Citation: Hu, W.; Ren, J.; Stepanyants, Y. Solitary Waves and Their Interactions in the Cylindrical Korteweg–De Vries Equation. Symmetry 2023, 15, 413. https:// doi.org/10.3390/sym15020413 Academic Editor: Mariano Torrisi Received: 31 December 2022 Revised: 15 January 2023 Accepted: 28 January 2023 Published: 3 February 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 Solitary Waves and Their Interactions in the Cylindrical Korteweg–De Vries Equation Wencheng Hu 1,2 , Jingli Ren 1 and Yury Stepanyants 3,4, * 1 Henan Academy of Big Data, School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China 2 College of Science, Zhongyuan University of Technology, Zhengzhou 450007, China 3 School of Mathematics, Physics and Computing, University of Southern Queensland, 487–535 West St., Toowoomba, QLD 4350, Australia 4 Department of Applied Mathematics, Nizhny Novgorod State Technical University, n.a. R.E. Alekseev, 24 Minin St., Nizhny Novgorod 603950, Russia * Correspondence: yury.stepanyants@usq.edu.au Abstract: We consider approximate, exact, and numerical solutions to the cylindrical Korteweg–de Vries equation. We show that there are different types of solitary waves and obtain the dependence of their parameters on distance. Then, we study the interaction of solitary waves of different types. Keywords: nonlinear wave; cylindrical Korteweg–De Vries equation; soliton; self-similar solitary wave 1. Introduction The study of weakly nonlinear cylindrical waves in dispersive media has a long history. In 1959 Lordansky derived the cylindrical version of the Korteweg–de Vries (cKdV) Equation [1] for surface waves in a fluid. A similar equation was later derived for water and plasma waves by various authors [2–8]. Currently, the cylindrical KdV equation is one of the basic equations of contemporary mathematical physics. In application to the description of outgoing waves with axisymmetric fronts, the equation in the proper physical coordinates reads: ∂u ∂r + 1 c ∂u ∂t − α c u ∂u ∂t − β 2c 5 ∂ 3 u ∂t 3 + u 2r = 0, (1) where c is the speed of long linear waves for which dispersion is negligible ( β = 0), α is the nonlinear coefficient, and β is the dispersive coefficient. Here r stands for the radial coordinate and t is time. The derivation of this equation is based on the assumption that the last three terms that describe the effects of weak nonlinearity, dispersion, and geometric divergence are relatively small (compared to the first two linear terms) and are of the same magnitude of smallness. The smallness of the geometric divergence presumes that the cKdV equation is valid at big distances from the centre of the polar coordinate frame where r ≫ Λ, and Λ is the characteristic width of a wave perturbation. A similar equation describing incoming waves can be also derived; it differs from Equation (1) only by the sign minus in front of the second term. In such a form the cKdV equation was used for the interpretation of physical experiments with plasma waves in laboratory chambers [4,6,9] (however, it becomes invalid when a wave approaches the origin). The importance of the cKdV equation in water wave problems is related to circular perturbations which can appear due to “point sources” produced by underwater earthquakes, volcanoes, atmospheric pressure, fallen meteorites, etc. Moreover, there are many observations when quasi-cylindrical internal waves are generated due to water intrusion in certain basins (see, for example, in the Internet numerous satellite images of internal waves generated by Atlantic water intrusions in the Mediterranean Sea due to the tide). The generalised cKdV equation was derived by McMillan and Sutherland [10] who considered the generation and evolution of solitary waves by intrusive gravity currents Symmetry 2023, 15, 413. https://doi.org/10.3390/sym15020413 https://www.mdpi.com/journal/symmetry