Eur. Phys. J. D 39, 49–57 (2006) DOI: 10.1140/epjd/e2006-00079-1 T HE EUROPEAN P HYSICAL JOURNAL D Modulational instability of dust acoustic waves in a dusty plasma with nonthermal electrons and ions A.P. Misra 1,2 and A. Roy Chowdhury 2, a 1 Department of Mathematics, Siksha Bhabana, Visva-Bharati University Santiniketan-731 235, India 2 High Energy Physics Division, Department of Physics, Jadavpur University Kolkata-700 032, India Received 31 October 2005 / Received in final form 19 December 2005 Published online 11 April 2006 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2006 Abstract. Effects of nonadiabaticity of variable dust charge, dust fluid temperature, trapped electrons as well as nonisothermality of ions on the amplitude modulation of dust acoustic waves in an unmagnetized dusty plasma are investigated. A modified nonlinear Schr¨odinger equation (NLSE) is obtained by the standard reductive perturbation technique and is solved numerically by the split-step Fourier method. The modulational instability and the envelope solitary wave structure are found to be modified somewhat by the effects of nonthermally distributed ions and trapped electrons. PACS. 52.35.Sb Solitons; BGK modes – 52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions – 52.30.Ex Two-fluid and multi-fluid plasmas – 52.35.-g Waves, oscillations, and instabilities in plasmas and intense beams 1 Introduction Theoretical as well as experimental investigations on mod- ulational instability of ion-acoustic and dust acoustic waves in a dispersive and nonlinear plasma medium have been increasing because of their possible applications in space physics, astrophysics and also in many laboratory situations. By incorporating the effects of harmonic gen- eration and ponderomotive nonlinearities, many authors have derived the nonlinear Schr¨ odinger equation (NLSE) which governs the dynamics of the nonlinear modulated acoustic wave packet and the structure of the envelope solitary wave [1–10]. In the NLSE the nonlinearities are in great balance with the wave group dispersion. Theoretical investigations suggest that the presence of dust fluid tem- perature, nonadiabatic dust charge variation can affect the modulational instability of the dust acoustic wave [6]. When streaming particles be injected in a plasma, it has been found that instead of developing into a turbu- lent one, they evolve towards a coherent trapped particle state [11]. When an amplitude of a nonlinear wave be- comes large, some electrons are trapped and carried out along with the wave. It is to be mentioned that the in- clusion of trapped electrons or ions gives rise nonlinear phenomena of waves. However, it has been found that the electron and ion distributions play a significant role in characterizing the physics of nonlinear waves [12–15]. They offer considerable increase in richness and varieties of wave motions which can exist in plasmas. Also, the inclusion of thermal effects affects the nature of wave- a e-mail: asesh r@yahoo.com particle interaction and possibility of having nonisother- mal electron distribution in the potential well instead of the usual Boltzmannian. When the amplitude of the non- linear wave becomes large, electrons are trapped in the potential trough, and as matter of fact the trapping of electrons is not a question of strength of the amplitude. The trapping of plasma particles which as a nonlinear pro- cess violates the linear wave ansatz is known from the early days of Plasma Physics [16] and is valid even for small amplitude limit, as proposed, probably for the first time by Holloway et al. [17] and later approved and con- firmed by Schamel [18]. The trapped particle structures, observed in the experiments of Risoe-group, have been in- terpreted in terms of electron hole equilibria [19,20]. The electron hole is a purely nonlinear phenomena and is due to the particle trapping. The electron hole has been found to be related with the slow electron-acoustic mode which is usually absent in plasma. It can exist due to nonlin- ear distortions of the electron distribution function. Like ion-acoustic solitons an upper limit for the strength of the electron trapping has been found [19]. Effects of trapped electrons, ions, two-temperature ions for the formation of KdV (Korteweg-de Vries) solitons, double layers are inves- tigated by a number of authors [13–15,21]. But up to now, no one has considered the effects of trapped electrons, non- thermal ions and nonadiabatic dust charge variations all together to study the modulational instability and enve- lope solitary structure of the dust acoustic waves (DAW). Our paper is organized as follows: in Section 2, the ba- sic equations describing the dust dynamics are presented and the modified NLSE is derived. Section 3 is devoted to study the modulational instability and the structure of the