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The formation and propagation of soliton wave profiles for the shynaray-iia equation

Khan Muhammad Ishfaq, Faridi Waqas Ali, Murad Muhammad Amin Sadiq, Iqbal Mujahid, Myrzakulov Ratbay, Umurzakhova Zhanar

Acta Mechanica et Automatica

2025

Vol. 19, no. 1

136--147

This study examines a novel use of the Jacobi elliptic function expansion method to solve the Shynaray-IIA equation, a significant nonlinear partial differential equation that arises in optical fiber, plasma physics, surface symmetry geometry, and many other mathematical physics domains. This kind of solution has never been attained in research prior to this study. Numerous properties of a particular class of solutions, called the Jacobi elliptic functions, make them useful for the analytical solution of a wide range of nonlinear problems. Using this powerful method, we derive a set of exact solutions for the Shynaray-IIA equation, shedding light on its complex dynamics and behaviour. The proposed method is shown to be highly effective in obtaining exact solutions in terms of Jacobi elliptic functions, such as dark, bright, periodic, dark-bright, dark-periodic, bright periodic, singular, and other various types of solitons. Furthermore, a detailed analysis is conducted on the convergence and accuracy of the obtained solutions. The outcomes of this study extend the applicability of the Jacobi elliptic function approach to a novel class of non-linear models and provide valuable insights into the dynamics of Shynaray-IIA equation. This study advances the creation of efficient mathematical instruments for resolving intricate nonlinear phenomena across a range of scientific fields.

The construction of analytical exact soliton waves of kuralay equation

Faridi Waqas Ali, Bakar Muhammed Abu, Myrzakulova Zhaidary, Myrzakulov Ratbay, Elamin Mawahib, Ragoub Lakhdar, Akinyemi Lanre

Acta Mechanica et Automatica

2024

Vol. 18, no. 4

603--615

The primary objective of this work is to examine the Kuralay equation, which is a complex integrable coupled system, in order to investigate the integrable motion of induced curves. The soliton solutions derived from the Kuralay equation are thought to be the supremacy study of numerous significant phenomena and extensive applications across a wide range of domains, including optical fibres, nonlinear optics and ferromagnetic materials. The inverse scattering transform is unable to resolve the Cauchy problem for this equation, so the analytical method is used to produce exact travelling wave solutions. The modified auxiliary equation and Sardar sub-equation approaches are used to find solitary wave solutions. As a result, singular, mixed singular, periodic, mixed trigonometric, complex combo, trigonometric, mixed hyperbolic, plane and combined bright–dark soliton solution can be obtained. The derived solutions are graphically displayed in 2-D and 3-D glances to demonstrate how the fitting values of the system parameters can be used to predict the behavioural responses to pulse propagation. This study also provides a rich platform for further investigation.