Novel optical solitons for the cubic–quintic nonlinear Schrö dinger equation with an additional anti-cubic nonlinear term using the symmetry group method

Mahmoud Gaballah; Rehab M. El-Shiekh; Mahmoud Gaballah; Rehab M. El-Shiekh

doi:10.3934/math.20251300

AIMS Mathematics

2025, Volume 10, Issue 12: 29595-29606. doi: 10.3934/math.20251300

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Novel optical solitons for the cubic–quintic nonlinear Schrö dinger equation with an additional anti-cubic nonlinear term using the symmetry group method

Mahmoud Gaballah ^{1
,
,},
Rehab M. El-Shiekh ^2,3

1.
Department of Physics, College of Science at Al-Zulfi, Majmaah University, Al-Majmaah City, 11952, Saudi Arabia
2.
Department of Mathematics, Faculty of Education, Ain Shams University, Cairo, Egypt
3.
College of Business Administration in Majmaah, Majmaah University, Al-Majmaah City, 11952, Saudi Arabia

Received: 01 September 2025 Revised: 15 November 2025 Accepted: 28 November 2025 Published: 15 December 2025
MSC : 35-XX, 35C08

This paper examines the cubic-quintic nonlinear Schrödinger equation (CQNLSE) with an additional anti-cubic nonlinear term, by using Stainberg's symmetry technique. The CQNLSE with the additional anti-cubic nonlinear term is a generalized model of higher-order nonlinear effects, offering a more accurate description of optical pulse propagation in nonlinear media with complex nonlinear responses, which makes the CQNLSE have a wide range of applications in several fields like optics, communications, spectroscopy, and computing. In our study, we used symmetry group analysis to derive a finite Lie group of transformations, and as a result, a novel similarity transformation, not previously reported in the literature, was obtained from this group. By using this transformation, the CQNLSE with the anti-cubic term was reduced to a nonlinear ordinary differential equation, which can be solved using the Jacobi elliptic expansion method, and a variety of wave solutions were obtained. These solutions include periodic waves, kink solitons, and bright solitons, contain other solutions, shown in the previous literature. We have also introduced a new solution, which has not been achived before in studies. The 3D and 2D plots of the periodic sn wave and its limit as a kink solitary wave were given to declare the dynamical behavior of the wave propagation by controlling the parameters contained in the solution.
Citation: Mahmoud Gaballah, Rehab M. El-Shiekh. Novel optical solitons for the cubic–quintic nonlinear Schrö dinger equation with an additional anti-cubic nonlinear term using the symmetry group method[J]. AIMS Mathematics, 2025, 10(12): 29595-29606. doi: 10.3934/math.20251300

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Abstract

This paper examines the cubic-quintic nonlinear Schrödinger equation (CQNLSE) with an additional anti-cubic nonlinear term, by using Stainberg's symmetry technique. The CQNLSE with the additional anti-cubic nonlinear term is a generalized model of higher-order nonlinear effects, offering a more accurate description of optical pulse propagation in nonlinear media with complex nonlinear responses, which makes the CQNLSE have a wide range of applications in several fields like optics, communications, spectroscopy, and computing. In our study, we used symmetry group analysis to derive a finite Lie group of transformations, and as a result, a novel similarity transformation, not previously reported in the literature, was obtained from this group. By using this transformation, the CQNLSE with the anti-cubic term was reduced to a nonlinear ordinary differential equation, which can be solved using the Jacobi elliptic expansion method, and a variety of wave solutions were obtained. These solutions include periodic waves, kink solitons, and bright solitons, contain other solutions, shown in the previous literature. We have also introduced a new solution, which has not been achived before in studies. The 3D and 2D plots of the periodic sn wave and its limit as a kink solitary wave were given to declare the dynamical behavior of the wave propagation by controlling the parameters contained in the solution.

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