Conservation Laws, Solitary Wave Solutions, and Lie Analysis for the Nonlinear Chains of Atoms

Abstract

Nonlinear chains of atoms(NCA) are complex systems with rich dynamics, influencing various scientific disciplines. Lie symmetry approach is considered to analyze the NCA. The Lie symmetry method is a powerful mathematical tool for analyzing and solving differential equations with symmetries, facilitating the reduction of complexity and obtaining solutions. After getting the entire vector field by using the Lie scheme, we find the optimal system of symmetries. Using the optimal system we have converted assumed PDE into nonlinear ODE. The new auxiliary scheme introduces novel approaches to complement existing techniques, enhancing accuracy and simplifying computations. Travelling wave solutions describe wave-like propagation in systems, while graphical behavior visually represents relationships and patterns in data or mathematical models. The multiplier method enables the identification of conservation laws, fundamental principles in physics that assert certain quantities remain constant over time. Understanding these concepts contributes to a deeper comprehension of nonlinear chains of atoms and their dynamics, fostering advancements in related fields.