A&A 401, 849-850 (2003)

DOI: 10.1051/0004-6361:20030189

*Research Note*

**P. K. Shukla ^{1,}^{}
- F. Verheest^{2}**

1 - Institut für Theoretische Physik IV, Ruhr-Universität Bochum,
44780 Bochum, Germany

2 - Sterrenkundig Observatorium, Universiteit Gent, Krijgslaan 281,
9000 Gent, Belgium

Received 5 December 2002 / Accepted 13 February 2003

**Abstract**

It is shown that nonlinear dynamics of modulated dust Alfvén wave packets
is governed by a derivative nonlinear Schrödinger equation.
The latter admits localized envelope solitons whose width *L* is larger
than
,
where
is the dust Alfvén
speed and
the dust gyrofrequency.
For dusty plasma parameters relevant to molecular clouds in interstellar
space, the soliton width is of the order of a fraction of a parsec.

**Key words: **plasmas - waves - ISM: clouds

Charged dust is ubiquitous in astrophysical plasmas such as those in cometary tails, interstellar molecular clouds, and planetary nebulae (see e.g. Nakano 1984; Bliokh et al. 1995; Hartquist et al. 1997; Kamaya & Nishi 1998; Zweibel 1999; Kamaya & Nishi 2000; Nakano et al. 2002; Mendis 2002). In magnetized astrophysical plasmas, charged dust grains are responsible for new wave modes, including obliquely (to the magnetic field direction) propagating dust-acoustic waves and magnetic-field aligned dust Alfvén waves (Mendis & Rosenberg 1994; Shukla 1999; Shukla & Mamun 2002), whose frequencies are much smaller than the ion gyrofrequency. In a nonuniform dusty magnetoplasma, the magnetic fields are decoupled from the plasma due to the violation of the frozen-in-field condition (Birk et al. 1996; Rudakov 2001). Here, one can have a possibility of nonlinear structures including shocks and vortices (Rudakov 2001; Shukla & Mamun 2001).

In this paper we consider the nonlinear propagation of magnetic field-aligned dispersive dust Alfvén (DDA) waves (Shukla & Mamun 2002) and show that the DA wave dynamics is governed by a derivative nonlinear Schrödinger (DNLS) equation. The latter admits localized envelope solitons, which can account for large scale structures in interstellar media containing small scale irregularities.

Let us consider right-hand circularly polarized (RHCP) DDA waves in a
magnetized plasma composed of electrons, ions, and negatively charged dust
grains.
At equilibrium we have

(1) |

where

where is the unit vector along the

which gives for the DDA wave frequency

Here is the dust Alfvén speed. Equation (4) shows that dust Alfvén waves are weakly dispersive.

Finite amplitude DDA waves interacting nonlinearly with background density
fluctuations produce envelope of DDA waves, which are governed by the DNLS
equation (Verheest 2000)

where is the perpendicular component of the wave magnetic field, and are the slowly varying time and space coordinates. We note that Eq. (5) can be readily derived from the dusty plasma magnetohydrodynamic equations (Shukla & Mamun 2002) by using a multiple-time-and-space-scale analysis (Verheest 2000).

Introducing
for circular polarization, we
obtain from (5)

Equation (6) admits localized envelope soliton solutions

(7) |

where

The length scale *L* thus obtained is large compared to the Jeans length
corresponding to the charged dust component, since the
dust Jeans frequency for the parameters quoted is
,
and at an
average dust temperature of
the dust thermal
velocity is
.
This results in a dust Jeans length of
.
Even if the dust-acoustic velocity
is used to compute the
dust Jeans length, that increases
with a factor 10^{2},
so that the same conclusion remains valid (Verheest & Cadez 2002).
All these estimates have to be taken as indications, of course, because of
the uncertainties about the parameters of charged dust component.

To summarize, we have considered the nonlinear propagation of magnetic field-aligned dispersive dust Alfvén waves in magnetized dusty plasmas. It is shown that the self-modulation of DDA waves occurs due to their nonlinear interaction with the ambient dusty plasma. The dynamics of modulated (by zero-frequency density fluctuations) DDA waves is governed by a derivative nonlinear Schrödinger equation, which admits envelope DDA solitons. For the plasma parameters in interstellar molecular clouds, the scale size of the DDA wave envelope solitons is a fraction of a pc. Hence, these nonlinear waves can account for large scale structures in astrophysical environments.

The work of PKS was supported by the Deutsche Forschungsgemeinschaft (Bonn) through the Sonderforschungsbereich 591. FV thanks the Fonds voor Wetenschappelijk Onderzoek (Vlaanderen) for a research grant.

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