Physica A 560 (2020) 125114 Contents lists available at ScienceDirect Physica A journal homepage: www.elsevier.com/locate/physa Computing solitary wave solutions of coupled nonlinear Hirota and Helmholtz equations Sudhir Singh a , Lakhveer Kaur b , R. Sakthivel c ,∗ , K. Murugesan a a Department of Mathematics, National Institute of Technology, Tiruchirappalli 620015, India b Department of Mathematics, Jaypee Institute of Information Technology, Noida, U.P., India c Department of Applied Mathematics, Bharathiar University, Coimbatore 641046, India article info Article history: Received 5 July 2019 Received in revised form 13 August 2020 Available online 26 August 2020 Keywords: Hirota equation Helmholtz equation exp (−φ(ε))–expansion method Soliton Periodic solution abstract In this article, we obtain the exact solutions two coupled models, one integrable system, namely coupled nonlinear Hirota (CNHI) equation and another non-integrable system, namely coupled nonlinear Helmholtz (CNHE) equation via the exp (−Φ(ε))–expansion method. The obtained travelling wave solutions are structured in rational, trigonometric and hyperbolic functions. These solutions lead to diverse types of solitary optical waves for free choices of parameters that guarantee the sustainability of such solutions. Also, 3D illustrations for the free choices of the physical parameters is provided to understand the physical explanation of the problems. These results further enrich and deepen the understanding of the dynamics of higher-dimensional soliton propagation. © 2020 Elsevier B.V. All rights reserved. 1. Introduction Investigation of nonlinear waves has an incredible history with several applications in various areas of physical sciences such as ultra-cold atomic systems, particle physics, fluid physics, plasma physics, nonlinear optics, biophysics, beam propagation, lightwave, plasma waves, protein folding, quantum field theory and condensed matter [1–3]. Over the past decades, the enthusiasm on nonlinear waves is exponentially expanding among the experts of different fields because of their omnipresent appearances in many physical systems. These nonlinear waves are modelled mostly through prototype linear/nonlinear ordinary and partial differential equations. Nonlinear partial differential equations (NLPDEs), especially of higher dimensions and coupled systems, play a pivotal role in the study of nonlinear waves. NLPDEs can be best understood by exploring their exact solutions. In particular, the exact solution provides pieces of information, which in results helps in modelling a more suitable/reliable model, and to study of higher dimensional soliton propagations and many more. Few of the outcomes of modelling of NLPDEs are fourth and fifth state of matter known as plasma and Bose–Einstein condensate, respectively. The Bose–Einstein condensate is modelled using integrable Manakov Model, coupled nonlinear Schrödinger (NLS) equation, Gross–Pitaevskii (GP) equation [4], and using the assumptions of plasma and the normalized governing hydrodynamic fluid equations various models are derived including Burgers, modified Burgers, Gardner and many more well-known equations and new integrable/non-integrable systems using reductive perturbation/extended Poincarè–Lighthill–Kuomethod techniques [5]. Therefore, many authors have been fascinated to investigate more and more about the exact solution of NLPDEs. Over the years, various powerful techniques are provided in the literature to obtain the exact solutions of NLPDEs such as inverse scattering transform, Hirota bilinear method, Bäcklund and Darboux transformation, ansatz approaches [6–22] etc. ∗ Corresponding author. E-mail address: krsakthivel0209@gmail.com (R. Sakthivel). https://doi.org/10.1016/j.physa.2020.125114 0378-4371/© 2020 Elsevier B.V. All rights reserved.