7th Asia-Pacific Conference on Plasma Physics, 12-17 Nov, 2023 at Port Messe Nagoya

Nonlinear Dispersion Relation for Dust Acoustic Wave based on Korteweg-deVries modelFarida Batool1, Ajaz Mir1 , Sachin Sharma1, Abhijit Sen2 and Sanat Tiwari1
1
Indian Institute of Technology Jammu, Jammu, J&K, 181221, India
2
Institute for Plasma Research, Gandhinagar, Gujarat, 382428, Indiae-mail: farida.batool@iitjammu.ac.in
The dust acoustic wave (DAW) is supported in theplasma due to the dust inertia and plasma pressurebalance. In dusty plasma experiments, these acoustic
waves can excite spontaneously or by external
perturbation [1] and may easily attain nonlinear
characteristics [2]. In most such studies, linear theoreticalmodels are proposed to explain the experimental
dispersion of DAWs missing the impact of plasma
nonlinearity. This paper proposes a theoretical model for
nonlinear dispersion relation (NLDR) of DAWs. The
model has been developed in the weak nonlinear
dynamical regime of dust fluid and is based upon theKorteweg de-Vries (KdV) equation.
The proposed nonlinear dispersion relation, obtained
from the exact solution of the KdV equation in which
nonlinearity is quantified by the nonlinearity factor (κ),
is given by:2

ω=4 (𝐾(κ))
2π2(κ3+ κ − 1) β𝑘

Here 𝑘 and ω represent the wavevector and frequency,respectively. β represents the dispersion of the medium
and can be written in terms of plasma parameters. The
value of κ ranges between 0 and 1. At κ = 0, theequation represents purely linear dispersion relation(LDR), and at κ = 1, it has a nonlinear continuum
dispersion representing a single soliton solution of theKdV equation. To check the effect of nonlinearity, wefirst spatio-temporarily evolved the solution of the KdV[3] and then took its Fourier transform (FT). The result isshown in Fig. 1, with bright spots. The first spot
represents the fundamental mode, while the other spots

Fig 1. Comparison of NLDR (red square) and LDR(yellow dash-dash). The bright spots are obtained bytaking the Fourier transform of the KdV solution.

represent the harmonics resulting from nonlinearity.
Further, using the KdV-based NLDR for two different
values of κ (0 and 0. 98), we found that the nonlinearity
leads to a shift in the frequency towards higher values.
This result is shown in Fig. 1, represented by the yellow
dash-dash (κ = 0) and red square (κ = 0. 98) lines.
We further compared the NLDR with the experimental
result [1] in the long wavelength limit 𝑘λ𝐷 << 1. We
also plot the NLDR for three different values of κ withdash-dotted (κ = 0), dash-dash (κ = 0. 7), and dotted (Fig. 2, indicating theκ = 0. 9) lines shown infrequency shifting toward higher values.
We conclude that nonlinearity causes a frequency shift
toward positive values. Therefore, there is a need toinclude the nonlinearity factor in the LDR of DAWs toproperly explain the experimental dispersion relations.
References:[1] Thompson, C., Barkan, A., D’Angelo, N., andMerlino, R. L.: Dust acoustic waves in a direct current
glow discharge. Phys. Plasmas 4, 2331 (1997)[2] V. V. Yaroshenko, S. A. Khrapak, M. Y. Pustylnik, H.
M. Thomas, O. F. Petrov, and V. E. Fortov, S. Jaiswal, A.
M. Lipaev, A. D. Usachev: Excitation of low-frequency
dust density waves in flowing complex plasmas. Phys.
Plasmas 26, 053702 (2019)[3] Mir, A. A., Tiwari, S. K., Goree, J., Sen, A., Crabtree,C., and Ganguli, G.: A forced Korteweg–de Vries modelfor nonlinear mixing of oscillations in a dusty plasma.
Phys. Plasmas 27, 113701 (2020)

Fig 2. Comparison of the experimental DAW (blackhexagon) [1] with LDR and the KdV NLDRs.