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Dynamic study of qualitative analysis, traveling waves, solitons, bifurcation, quasiperiodic, and chaotic behavior of integrable kuralay equations

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Abstract

In this work, we implement the bifurcation theory for the planar dynamical system and the complete discrimination system method of polynomials to the integrable Kuralay equations. Solitons generated by these equations are found to be relevant in diverse fields such as ferromagnetic materials, nonlinear optics, and optical fibers. Through specific wave transformations, it undergoes conversion into an ordinary differential equation (ODE), which is subsequently transformed into a planar dynamical system associated with a one-dimensional Hamiltonian function. As per the qualitative theory of the planar dynamical system, phase portraits of the Hamiltonian system are plotted and used to construct some new traveling wave solutions. Numerical examination reveals diverse nonlinear structures in the analytical solutions, encompassing solitary waves, kink waves, and periodic wave profiles. Additionally, the integrable Kuralay equation employs the complete discrimination system method of polynomial for the first time, yielding solutions expressed in trigonometric, exponential, hyperbolic, and Jacobi elliptic functions. Visual representations of the derived solutions include 3-D, 2-D, and contour plots. The reliability and effectiveness are affirmed through the numerical graphs of the solutions. Furthermore, we conducted a numerical investigation into the chaotic and quasiperiodic behavior of the perturbed system by introducing a specific periodic force into the primary system.

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Acknowledgements

The Editor and the referees are greatly appreciated by the authors for their thoughtful and instructive comments. For financial support in conducting this research under the Faculty Research Programme Grant-IoE via Ref. No./IoE/2023-24/12/FRP, the author, Sachin Kumar, would like to acknowledge the Institution of Eminence, University of Delhi, India.

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  1. Department of Mathematics, Faculty of Mathematical Sciences, University of Delhi, Delhi, 110007, India

    Sachin Kumar & Nikita Mann

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Kumar, S., Mann, N. Dynamic study of qualitative analysis, traveling waves, solitons, bifurcation, quasiperiodic, and chaotic behavior of integrable kuralay equations. Opt Quant Electron 56, 859 (2024). https://doi.org/10.1007/s11082-024-06701-3

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