Painlevé integrability and a collection of new wave structures related to an important model in shallow water waves

Abstract

This paper investigates the perturbed Boussinesq equation that emerges in shallow water waves. The perturbed Boussinesq equation describes the properties of longitudinal waves in bars, long water waves, plasma waves, quantum mechanics, acoustic waves, nonlinear optics, and other phenomena. As a result, the governing model has significant importance in its own right. The singular manifold method and the unified methods are employed in the proposed model for extracting hyperbolic, trigonometric, and rational function solutions. These solutions may be useful in determining the underlying context of the physical incidents. It is worth noting that the executed methods are skilled and effective for examining nonlinear evaluation equations, compatible with computer algebra, and provide a wide range of wave solutions. In addition to this, the Painlevé test is also used to check the integrability of the governing model. Two-dimensional and three-dimensional plots are made to illustrate the physical behavior of the newly obtained exact solutions. This makes the study of exact solutions to other nonlinear evaluation equations using the singular manifold method and unified technique prospective and deserving of further study.