Eur. Phys. J. D 28, 109–117 (2004) DOI: 10.1140/epjd/e2003-00292-4 T HE EUROPEAN P HYSICAL JOURNAL D Finite ion temperature effects on oblique modulational stability and envelope excitations of dust-ion acoustic waves I. Kourakis a and P.K. Shukla Institut f¨ ur Theoretische Physik IV, Fakult¨at f¨ ur Physik und Astronomie, Ruhr–Universit¨at Bochum, 44780 Bochum, Germany Received 2 September 2003 Published online 21 October 2003 – c  EDP Sciences, Societ`a Italiana di Fisica, Springer-Verlag 2004 Abstract. Theoretical and numerical investigations are carried out for the amplitude modulation of dust- ion acoustic waves (DIAW) propagating in an unmagnetized weakly coupled collisionless fully ionized plasma consisting of isothermal electrons, warm ions and charged dust grains. Modulation oblique (by an angle θ) to the carrier wave propagation direction is considered. The stability analysis, based on a nonlinear Schr¨ odinger-type equation (NLSE), exhibits a sensitivity of the instability region to the modulation angle θ, the dust concentration and the ion temperature. It is found that the ion temperature may strongly modify the wave’s stability profile, in qualitative agreement with previous results, obtained for an electron- ion plasma. The effect of the ion temperature on the formation of DIAW envelope excitations (envelope solitons) is also discussed. PACS. 52.27.Lw Dusty or complex plasmas; plasma crystals – 52.35.Fp Electrostatic waves and oscil- lations (e.g., ion-acoustic waves) – 52.35.Mw Nonlinear phenomena: waves, wave propagation, and other interactions (including parametric effects, mode coupling, ponderomotive effects, etc.) – 52.35.Sb Solitons; BGK modes 1 Introduction In the recent few years, dusty plasmas (DP) have at- tracted increasing attention due to a realm of new phe- nomena associated to them and the exciting novel physics involved in their description [1]. Of particular interest was the theoretical prediction [2,3] and subsequent ex- perimental confirmation [4–6] of the existence of new DP oscillatory modes, namely the dust-acoustic wave (DAW) and the dust-ion acoustic wave (DIAW) [1,7]. The lat- ter, which is the object of this study, relies on a physical mechanism quite analogous to that of the ion acoustic wave (IAW): inertialess thermalized electrons provide the restoring force, while massive ions provide the inertia. The DIAW is characterized by a phase velocity which is much smaller (larger) than the ion (electron) thermal speed, and a frequency which is higher than the dust plasma frequency ω p,d ; therefore, on the timescale of relevance, stationary dust does not participate in the wave dynam- ics. In fact, the DIAW phase velocity is higher than that of IA waves, because of the electron density depletion in the background plasma when dust grains are negatively charged; remarkably, this fact results in suppression of the a On leave from: U.L.B., Universit´ e Libre de Bruxelles, Fac- ult´ e des Sciences Apliqu´ ees, C.P. 165/81 Physique G´ en´ erale, avenue F.D. Roosevelt 49, 1050 Brussels, Belgium. e-mail: ioannis@tp4.rub.de Landau damping mechanism [1], which is known to prevail over the IAW propagation in an electron-ion plasma [8,9]. Wave propagation in a nonlinear medium like plasma is generically subject to amplitude modulation due to the carrier wave self-interaction, related to the harmonic generation. The standard reductive perturbation tech- nique [10, 11] used to study this mechanism, leads to a non- linear Schr¨ odinger-type equation (NLSE), which describes the evolution of the carrier wave envelope. Modulated waves may develop a Benjamin-Feir-type (modulational) instability (MI), i.e. envelope collapses when subjected to external perturbations, a mechanism which often favors energy localization via the formation of envelope localized structures (envelope solitons), as is known from a variety of physical contexts [12–15]. These long-lived localized ex- citations are sustained by a mutual compensation of dis- persion and nonlinearity and can propagate in the medium over long distances, remarkably surviving impacts with each other. Plasma electrostatic modes have been widely studied in this respect [10,11,16–24]. In a rather general fash- ion [25], these studies have revealed the existence of a (carrier wavenumber) instability threshold, which is found to change once oblique modulation [19–21] or temper- ature effects [22–24] are taken into account. As far as DP electrostatic modes are concerned, studies of both DAW [26,27] and DIAW [26,29,30] have been carried out,