Novel constructed dynamical analytical solutions and conserved quantities of the new (2+1)-dimensional KdV model describing acoustic wave propagation

Abstract In this study, we delve into the exploration of the new (2+1)-dimensional KdV model, a mathematical model essential to the understanding of various nonlinear phenomena such as ion acoustic waves and harmonic crystals. The primary objective of this study is to conduct a comprehensive symmetry analysis of the model, a method that has not been previously applied to the model. This approach aims to uncover new exact solutions, which will not only enrich the existing literature but also provide valuable insights for researchers specializing in symmetry analysis and the broader scientific community. To achieve this goal, we employ a comprehensive analytical methodology that combines Lie symmetry analysis technique with Kudryashov’s method, the simplest equation technique, direct integration, Jacobi elliptic expansion technique, and the power series method. Through the utilization of these diverse analytical approaches, we successfully derive solutions for the model in various functional forms, including exponential, trigonometric, hyperbolic, Jacobi elliptic, and rational functions. These solutions exhibit broad applicability across diverse disciplines within nonlinear science and engineering. To provide a deeper understanding of the obtained solutions, we present our findings through visual representations, utilizing three-dimensional, two-dimensional, and density plots. By selecting appropriate ranges for the involved arbitrary constants, we effectively illustrate the dynamical wave behaviours inherent in the solutions. Notably, our visual representations reveal discernible patterns characterized by periodic and kink-shaped structures, shedding light on the intricate dynamics encapsulated within the solutions. Furthermore, we endeavour to identify conserved vectors associated with the model under investigation. To achieve this, we employ the multiplier method and Ibragimov’s theorem. The results are expected to enhance our understanding of the model’s physical implications and contribute to advancing the field.