Bifurcation analysis and stochastic optical solutions to the stochastic nonlinear Kodama equation in nonlinear optics

Sofian T. Obeidat; Doaa Rizk; Wael W. Mohammed; Adel Elmandouh; Sofian T. Obeidat; Doaa Rizk; Wael W. Mohammed; Adel Elmandouh

doi:10.3934/math.2025504

AIMS Mathematics

2025, Volume 10, Issue 5: 11111-11130. doi: 10.3934/math.2025504

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Bifurcation analysis and stochastic optical solutions to the stochastic nonlinear Kodama equation in nonlinear optics

1.
Department of Mathematics, College of Science, University of Ha'il, Ha'il 2440, Saudi Arabia
2.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
3.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
4.
Department of Mathematics, Faculty of Science, Mansoura University, Mansoura 35516, Egypt

Received: 14 January 2025 Revised: 03 May 2025 Accepted: 08 May 2025 Published: 15 May 2025
MSC : 35A20, 35Q51, 60H10, 60H15

We consider the stochastic nonlinear Kodama equation (SNLKE) driven by multiplicative white noise. A specific wave transformation is applied to convert this system into a one-dimensional conservative Hamiltonian system. We analyze the bifurcation of this system and present its phase portrait. Additionally, a brief description of the phase portrait is provided, along with an illustration of the phase orbit degeneracy depending on the bifurcation parameter. Bifurcation allows us to deduce that changing the parameter values can have a substantial influence on nonlinear optics and mathematical physics as well as the dynamics of the optical soliton solutions of the Kodama equation. Using the conserved quantity, we derive new traveling wave solutions for the SNLKE. In the absence of noise, we recover certain wave solutions for the deterministic case. Furthermore, we examine the influence of multiplicative white noise on the exact solutions of the SNLKE, with some of the obtained solutions visualized graphically.
- exact solutions,
- simulation,
- mathematical model,
- qualitative theory,
- white noise,
- stabilizes by noise
Citation: Sofian T. Obeidat, Doaa Rizk, Wael W. Mohammed, Adel Elmandouh. Bifurcation analysis and stochastic optical solutions to the stochastic nonlinear Kodama equation in nonlinear optics[J]. AIMS Mathematics, 2025, 10(5): 11111-11130. doi: 10.3934/math.2025504

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Abstract

We consider the stochastic nonlinear Kodama equation (SNLKE) driven by multiplicative white noise. A specific wave transformation is applied to convert this system into a one-dimensional conservative Hamiltonian system. We analyze the bifurcation of this system and present its phase portrait. Additionally, a brief description of the phase portrait is provided, along with an illustration of the phase orbit degeneracy depending on the bifurcation parameter. Bifurcation allows us to deduce that changing the parameter values can have a substantial influence on nonlinear optics and mathematical physics as well as the dynamics of the optical soliton solutions of the Kodama equation. Using the conserved quantity, we derive new traveling wave solutions for the SNLKE. In the absence of noise, we recover certain wave solutions for the deterministic case. Furthermore, we examine the influence of multiplicative white noise on the exact solutions of the SNLKE, with some of the obtained solutions visualized graphically.

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