Citation: Manikandan, K.; Serikbayev, N.; Vijayasree, S.P.; Aravinthan, D. Controlling Matter-Wave Smooth Positons in Bose–Einstein Condensates. Symmetry 2023, 15, 1585. https:// doi.org/10.3390/sym15081585 Academic Editor: Sergei D. Odintsov Received: 5 July 2023 Revised: 10 August 2023 Accepted: 10 August 2023 Published: 14 August 2023 Copyright: © 2023 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/). symmetry S S Article Controlling Matter-Wave Smooth Positons in Bose–Einstein Condensates Kannan Manikandan 1 , Nurzhan Serikbayev 2,3, *, Shunmuganathan P. Vijayasree 4 and Devarasu Aravinthan 1 1 Center for Computational Modeling, Chennai Institute of Technology, Chennai 600 069, Tamilnadu, India; manikandank@citchennai.net or manikandan.cnld@gmail.com (K.M.); aravinthand@citchennai.net (D.A.) 2 Department of General and Theoretical Physics, L. N. Gumilyov Eurasian National University, Astana 010008, Kazakhstan 3 Laboratory for Theoretical Cosmology, International Center of Gravity and Cosmos, Tomsk State Univerity of Control Systems and Radio Electronics (TUSUR), 634050 Tomsk, Russia 4 Department of Computer Science and Engineering, Chennai Institute of Technology, Chennai 600 069, Tamilnadu, India; vijayasreeshunmuganathan@gmail.com * Correspondence: serikbayev_ns@enu.kz Abstract: In this investigation, we explore the existence and intriguing features of matter-wave smooth positons in a non-autonomous one-dimensional Bose–Einstein condensate (BEC) system with attractive interatomic interactions. We focus on the Gross–Pitaevskii (GP) equation/nonlinear Schrödinger-type equation with time-modulated nonlinearity and trap potential, which govern nonlinear wave propagation in the BEC. Our approach involves constructing second- and third-order matter-wave smooth positons using a similarity transformation technique. We also identify the con- straints on the time-modulated system parameters that give rise to these nonlinear localized profiles. This study considers three distinct forms of modulated nonlinearities: (i) kink-like, (ii) localized or sech-like, and (iii) periodic. By varying the parameters associated with the nonlinearity strengths, we observe a rich variety of captivating behaviors in the matter-wave smooth positon profiles. These behaviors include stretching, curving, oscillating, breathing, collapsing, amplification, and suppres- sion. Our comprehensive studies shed light on the intricate density profile of matter-wave smooth positons in BECs, providing valuable insights into their controllable behavior and characteristics in the presence of time-modulated nonlinearity and trap potential effects. Keywords: matter waves; positons; Bose–Einstein condensates; Gross–Pitaevskii equation; similarity transformation 1. Introduction Theoretical investigations into the nonlinear collective excitations of matter waves have emerged as a highly intriguing and pertinent field, especially in light of the experimen- tal observations of Bose–Einstein condensation (BEC) in vapors of alkali metal atoms [1,2]. Among the captivating manifestations of localized waves in atomic matter, solitons inspire particular interest. The concept of a soliton was initially introduced to describe nonlinear solitary waves that exhibit remarkable properties, such as non-dispersive behavior, pre- serving their localized form and speeds both during propagation and after collisions [3–9]. These inherent advantages of solitons have sparked significant interest in the study of non- linear systems across various fields of physics, particularly in high-rate telecommunications involving optical fibers, fluid dynamics, capillary waves, hydrodynamics, plasma physics, and so on [4,6,10–12]. On the other hand, BECs emit Faraday and resonant density waves when subjected to harmonic driving [13]. The characteristics of density waves in dipolar condensates at a temperature of absolute zero have been investigated using both mean-field variational and full numerical approaches [13]. The breaking of symmetry resulting from the anisotropy of the dipole–dipole interaction was found to be a crucial factor in this phenomenon. Symmetry 2023, 15, 1585. https://doi.org/10.3390/sym15081585 https://www.mdpi.com/journal/symmetry