) and kinks (monotonic fronts). In the second case, in addition to RWs, there are traveling wave solutions approximating both classical (Lax) and non-classical (undercompressive) shock waves. Despite the singular nature of the zero-diffusion limit and rather differing analytical approaches employed in the descriptions of dispersive and diffusive-dispersive regularization, the resulting comparison of the two cases reveals a number of striking parallels. In contrast to the case of convex flux, the mKdVB to mKdV mapping is not one-to-one. The mKdV kink solution is identified as an undercompressive DSW. Other prominent features, such as shock-rarefactions, also find their purely dispersive counterparts involving special contact DSWs, which exhibit features analogous to contact discontinuities. This review describes an important link between two major areas of applied mathematics, hyperbolic conservation laws and nonlinear dispersive waves.

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Section: Overview Of This Workmentioning

confidence: 68%

Section: Mkdvb Approximationmentioning

confidence: 99%

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

El¹,

Hoefer²,

Shearer³

2017

SIAM Rev.

108

150

View full text Add to dashboard Cite

show abstract

“…The dynamics of modulationally unstable packets described by the mKdV equation has been extensively studied (see, e.g., [32]). Recently a similar study has been carried out for modulationally unstable packets described by the Gardner equation [31].…”

Section: Dynamics Of Modulationally Unstable Ion-acoustic Wavesmentioning

confidence: 98%

“…In that case the dynamics of nonlinear ion-acoustic waves is described by the Gardner equation [30,31],…”

Section: Gardner Equation For Nonlinear Wavesmentioning

confidence: 99%

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Freak waves in laboratory and space plasmas

Рудерман

2010

Eur. Phys. J. Spec. Top.

137

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Abstract. Generation of large-amplitude short-lived wave groups from small-amplitude initial perturbations in plasmas is discussed. Two particular wave modes existing in plasmas are considered. The first one is the ion-sound wave. In a plasmas with negative ions it is described by the Gardner equation when the negative ion concentration is close to critical. The results of numerical solution of the Gardner equation with the modulationally unstable initial condition are presented. These results clearly show the possibility of generation of freak ionacoustic waves due to the modulational instability. The second wave mode is the Alfvén wave. When this wave propagates at a small angle with respect to the equilibrium magnetic field, and its wave length is comparable with the ion inertia length, it is described by the DNLS equation. Studying the evolution of an initial perturbation using the linearized DNLS equation shows that the generation of freak Alfvén waves is possible due to linear dispersive focusing. The numerical solution of the DNLS equation reveals that the nonlinear dispersive focusing can also produce freak Alfvén waves.

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Breather structures in a plasma with warm ions and Cairns electrons

Shan

2018

Contributions to Plasma Physics

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The modulational instability (MI) of ion-acoustic waves (IAWs) in a two-component non-Maxwellian hot plasma with Cairns distributed electrons is investigated. Using the derivative expansion method, a non-linear Schrödinger equation (NLSE) is obtained. The criterion for MI, which occurs for small values of the carrier wavenumber, is obtained. It is found that non-thermal parameter modifies the criteria for MI instability. The analytical solutions of NLSE in the form of various types of freak waves, including the Peregrine soliton, the Akhmediev breather, and the Kuznetsov-Ma breather, are examined. The effects of the non-thermal parameter as well as ion-to-electron temperature ratio on the rogue wave profile are discussed. Our results should be relevant to wave propagation stability. Non-thermal models may play an increasingly important role in predicting complex plasma behaviour and understanding the underlying physical processes. KEYWORDSAkhmediev breather, breathers, Kuznetsov-Ma breather, Peregrine soliton 1

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Dynamics of modulationally unstable ion-acoustic wavepackets in plasmas with negative ions

Cited by 102 publications

References 38 publications

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

Dispersive and Diffusive-Dispersive Shock Waves for Nonconvex Conservation Laws

Freak waves in laboratory and space plasmas

Breather structures in a plasma with warm ions and Cairns electrons

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