Nonlinear wave propagation in plasma and spin chains: New analytical solutions
Abstract
In this study, we investigated two significant nonlinear evolution models, namely the New Hamiltonian Amplitude Equation (NHAE) and the Heisenberg Ferromagnetic Spin Chain Equation (HFSCE), which are used to describe wave propagation phenomena arising in plasma physics, nonlinear optics, fluid dynamics, magnetism, and spin chain systems. Investigating the exact solutions of these models is important for understanding the physical mechanisms of nonlinear wave interactions and predicting complex dynamical behaviors in applied sciences. To this end, the governing partial differential equations were transformed into nonlinear ordinary differential equations through a traveling wave transformation, and the -expansion method was employed to construct new exact analytical solutions. As a result, several previously unreported wave structures were obtained for both models. The solutions for the NHAE include rational-exponential forms, periodic waves, and localized complex profiles with phase-dependent amplitude modulation. The obtained solutions for the HFSCE represent oscillatory spin-wave modes, kink-type transitions, and exponentially varying magnetic excitations with modulated envelopes. These findings demonstrate that both equations possess rich families of bounded, singular, periodic, and modulated traveling waves, reflecting the diverse nonlinear dynamics supported by these models. Three-dimensional surface plots and contour illustrations are provided to visualize the propagation characteristics of the derived solutions. A comparison with earlier studies confirms that the obtained results are new, more general in several parameter regimes, and extend the existing literature by presenting additional exact wave structures through an efficient analytical framework.
Keywords
New Hamiltonian amplitude equation, Heisenberg ferromagnetic spin chain equation, (m+1/G')-expansion method, Exact wave solution
References
- Akkilic, A. N., & Bulut, H. (2021). Soliton solutions of Hirota equation and Hirota-Maccari system by the (m + 1/G′)-expansion method. Advanced Mathematical Models and Applications, 6(1), 22–30.
- Aljahdaly, N. H., & Behera, S. (2023). Nonlinear evolution equations and their traveling wave solutions in fluid media by modified analytical method. Pramana – Journal of Physics, 97(3), 130.
- Almatrafi, M. B. (2023). Abundant traveling wave and numerical solutions for Novikov-Veselov system with their stability and accuracy. Applicable Analysis, 102(8), 2389–2402.
- Alsharidi, A. K., & Bekir, A. (2023). Discovery of new exact wave solutions to the M-fractional complex three coupled Maccari’s system by Sardar sub-equation scheme. Symmetry, 15(8), 1567.
- Bashar, M. H., Islam, S. R., & Kumar, D. (2021). Construction of traveling wave solutions of the (2+1)-dimensional Heisenberg ferromagnetic spin chain equation. Partial Differential Equations in Applied Mathematics, 4, 100040.
- Bulut, H., & Ismael, H. F. (2022). Exploring new features for the perturbed Chen-Lee-Liu model via (m + 1/G′)-expansion method. Proceedings of the Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 48(1), 164–173.
- Bulut, H., & Khalid, B. J. (2020). Optical soliton solutions of Fokas-Lenells equation via (m + 1/G′)-expansion method. Journal of Advanced Applied & Computational Mathematics, 7, 20–24.
- Bulut, H., Sulaiman, T. A., & Baskonus, H. M. (2018). Dark, bright and other soliton solutions to the Heisenberg ferromagnetic spin chain equation. Superlattices and Microstructures, 123, 12–19.
- Chen, Y., Li, B., & Zhang, H. (2004). Explicit exact solutions for a new generalized Hamiltonian amplitude equation with nonlinear terms of any order. Zeitschrift für Angewandte Mathematik und Physik, 55, 983–993.
- Das, N., & Ray, S. S. (2023). Exact traveling wave solutions and soliton solutions of conformable M-fractional modified nonlinear Schrödinger model. Optik, 171060.