Eur. Phys. J. Plus (2021) 136:1061 https://doi.org/10.1140/epjp/s13360-021-02063-x Regular Article Modulation of multi-dimensional waves in anisotropic magnetized plasma Muhammad Khalid 1,2 , F. Hadi 3 , Ata ur Rahman 1,a 1 Theoretical Plasma Physics Group, Islamia College Peshawar, Peshawar 25120, Pakistan 2 Department of Physics, Government Post Graduate College Mardan, Mardan 23200, Pakistan 3 Department of Applied Physical and Material Sciences, University of Swat, Swat 19200, Pakistan Received: 8 January 2021 / Accepted: 10 October 2021 © The Author(s), under exclusive licence to Società Italiana di Fisica and Springer-Verlag GmbH Germany, part of Springer Nature 2021 Abstract One of the generic nonlinear mechanism which was so far unexplored is the oblique amplitude modulation of ion acoustic waves (IAWs) in anisotropic magnetoplasma with positive ions having anisotropic thermal pressure and Maxwellian electrons. The three- dimensional (3D) nonlinear Schrödinger equation (NLSE) for the wave potential is derived via a multi-scales perturbation technique. The criteria under which the 3D modulational insta- bility (MI) develops are discussed. The upshots of relevant plasma parameters, specifically, the ion pressure anisotropy on the dynamics of 3D MI of IAWs as well as the instability growth rate are investigated in detail. The bearing to peculiar plasma environments is also highlighted. 1 Introduction One of many standard nonlinear mechanisms dominating wavepacket propagation in plas- mas is amplitude modulation, related to the nonlinear self-interaction of the carrier wave, resulting in the generation of higher harmonics. This mechanism is described by a nonlinear Schrödinger-type equation, derived from the basic fluid equations via a standard multiple (space and time) scale technique [1, 2]. Through analytical and numerical investigations, it has been identified that the modulated wavepackets have connection with modulational insta- bility (MI), which may ultimately, in turn give rise to the formation of envelope structures, whose characteristics depend on the same criteria as is necessary for the occurrence of MI. Owing to the valuable importance in stable wave propagation, the MI of various wave modes in a nonlinear dispersive plasma is an intense area of research in the recent past [3–5]. In 1977 Watanabe was the first one who made the experimental confirmation of the MI of monochromatic ion acoustic waves (IAWs) [6]. As regards, the theoretical study of MI of electrostatic nonlinear waves, a few papers are highlighted herein [3–5, 7–14] . Islami and Mottaghizadeh [15] studied the MI of electron acoustic waves (EAWs) in the presence of an external magnetic field and derived a three-dimensional (3D) NLSE. It was found that the conditions of MI, associated threshold, and the growth rate are modified owing to the presence of superthermal electrons. Dalui and Bandyopadhyay [16] also looked for the MI of a e-mail: ata797@yahoo.com (corresponding author) 0123456789().: V,-vol 123