Öz faz modülasyonunun Kerr Yasası formuna sahip yüksek mertebeden doğrusal olmayan Schrödinger denkleminin optik soliton çözümleri

Year 2025, Volume: 13 Issue: 1, 37 - 45, 30.06.2025

Selvi Altun Durmuş

https://doi.org/10.18586/msufbd.1636803

Abstract

Bu makalenin temel amacı, fiber optiklerde dispersif darbelerin yayılımını tanımlayan, kendi kendine faz modülasyonu ve Kerr yasası doğrusal olmayanlığı ile karakterize edilen yüksek mertebeden doğrusal olmayan Schrödinger denklemini incelemek ve birleşik Riccati denklemi genişletme yöntemini kullanmaktır. Amacımız yalnızca makalede önerilen teknikle çeşitli soliton çözümleri elde etmekle sınırlı değildir; aynı zamanda incelenen modelde yüksek mertebeden dispersiyon terimlerinin soliton yayılımı üzerindeki etkisini araştırmaktır. Yüksek mertebeden dispersiyon terimleri ve Kerr yasası ortamının etkisiyle incelenen modele ait yeni ve önemli fiziksel özellikler literatüre kazandırılmıştır. Ayrıca, model parametrelerinin uygun değerleri için bazı soliton çözümlerinin dinamik davranışları ve fiziksel önemi gösterilmiştir. Elde edilen soliton çözümleri arasında karanlık ve M-şekilli solitonlar bulunmaktadır. Bu çözümlerin fiziksel önemini yorumlamak amacıyla 3D, 2D ve kontur grafikleri sunulmuştur. Önerilen yaklaşımın, çeşitli karmaşık doğrusal olmayan problemlere optik çözümler elde etmek için etkili bir analitik yöntem olduğu gözlemlenmiştir.

Keywords

Fiber optik , Parametre etkisi , Yüksek mertebeden dağılım , M-şekilli soliton

Optical soliton solutions of the higher-order nonlinear Schrödinger equation with Kerr law nonlinearity of self-phase modulation

Abstract

The primary aim of this article is to examine the higher-order nonlinear Schrödinger equation, which describes the propagation of dispersive pulses in fiber optics, characterized by self-phase modulation and Kerr law nonlinearity, and to employ the unified Riccati equation expansion method. Our objective is not only limited to obtaining various soliton solutions using the technique proposed in the article but also to investigate the impact of higher-order dispersion terms on soliton propagation in the examined model. New and significant physical features of the investigated model, influenced by higher-order dispersion terms and Kerr law media, have been added to the literature. Furthermore, the dynamic behaviors and physical significance of some soliton solutions for appropriate values of the model parameters have been demonstrated. The obtained soliton solutions include dark and M-shaped solitons. The 3D, 2D, and contour plots of these solutions are presented to interpret the physical significance of the problem under consideration. It has been observed that the proposed approach is an effective analytical method that can be utilized to obtain optical solutions to various complex nonlinear problems.

Keywords

Optical fibers , Parameter effect , High order dispersion , M-shape soliton

References

• [1] Akgül A., Baleanu D., On solutions of variable-order fractional differential equations, An International Journal of Optimization and Control: Theories & Applications (IJOCTA), 7 (1), 112–116, 2017.Lee J.H., Lee J.M., Kim H.J., Moon Y. S. Machine vision system for automatic inspection of bridges, Congress on Image and Signal Processing (CISP). 3 363-366, 2008.
• [2] Özisik Ö., Akbarov S., Rayleigh-wave propagation in a half-plane covered with a prestressed layer under complete and incomplete interfacial contact, Mechanics of Composite Materials, 39, 177–182, 2003.
• [3] Akbarov S., Mehdiyev M., Özisik M., Three-dimensional dynamics of the moving load acting on the interior of the hollow cylinder surrounded by the elastic medium, Structural Engineering and Mechanics, 67 (2), 185–206, 2018.
• [4] Kudryashov N.A., Method for finding optical solitons of generalized nonlinear Schrödinger equations, Optik, 261, 169163, 2022.
• [5] Ghanbari B., Baleanu D., Applications of two novel techniques in finding optical soliton solutions of modified nonlinear Schrödinger equations, Results in Physics, 44, 106171, 2023.
• [6] Rezazadeh H., Ullah N., Akinyemi L., Shah A., Mirhosseini-Alizamin S. M., Chu Y.-M., Ahmad H., Optical soliton solutions of the generalized non-autonomous nonlinear Schrödinger equations by the new Kudryashov’s method, Results in Physics, 24, 104179, 2021.
• [7] Islam S. R., Arafat S. Y., Alotaibi H., Inc M., Some optical soliton solutions with bifurcation analysis of the paraxial nonlinear Schrödinger equation, Optical and Quantum Electronics, 56 (3), 379, 2024.
• [8] Gu . N., Aminakbari N., New optical soliton solutions for the variable coefficients nonlinear Schrödinger equation, Optical and Quantum Electronics, 54 (4), 255, 2022.
• [9] Ahmad J., Akram S., Noor K., Nadeem M., Bucur A., Alsayaad Y., Soliton solutions of fractional extended nonlinear Schrödinger equation arising in plasma physics and nonlinear optical fiber, Scientific Reports, 13 (1), 10877, 2023.
• [10] Younas U., Sulaiman T. A., Ren J., On the study of optical soliton solutions to the three-component coupled nonlinear Schrödinger equation: applications in fiber optics, Optical and Quantum Electronics, 55 (1), 72, 2023.
• [11] Gao W., Ismael H. F., Husien A. M., Bulut H., Baskonus H. M., Optical soliton solutions of the cubic-quartic nonlinear Schrödinger and resonant nonlinear Schrödinger equation with the parabolic law, Applied Sciences, 10 (1), 219, 2019.
• [12] Awan A., Rehman H., Tahir M., Ramzan M., Optical soliton solutions for resonant Schrödinger equation with anti-cubic nonlinearity, Optik, 227, 165496, 2021.
• [13] Ali K. K., Karakoç S. B. G., Rezazadeh H., Optical soliton solutions of the fractional perturbed nonlinear Schrödinger equation, TWMS Journal of Applied and Engineering Mathematics, 2020.
• [14] Inan E., Inc M., Rezazadeh H., Akinyemi L., Optical solitons of (3+1) dimensional and coupled nonlinear Schrödinger equations, Optical and Quantum Electronics, 54 (4), 261, 2022.
• [15] Zayed E., Al-Nowehy A.-G., Exact solutions and optical soliton solutions for the (2+1)-dimensional hyperbolic nonlinear Schrödinger equation, Optik, 127 (12), 4970–4983, 2016.
• [16] Okposo N. I., Veeresha P., Okposo E. N., Solutions for time-fractional coupled nonlinear Schrödinger equations arising in optical solitons, Chinese Journal of Physics, 77, 965–984, 2022.
• [17] Chowdury A., Kedziora D., Ankiewicz A., Akhmediev N., Soliton solutions of an integrable nonlinear Schrödinger equation with quintic terms, Physical Review E, 90 (3), 032922, 2014.
• [18] Zhang K., Han T., The optical soliton solutions of nonlinear Schrödinger equation with quintic non-Kerr nonlinear term, Results in Physics, 48, 106397, 2023.
• [19] Rizvi S., Seadawy A. R., Younis M., Iqbal S., Althobaiti S., El-Shehawi A. M., Various optical soliton for a weak fractional nonlinear Schrödinger equation with parabolic law, Results in Physics, 23, 103998, 2021.
• [20] Kaplan M., Hosseini K., Samadani F., Raza N., Optical soliton solutions of the cubic-quintic nonlinear Schrödinger’s equation including an anti-cubic term, Journal of Modern Optics, 65 (12), 1431–1436, 2018.
• [21] Rizvi S. T., Seadawy A. R., Akram U., New dispersive optical soliton for a nonlinear Schrödinger equation with Kudryashov law of refractive index along with P-test, Optical and Quantum Electronics, 54 (5), 310, 2022.
• [22] Altun Durmuş S., Optical soliton solutions of stochastic the third-order nonlinear Schrödinger equation with multiplicative white noise via Itô calculus, Optical and Quantum Electronics, 56 (5), 779, 2024.
• [23] Durmuş S. A., Ozdemir N., Secer A., Ozisik M., Bayram M., Bright soliton of the third-order nonlinear Schrödinger equation with power law of self-phase modulation in the absence of chromatic dispersion, Optical and Quantum Electronics, 56 (5), 794, 2024.
• [24] Biswas A., Ekici M., Sonmezoglu A., Belic M. R., Highly dispersive optical solitons with Kerr law nonlinearity by F-expansion, Optik, 181, 1028–1038, 2019.
• [25] Zhang J., Yang Q., Dai C., Optical quasi-soliton solutions for higher-order nonlinear Schrödinger equation with variable coefficients, Optics Communications, 248 (1-3), 257–265, 2005.
• [26] Tang L., Optical solitons and traveling wave solutions for the higher-order nonlinear Schrödinger equation with derivative non-Kerr nonlinear terms, Optik, 271, 170115, 2022.
• [27] Ma G., Zhao J., Zhou Q., Biswas A., Liu W., Soliton interaction control through dispersion and nonlinear effects for the fifth-order nonlinear Schrödinger equation, Nonlinear Dynamics, 106, 2479–2484, 2021.
• [28] İnc M., Yusuf A., Aliyu A. I., Baleanu D., Optical soliton solutions for the higher-order dispersive cubic-quintic nonlinear Schrödinger equation, Superlattices and Microstructures, 112, 164–179, 2017.
• [29] Nakkeeran K., Porsezian K., Sundaram P. S., Mahalingam A., Optical solitons in N-coupled higher order nonlinear Schrödinger equations, Physical Review Letters, 80 (7), 1425, 1998.
• [30] Guo R., Hao H.-Q., Breathers and multi-soliton solutions for the higher-order generalized nonlinear Schrödinger equation, Communications in Nonlinear Science and Numerical Simulation, 18 (9), 2426–2435, 2013.
• [31] Hosseini K., Sadri K., Mirzazadeh M., Chu Y., Ahmadian A., Pansera B., Salahshour S., A high-order nonlinear Schrödinger equation with the weak non-local nonlinearity and its optical solitons, Results in Physics, 23, 104035, 2021.
• [32] Yang C., Liu W., Zhou Q., Mihalache D., Malomed B. A., One-soliton shaping and two-soliton interaction in the fifth-order variable-coefficient nonlinear Schrödinger equation, Nonlinear Dynamics, 95, 369–380, 2019.
• [33] Triki H., Choudhuri A., Porsezian K., Dinda P. T., Dark solitons in an extended nonlinear Schrödinger equation with higher-order odd and even terms, Optik, 164, 661–670, 2018.
• [34] Hosseini K., Matinfar M., Mirzazadeh M., Soliton solutions of high-order nonlinear Schrödinger equations with different laws of nonlinearities, Regular and Chaotic Dynamics, 26, 105–112, 2021.
• [35] Seadawy A. R., Modulation instability analysis for the generalized derivative higher order nonlinear Schrödinger equation and its the bright and dark soliton solutions, Journal of Electromagnetic Waves and Applications, 31(14), 1353–1362, 2017.
• [36] Nawaz B., Ali K., Abbas S. O., Rizvi S. T. R., Zhou Q., Optical solitons for non-Kerr law nonlinear Schrödinger equation with third and fourth order dispersions, Chinese Journal of Physics, 60, 133–140, 2019.
• [37] Sirendaoreji, Unified Riccati equation expansion method and its application to two new classes of Benjamin-Bona-Mahony equations, Nonlinear Dynamics, 89, 333–344, 2017.