Citation: Razzaq, W.; Zafar, A.; Alsharidi, A.K.; Alomair, M.A. New Three Wave and Periodic Solutions for the Nonlinear (2+1)-Dimensional Burgers Equations. Symmetry 2023, 15, 1573. https://doi.org/10.3390/ sym15081573 Academic Editors: Calogero Vetro, Evren Hınçal, Kamyar Hosseini and Mohammad Mirzazadeh Received: 15 June 2023 Revised: 2 August 2023 Accepted: 4 August 2023 Published: 12 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 New Three Wave and Periodic Solutions for the Nonlinear (2+1)-Dimensional Burgers Equations Waseem Razzaq 1,2, * , Asim Zafar 1 , Abdulaziz Khalid Alsharidi 3 and Mohammed Ahmed Alomair 4 1 Department of Mathematics, COMSATS University Islamabad, Vehari Campus, Vehari 61100, Pakistan; asimzafar@cuivehari.edu.pk 2 Math Center, Rahmanyia Colony, Vehari 61120, Pakistan 3 Department of Mathematics and Statistics, College of Science, King Faisal University, Al-Hasa 31982, Saudi Arabia; akalsharidi@kfu.edu.sa 4 Department of Quantitative Methods, School of Business, King Faisal University, Al-Hasa 31982, Saudi Arabia; ma.alomair@kfu.edu.sa * Correspondence: mathcentr@gmail.com Abstract: This research paper is about the new three wave, periodic wave and other analytical wave solutions of (2+1)-Dimensional Burgers equations by utilizing Hirota bilinear and extended sinh- Gordon equation expansion (EShGEE) schemes. Achieved solutions are verified and demonstrated by different plots with the use of Mathematica 11.01 software. Some of the achieved solutions are also described graphically by two-dimensional, three-dimensional and contour plots. The gained solutions are helpful for the future study of concerned models. Finally, these two schemes are simple, fruitful and reliable to handle the nonlinear PDEs. Keywords: (2+1)-dimensional Burgers equations; Hirota bilinear scheme; extended sinh-Gordon equation expansion scheme; new three wave; new periodic wave; analytical wave solutions 1. Introduction Symmetry is used in real life to simplify calculations and solve problems more eas- ily. Symmetry also offers human beings an additional extension to their capabilities. Applications of symmetry are determining the orbital overlap for molecular orbital, un- derstanding the spectroscopic properties of molecules and identifying chiral molecular species, etc. Symmetry is a frequently recurring theme in mathematics, nature, science, etc. In mathematics, its most familiar manifestation appears in geometry [1,2]. The nonlinear partial differential equations (NLPDEs) arise in various types of physical problems such as fluid dynamics, plasma physics, quantum field theory, etc. The system of nonlinear partial differential Equations has been observed in chemical, biological, engineering and other areas of applied sciences. A lot of research has been performed in these areas to find the numerical and analytical results of NLPDEs. Vari- ous schemes have been developed for this purpose. For example, the auxiliary rational method [3], Kudryashov technique [4], two variable (G ′ /G,1/G)-expansion technique [5], mapping method [6], generalized auxiliary equation method [7], modified F-expansion technique [8], unified method [9], modified extended tanh expansion method [10], modi- fied simplest equation technique [11], extended Jacobi elliptic function scheme [12], He’s semi-inverse and Riccati equation mapping schemes [13], the tanh-coth technique [14], exp(−ϕ(μ))-expansion scheme [15], etc. Except for these schemes, there are two other simple, useful, and significant schemes: the Hirota bilinear scheme and the extended sinh-Gordon equation expansion scheme. The Hirota bilinear method can be used to search for new integrable evolution equations. Solutions obtained through the Hirota bilinear method have distinct structures, but all of them have emerged under the banner of the same scheme. This scheme solves solutions Symmetry 2023, 15, 1573. https://doi.org/10.3390/sym15081573 https://www.mdpi.com/journal/symmetry