A Tripartite Analytical Framework for Nonlinear (1+1)-Dimensional Field Equations: Painlevé Analysis, Classical Symmetry Reduction, and Exact Soliton Solutions

This study presents a tripartite analytical framework for the (1+1)-dimensional nonlinear Klein–Fock–Gordon equation, a key model for spinless particles in relativistic quantum mechanics. The investigation begins with a Painlevé analysis showing that the equation is completely integrable via the Painlevé test by using Maple. Subsequently, classical Lie symmetry analysis is employed to derive the infinitesimal generators of the equation. A Lagrangian formulation is constructed for these generators, from which similarity variables are systematically obtained. This framework enables a complete similarity reduction, transforming the complex nonlinear partial differential equation into a more tractable ordinary differential equation. To solve this reduced ordinary differential equation and to obtain a spectrum of soliton solutions, we implement the new generalized exponential differential rational function method. This advanced technique utilizes a rational trial function based on the ith derivatives of exponentials, generating a diverse spectrum of closed-form soliton solutions through strategic choices of arbitrary constants. The novelty of this approach provides a unified framework for handling higher-order nonlinearities, yielding solutions such as multi-peakons and lump solitons, which are vividly characterized using Mathematica-generated 3D, 2D, and contour plots. These findings provide significant insights into nonlinear wave dynamics with potential applications in quantum field theory, nonlinear optics, plasma physics, etc.