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Article

Optical Solitons for the Concatenation Model by Lie Symmetry and Kumar–Malik Approach

by
Rajveer Singh
1,
Sachin Kumar
1,
Ahmed H. Arnous
2,3,
Oswaldo González-Gaxiola
4,
Lina S. Calucag
5 and
Anjan Biswas
6,7,8,*
1
Department of Mathematics and Statistics, Central University of Punjab, Bathinda 151401, Punjab, India
2
Department of Mathematical Sciences, Saveetha School of Engineering, SIMATS, Chennai 602105, Tamilnadu, India
3
Research Center of Applied Mathematics, Khazar University, Baku AZ-1096, Azerbaijan
4
Applied Mathematics and Systems Department, Universidad Autonoma Metropolitana–Cuajimalpa, Vasco de Quiroga 4871, Mexico City 05348, Mexico
5
Department of Mathematics and Science, University of Technology Bahrain, Salmabad P.O. Box 18041, Bahrain
6
Department of Mathematics & Physics, Grambling State University, Grambling, LA 71245, USA
7
Department of Physics and Electronics, Khazar University, Baku AZ-1096, Azerbaijan
8
Department of Mathematics and Applied Mathematics, Sefako Makgatho Health Sciences University, Pretoria 0204, South Africa
*
Author to whom correspondence should be addressed.
Symmetry 2026, 18(1), 9; https://doi.org/10.3390/sym18010009
Submission received: 18 November 2025 / Revised: 8 December 2025 / Accepted: 16 December 2025 / Published: 19 December 2025
(This article belongs to the Special Issue Quantum Optics and Symmetry)
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Abstract

The current paper retrieves optical 1–soliton solution to the concatenation model with the Kerr law of self-phase modulation structure. The primary mode of integration is the Lie symmetry analysis. The subsequent ordinary differential equations (ODEs) are formulated by the recently proposed Kumar–Malik scheme. This leads to the emergence of a variety of 1–soliton solutions along with their respective existence criteria. The numerical simulations support the analytical solutions that are yielded by the joint application of the two schemes.
Keywords: solitons; Lie symmetry; Kumar–Malik method solitons; Lie symmetry; Kumar–Malik method

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MDPI and ACS Style

Singh, R.; Kumar, S.; Arnous, A.H.; González-Gaxiola, O.; Calucag, L.S.; Biswas, A. Optical Solitons for the Concatenation Model by Lie Symmetry and Kumar–Malik Approach. Symmetry 2026, 18, 9. https://doi.org/10.3390/sym18010009

AMA Style

Singh R, Kumar S, Arnous AH, González-Gaxiola O, Calucag LS, Biswas A. Optical Solitons for the Concatenation Model by Lie Symmetry and Kumar–Malik Approach. Symmetry. 2026; 18(1):9. https://doi.org/10.3390/sym18010009

Chicago/Turabian Style

Singh, Rajveer, Sachin Kumar, Ahmed H. Arnous, Oswaldo González-Gaxiola, Lina S. Calucag, and Anjan Biswas. 2026. "Optical Solitons for the Concatenation Model by Lie Symmetry and Kumar–Malik Approach" Symmetry 18, no. 1: 9. https://doi.org/10.3390/sym18010009

APA Style

Singh, R., Kumar, S., Arnous, A. H., González-Gaxiola, O., Calucag, L. S., & Biswas, A. (2026). Optical Solitons for the Concatenation Model by Lie Symmetry and Kumar–Malik Approach. Symmetry, 18(1), 9. https://doi.org/10.3390/sym18010009

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