Analytical extensions of Chen-Lee-Liu equations with cubic-quintic nonlinearity and resonant effects

Ali Althobaiti; Ali Althobaiti

doi:10.3934/math.2026917

AIMS Mathematics

2026, Volume 11, Issue 7: 22724-22747. doi: 10.3934/math.2026917

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Analytical extensions of Chen-Lee-Liu equations with cubic-quintic nonlinearity and resonant effects

Ali Althobaiti ^{1,2
,
,}

1.
Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
2.
Research Center of Basic Sciences, Engineering and High Altitude, Taif University, Taif, Saudi Arabia

Received: 15 May 2026 Revised: 08 July 2026 Accepted: 14 July 2026 Published: 28 July 2026
MSC : 35A09, 35A24, 35C07, 35C08

Extended and high-order complex-valued evolution equations are found ideally to model complex nonlinear scenarios, having perfectly featured the underlined physical assumptions. Thus, with the identified applications of the classical Chen-Lee-Liu equation, in particular, in different realms of nonlinear sciences, this study proposes several extensions to the Chen-Lee-Liu equation and its perturbed form by incorporating the cubic-quintic nonlinearity law and a resonant term with the view to extending the applicability spectrum of the classical model. Method-wise, the modified extended tanh method, in favor of its mathematical flexibility and effectiveness in handing both the real- and complex-valued evolution equations, is utilized as the primary tool for deriving exact solutions for the new models, in addition to being complemented by the modified Kudryashov method to extract additional solitonic expressions. The study also performs a linearization of the generalized model to conduct dispersion relation and modulation instability analyses. Several periodic and solitonic solutions are obtained analytically, and the characteristic equation governing the overall wave dynamics is derived. A numerical investigation of the involved parameters reveals that the inclusion of the resonant term enhances wave dispersion in the medium, while the added perturbation terms act to suppress the dispersion of harmonic waves in the complex system. This way, one notes that these new models provide an avenue to controlling transmission of field in the governing media, including fluid medium, optical fibers, or even in optical communication processes where pulsification trend is of paramount importance. Overall, the findings of this research are expected to have significant implications in fluid dynamics, nonlinear optics, quantum physics, and optoelectronics, among other fields in nonlinear science.
- nonlinear evolution equation,
- Chen-Lee-Liu equations,
- cubic-quintic nonlinearity,
- resonant force,
- extensional analysis,
- analytical study,
- modulation instability analysis
Citation: Ali Althobaiti. Analytical extensions of Chen-Lee-Liu equations with cubic-quintic nonlinearity and resonant effects[J]. AIMS Mathematics, 2026, 11(7): 22724-22747. doi: 10.3934/math.2026917

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Abstract

Extended and high-order complex-valued evolution equations are found ideally to model complex nonlinear scenarios, having perfectly featured the underlined physical assumptions. Thus, with the identified applications of the classical Chen-Lee-Liu equation, in particular, in different realms of nonlinear sciences, this study proposes several extensions to the Chen-Lee-Liu equation and its perturbed form by incorporating the cubic-quintic nonlinearity law and a resonant term with the view to extending the applicability spectrum of the classical model. Method-wise, the modified extended tanh method, in favor of its mathematical flexibility and effectiveness in handing both the real- and complex-valued evolution equations, is utilized as the primary tool for deriving exact solutions for the new models, in addition to being complemented by the modified Kudryashov method to extract additional solitonic expressions. The study also performs a linearization of the generalized model to conduct dispersion relation and modulation instability analyses. Several periodic and solitonic solutions are obtained analytically, and the characteristic equation governing the overall wave dynamics is derived. A numerical investigation of the involved parameters reveals that the inclusion of the resonant term enhances wave dispersion in the medium, while the added perturbation terms act to suppress the dispersion of harmonic waves in the complex system. This way, one notes that these new models provide an avenue to controlling transmission of field in the governing media, including fluid medium, optical fibers, or even in optical communication processes where pulsification trend is of paramount importance. Overall, the findings of this research are expected to have significant implications in fluid dynamics, nonlinear optics, quantum physics, and optoelectronics, among other fields in nonlinear science.

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