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

Nan Xing, Umair Asghar (), Khaleel Ahmad and Luminita-Ioana Cotirla
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Nan Xing: Department of Mathematics and Physics, Changchun Guanghua University, Changchun 130033, China
Umair Asghar: Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
Khaleel Ahmad: Department of Mathematics, University of Management and Technology, Lahore 54770, Pakistan
Luminita-Ioana Cotirla: Department of Mathematics, Technical University of Cluj-Napoca, 400114 Cluj-Napoca, Romania

Mathematics, 2025, vol. 13, issue 23, 1-26

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.

Keywords: convective–diffusive Cahn–Hilliard equation; modified Sardar sub-equation method; closed form solutions; thin film of copper oxide; applications 35A25; 37K10; 35C08; 76B25; 74J35 (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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