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Mathematics
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26 November 2025

Nonlinear Optical Characteristics of Copper Oxide Thin Films Interpreted Through Soliton Solutions of the Convective–Diffusive Cahn–Hilliard Equation

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1
Department of Mathematics and Physics, Changchun Guanghua University, Changchun 130033, China
2
Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
3
Center for Theoretical Physics, Khazar University, 41 Mehseti Str., Baku AZ1096, Azerbaijan
4
Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania
Mathematics2025, 13(23), 3799;https://doi.org/10.3390/math13233799 
(registering DOI)
This article belongs to the Special Issue Advances in Computational Mathematics and Applied Mathematics, 2nd Edition

Abstract

This study investigates the convective–diffusive Cahn–Hilliard equation, a nonlinear model which is used in real-world applications to phase separation and material pattern formation. Using the modified Sardar sub-problem technique, which is an extension of the Sardar sub-equation approach, we derive multiple classes of exact soliton solutions, including bright, dark, kink, and periodic forms. The parametric behaviors of these solutions are examined and visualized through analytical plots generated in Mathematica and Maple. Furthermore, UV–Vis spectrophotometry is employed to examine the optical response of copper oxide (CuO) thin films. The films exhibited a sharp absorption edge around 380–410 nm and an optical band gap of approximately 2.3 eV, confirming their semiconducting nature. The experimentally observed periodic transmission characteristics are linked with the theoretical soliton profiles predicted by the model. Overall, the proposed analytical and experimental framework establishes a clear connection between nonlinear wave theory and thin-film optical characterization, providing new insights into soliton transformation phenomena in complex material systems.

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