  Citation: Chaudhuri, S.; Chowdhury, A.R.; Ghosh, B. 3D-Modulational Stability of Envelope Soliton in a Quantum Electron–Ion Plasma—A Generalised Nonlinear Schrödinger Equation. Plasma 2022, 5, 60–73. https://doi.org/10.3390/ plasma5010005 Academic Editor: Andrey Starikovskiy Received: 15 November 2021 Accepted: 28 December 2021 Published: 17 January 2022 Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affil- iations. Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). plasma Article 3D-Modulational Stability of Envelope Soliton in a Quantum Electron–Ion Plasma—A Generalised Nonlinear Schrödinger Equation Shatadru Chaudhuri 1 , Asesh Roy Chowdhury 2, * and Basudev Ghosh 1 1 Department of Physics, Jadavpur University, Kolkata 700 032, India; shatadru.chaudhuri12@gmail.com (S.C.); bsdvghosh@gmail.com (B.G.) 2 High Energy Physics Division, Department of Physics, Jadavpur University, Kolkata 700 032, India * Correspondence: arc.roy@gmail.com Abstract: In physical reality, the phenomena of plasma physics is actually a three-dimensional one. On the other hand, a vast majority of theoretical studies only analyze a one-dimensional prototype of the situation. So, in this communication, we tried to treat the quantum electron–ion plasma in a full 3D setup and the modulational stability of envelope soliton was studied in a quantum electron– ion plasma in three dimensions. The Krylov–Bogoliubov–Mitropolsky method was applied to the three-dimensional plasma governing equations. A generalized form of the nonlinear Schrödinger (NLS) equation was obtained, whose dispersive term had a tensorial character, which resulted in the anisotropic behavior of the wave propagation even in absence of a magnetic field. The stability condition was deduced ab initio and the stability zones were plotted as a function of plasma parameters. The modulational stability of such a three-dimensional NLS equation was then studied as a function of plasma parameters. It is interesting to note that the nonlinear excitation of soliton took place again here due to the balance of nonlinearity and dispersion. The zones of contour plots are given in detail. Keywords: electron–ion plasma; quantum plasma; 3D-NLS; modulational instability; envelope soliton; K–B–M method PACS: 52.27.Aj; 52.35.Mw; 52.35.Sb; 52.65.Vv; 52.35.-g 1. Introduction The analysis of nonlinear wave propagation in plasma is one of the most important topics of present-day research. Though the physical situation is three-dimensional, for mathematical simplicity, many a times the study is done in one dimension only. However, one should have an idea of the actual theoretical prediction in 3D. With this idea in mind, we treated a quantum electron–ion plasma in three dimensions; we found that we arrived at an new type of generalized nonlinear Schrödinger equation (NLSE). The study of quantum plasma was initiated mainly by the elegant works of Haas [1], Manfredi [2], Shukla [3,4] and Brodin [5]. Up until now, many different situations have been analyzed by various researchers, but all are mainly in one space–time dimension. The basis of the formulation of quantum plasma lies in the unique phase-space quantization advocated by E.P. Wigner [6] long ago. For a long time, this methodology was ignored due to the conceptual difficulty of phase space in quantum physics. However, with its successful application in plasma physics, interest has been invoked and many new observations have been conducted. In this context, one may note that many different situations of quantum plasma have been investigated in relation to the study of a nonlinear Schrödinger (NLS) equation and the existence of an envelope soliton; however, all are mainly in two dimensions. Even the NLS equation has been derived in many other situations of physics and mathematics, viz., nonlinear optics, pulse propagation, etc. [7–16], but all were in two dimensions. In Plasma 2022, 5, 60–73. https://doi.org/10.3390/plasma5010005 https://www.mdpi.com/journal/plasma