A Study of The Exact Solutions and Conservation Laws of The Classical Lonngren Wave Equation for Communication Signals

Abstract

This study undertakes a comprehensive examination of the classical Lonngren wave equation, a fundamental computational model used for simulating electrical signals in semiconductor materials, with specific emphasis on the tunnel diode. {The primary objective of this study is to attain novel and more comprehensive solutions beyond those documented in existing literature. To achieve this goal, we have employed well-established mathematical methods, specifically analysis via Lie symmetry, coupled with other specialized techniques such as the power series method and Jacobi elliptic expansion technique. Notably, this marks the inaugural application of these methodologies to the classical Lonngren wave equation, signifying a pioneering endeavor in the exploration of this equation using these analytical tools.} These methodologies yield solutions characterized by elliptic functions. The results are visually presented through 3D, 2D, and density plots, effectively illustrating the characteristics of these solutions. The visual representations reveal a range of patterns, including periodic and singular periodic solutions. Furthermore, the paper applies the multiplier method and leverages the conservation theorem introduced by Ibragimov to derive conserved vectors. These conserved vectors play a pivotal role in the examination of physical quantities, such as energy and momentum conservation, thereby enhancing our understanding of the underlying physics within the system.